An Elsevier journal has angered an author by removing his study without telling him.
After spending months asking the journal why it removed the paper — about a heavily debated theorem in physics — and getting no response, the author threatened to seek damages from the journal and publisher for “permanently stigmatizing” his work. Yesterday, an Elsevier representative told the author what happened: Experts told the journal the paper had a major mistake, so the journal decided to withdraw the study, but failed to tell the author due to an “internal error.”
That explanation didn’t satisfy study author Joy Christian, scientific director of the Einstein Centre for Local-Realistic Physics in Oxford, UK, who has demanded the journal either republish the article or remove it and return the copyright to him, or he will pursue legal action.
Here’s the cryptic publisher’s note for “Local causality in a Friedmann–Robertson–Walker spacetime:”
This article was erroneously included in this issue. We apologize for any inconvenience this may cause.
The study — published in the Annals of Physics — undermines Bell’s theorem, a cornerstone of modern quantum mechanics, which has divided physicists for years. It has two active PubPeer threads for it: One for its earlier preprint version, and another for its online version, both of which were started by the same researcher, Richard Gill from Leiden University in the Netherlands. Gill told us he and others contacted the journal to raise concerns about the study, but he doesn’t think the paper should have been removed.
After months of asking the journal why the paper had been removed, yesterday Christian received a response from Marc Chahin, an executive publisher at Elsevier, saying that his paper had been withdrawn from the journal, adding:
Unfortunately, we failed to inform you about this decision due to an internal error and I apologize for that.
This is the second time in one week we’ve reported that an Elsevier journal removed a paper without telling the author.
Chahin’s email includes another letter, which he said was formulated by the journal’s editorial board, but not sent out due to an “internal error.” It reads:
In your case, soon after the acceptance of your paper was announced, several experts in the field have sent us a correspondence to report the error in your manuscript.
The letter goes on to say:
After our editorial meeting, we have concluded that your result is in obvious conflict with a proven scientific fact, i.e., violation of local realism that has been demonstrated not only theoretically but experimentally in recent experiments, and thus your result could not be generally accepted by the physics community. On this basis, we have made such a decision to withdraw your paper.
In his reply to Chahin, Christian writes that he believes the article was removed for political reasons, as “there is absolutely no scientific basis” for the decision. As such:
I demand that my article is either (1) published again in its final form (cf. the attachment), or (2) completely removed from all your publicly accessible websites without any trace, reverting all copyrights back to me as soon as possible. I am willing to forget the damage Annals of Physics and Elsevier has already caused (including the loss of my ten months in the review process) if you are able to satisfy my demand (1) or (2) above. In case you are unable to satisfy either of my demands (1) or (2), then I will have no choice but to seek legal action.
The journal is edited by Brian Greene, a prominent theoretical physicist at Columbia University in New York. He has not responded to requests for comment.
We initially learned of this story after receiving a chain of correspondence between Christian and the journal, as he sought answers for why the paper had disappeared. In an email to journal officials dated September 28, Christian wrote:
…my article was actually published online for about a month, from 30 June 2016 onwards, and has been downloaded and cited by me and other scientists over the past few months, in accordance with the DOI and related instructions provided by the publisher on the previously purchasable and downloadable article. But then it was mysteriously removed from the journal’s website without even a hint of notification to me.
Christian went on to point out that the current PDF version of the publisher’s note is followed by 12 blank pages, adding:
When I signed the copyright agreement requested by Elsevier it was with the understanding that my article will be either accepted for publication or rejected, not that it will be replaced by blank pages with permanent stigma attached to it for anyone to exploit for eternity, and in a manner that would prevent me from publishing it elsewhere with any scientific credibility, or seek acceptance from my peers otherwise.
Gill said he critiqued the study for containing controversial claims that downplay Bell’s theorem, and as well as “elementary” mathematical errors and “self-contradictions,” but told us he didn’t think the paper should have been removed. Doing so will trigger further “conspiracy” and bring more attention to it, he said, whereas if it remained published, it would have been “forgotten” or “ignored.”
Two more of Christian’s papers are also being questioned on PubPeer; one of these threads was started by Gill, and contains more than 750 comments.
Christian and Gill have a troubled history, which has played out on physics message boards.
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I truly empathize with Dr. Christian’s frustration with Elsevier. My advice, even though you are 100% right, control that anger, of ace being banned (victimization as a result of editorial incompetence is ranked way below political correctness)!
“of ace” should read “or face” (my apologies)
Thank you, Jaime. The full paper is freely available anyways:
https://arxiv.org/abs/1405.2355
In the end we are concerned here with a local hidden variable theories. In particular, my 3-sphere model is a local hidden variable model. From that perspective the sign issue wrongly contested by Lockyer, Gill and others for the past six years is a non-issue. For one can simply define the local hidden variable lambda = +/-1 of my model by the following pair of mathematically proven identities:
L(a, lambda = +1) L(b, lambda = +1) = D(a) D(b)
and
L(a, lambda = -1) L(b, lambda = -1) = D(b) D(a) .
Since the above identities are now a part of the definition of the local hidden variable lambda of my model, substituting them into equation (72) of the paper in question, namely https://arxiv.org/abs/1405.2355 — i.e., substituting the above identities into
1/2 { L(a, lambda = +1) L(b, lambda = +1) + L(a, lambda = -1) L(b, lambda = -1) } ,
immediately leads to the strong correlation derived in (79):
E(a, b) = 1/2 { D(a) D(b) + D(b) D(a) } = -1/2 { a b + b a } = -a.b . That is all there is to it.
There is no need to inform the authors about retractions or even the plan to retract a paper. Although this is recommended, at the end of the day publishers/journals are free to publish in their own space everything they want. This is how it works.
Also, journals usually require copyright transfer from authors to the publisher when an article is accepted (as Elsevier does). This mean that the publisher is the legal owner of such content and of course they can handle with them as desire – including unilateral action like this case.
Editorial freedom is also widely protected by several laws (such US 1st amendment) and recently confirmed by multiple courts in several lawsuit. Please read:
https://popehat.com/2015/02/26/dr-mario-j-a-saad-tries-and-fails-to-censor-american-diabetes-association/
YML, I beg to differ. In my view what Elsevier has done amounts to outright deception, worse than what a used car dealer may attempt. They promised to publish one article in my name, but published blank pages in my name instead, which I did not give them permission to publish. I spell out my detailed view on this matter at the following link:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=283#p6766
“…They promised to publish one article in my name, but published blank pages in my name instead..”
Joy,
I understand that you live in the UK. Where your moral rights as an author are protected by copyright law.
By publishing blank pages in your name, the publisher has distorted your work, and therefore you have every right to object to this derogatory treatment. Unless, of course, you waived your rights when you transferred your copyright to the publisher. Still, why not try?
Thanks, aceil. I have no intention to let them get away with this injustice, both to me and to physics. After all, they are making millions off our intellectual properties.
YML, you seem to be confusing editorial liberty with editorial abuse. As equally as editors have the (legal) right (and freedom) to exercise editorial independence, so too do authors have the right to challenge those decisions when they appear to be unfair, impositional, biased, or incorrect/false. The sad state of publishing is that authors (and their intellect) have been reduced to traded commodities, and publishers hide their trade model cowardly behind the editorial firewall. More and more, authors have reduced rights [1]. They place editors on the front-line of the battle against authors (in some cases) while they peer over editors’ shoulders to see how things pan out. Even if there was some sort of an “internal error”, that already right there says volumes about the efficiency of editorial processing in the world’s No. 1 publisher. In addition, Chahin deserves criticism when he states “After our editorial meeting”, simply because if this was so obvious, this should have been noted before acceptance, during peer review. So, one can state emphatically that there was editorial failure [2]. At least he had the courtesy of offering some sort of an apology to Dr. Christian.
[1] Al-Khatib, A., Teixeira da Silva, J.A. (2016) What rights do authors have? Science and Engineering Ethics
http://link.springer.com/article/10.1007/s11948-016-9808-8
DOI: 10.1007/s11948-016-9808-8
[2] Teixeira da Silva, J.A. (2016) Retractions represent failure. Journal of Educational and Social Research 6(3): 11-12.
http://www.mcser.org/journal/index.php/jesr/article/view/9481
DOI: 10.5901/jesr.2016.v6n3p11
It is worth noting that Richard D. Gill is by no means an “expert” on the physics and mathematics discussed in my paper. He is a statistician, not a physicist. My paper, on the other hand, is a physics paper, which was published in Annals of Physics, which is a physics journal. My paper is based on Clifford algebra and a cosmological solution of Einstein’s theory of gravity. Gill has absolutely no knowledge or competence in either of these subjects. For example, he has never published a single peer-reviewed paper in either of these subjects, even on the preprint arXiv, as anyone can easily verify. What is more, his elementary mathematical mistakes in his supposed critique of my work have been repeatedly exposed by me and other genuine experts in the field — see for example this paper:
https://arxiv.org/abs/1501.03393 .
It is therefore astonishing that Annals of Physics seems to have taken his word without checking his background. At the least, this is highly unprofessional behaviour for any Editorial Board of a physics journal. To my eyes this fact alone proves beyond doubt that the secret removal of my paper from their website was entirely politically motivated, without any scientific basis whatsoever.
It doesn’t matter whether JC considers Gill an expert or not. What matters is whether his demonstration of JC’s obvious mathematical errors is correct. I encourage everybody to read this full thread at JC’s forum to get a useful perspective on the dispute between Christian and Gill.
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=283
Donald Graft, that is quite easy to settle. Gill’s claim of any error in my work is easily seen to be completely bogus by any genuine expert in Clifford algebra and general relativity. See, for example, the following links:
https://arxiv.org/abs/1203.2529
https://arxiv.org/abs/1501.03393
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=222#p5813
By the way, that is not my forum. I don’t have a forum, but I do direct a Centre:
http://einstein-physics.org/
Yes, it is formally Fred Diether’s forum. Personally, I have carefully considered both sides, including your cited links and others, and find Gill’s criticisms decisive, as have many others. AoP has also come to this conclusion.
We all know that you created your own “Centre” and appointed yourself the “Director”.
May I ask, Donald Graft, what qualification do you have to judge my work based on Clifford algebra and general relativity? For example, can you tell us how many papers have you published on Clifford algebra and general relativity?
Contrary to your claim, after seven months of rigorous review process the Editorial Board of AoP accepted my paper and published it, providing me with a very positive and technically detailed referee report. Now I am still waiting, after more than two months, for the same Editorial Board to provide me a detailed scientific report showing exactly where Gill claims the so-called error is in my paper. Why do you think, Donald Graft, the Editorial Board of AoP is so shy in publishing Gill’s report on their website? Why are they hiding that report?
And yes, I am indeed the Founding Director of Einstein Centre for Local-Realistic Physics in Oxford.
Everybody please note the august scientific advisory board of the “Centre”:
http://einstein-physics.org/staff-and-affiliates/
Simple demonstrations of mathematical contradictions cannot be negated by obfuscatory appeals to “parallelized 3-spheres” and other nonsense.
Perhaps it is worth noting that Donald Graft did not answer my questions.
Also note that Jay R. Yablon is no longer at MIT as claimed and it appears he did only undergrad work there. Will JC correct the web page? I have saved it in case he decides that it is inconvenient to his cause.
Correction: he may indeed have a PhD from MIT but he currently has no affiliation there.
So back to the topic of this thread, I wonder why Annals of Physics is avoiding to publish the supposed error in my work identified by Richard D. Gill.
Yablon claims a Juris Doctor degree. He claims to have “opted away from physics” after completing his MIT undergraduate degree. JC’s page is therefore incorrect and misleading. Will JC correct it?
Juris Doctor degree from the State University of New York at Buffalo.
Well, this is degenerating fast into another useless slugfest that is unlikely to ever be resolved. Hopefully the RW staff will reconsider many of the messages above.
The only relevant question here remains why exactly the journal retracted the paper, and I personally think “because Richard Gill complained” is at best a very small part of the answer.
At the link below I respond to the “critique” by Richard D. Gill which he has been quoted to have sent to Annals of Physics. I will leave it to the readers to decide whether the journal was justified in removing my published article based on Gill’s “critique.”
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=271&p=6808#p6808
The following is my second reply to someone who claims to be Elsevier’s Executive Publisher responsible for the publication of Annals of Physics.
Dear Marc N. Chahin
(Bcc: undisclosed recipients)
As I stressed in my previous email, there are no errors of any kind in my paper entitled “Local causality in a Friedmann-Robertson-Walker spacetime.” Neither are my results in any conflict with proven scientific facts. You have failed to provide any scientific proof or demonstration of your fallacious claims, or the claims of your so-called unsolicited “experts” who supposed to have reported “errors” in my error-free paper. No one in their right mind would consider a third-rate statistician without a single peer-reviewed publication on Clifford algebra or general relativity an “expert” qualified to understand, let alone criticize the arguments presented in my paper.
As I stressed in my previous email, your unjust action against me and my scientifically and mathematically impeccable paper are purely politically and ideologically motivated. You will find my detailed scientific response to the false claims by your unsolicited “experts” at the following link:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=271&p=6813#p6808
Every word I wrote in my previous email to you still stands.
Sincerely,
Joy Christian
Dr. Christian, a suggestion: drop the ad hominems. Richard Gill may be pretty good at ignoring them, but others will not.
To the best of my knowledge, you have no documented expertise in the area of statistics (no peer-reviewed paper published in the field of statistics) that allows you to claim with such confidence that Gill is “a third-rate statistician”. In contrast, the KNAW clearly begs to differ with your claims, as he is a KNAW member, with the KNAW quite selective in its membership. That does not mean he is right about your work, but, again, you may want to refrain from these types of ad hominems, as they suggest to people like me, with no expertise in the field and thus no ability to assess the validity of the claims of either side, that it is more likely that Gill has put his finger on a sore spot, which you prefer to ignore.
And unless the journal explicitly told you, I doubt its “unsolicited experts” (plural) consists solely of Richard Gill, and thus you cannot refer to your response to Gill as a “detailed scientific response to the false claims by your unsolicited “experts””.
I find it telling that JC has not responded to the fact that Yablon is misrepresenting his affiliation. He could easily say either “Thanks for pointing that out – it has been corrected” or “It is not a misrepresentation. He actually is affiliated with MIT – proof is found here”. The fact that instead he completely ignores the issue while responding to other comments leads me to suspect that his Centre is not concerned with misleading representations.
Excuse me, I find it telling that people who are arguing with Joy Christian have decided to also start taking collateral shots me at other people who have any sort of relationship with him. That has noting to do with his retracted AOP paper. I have public blog postings where I have very clearly laid out my journey away from and back to physics, and have never been shy about leaving an internet footprint that can be easily researched by anybody who wishes to do so, for example at https://jayryablon.wordpress.com/, and in many posts at http://www.sciphysicsforums.com/spfbb1/viewforum.php?f=6. I am an alumnus of MIT, and to be crystal clear about my scientific background and present affiliations, I have separately emailed Joy to request a modification to my listing.
Because my background has been dragged into this debate by others, let me also respond by listing a bibliography of some 2013-2014 papers and an international patent application:
1. Yablon, J. R., Why Baryons Are Yang-Mills Magnetic Monopoles, Hadronic Journal, Volume 35, Number 4, 399-467 https://jayryablon.files.wordpress.com/2013/03/hadronic-journal-volume-35-number-4-399-467-20121.pdf (2012)
2. Yablon. J. R., Predicting the Binding Energies of the 1s Nuclides with High Precision, Based on Baryons which are Yang-Mills Magnetic Monopoles, Journal of Modern Physics Vol.4 No.4A, pp. 70-93. doi: 10.4236/jmp.2013.44A010, http://www.scirp.org/journal/PaperDownload.aspx?paperID=30817 (2013)
3. Yablon, J. R., Predicting the Neutron and Proton Masses Based on Baryons which Are Yang-Mills Magnetic Monopoles and Koide Mass Triplets, Journal of Modern Physics, Vol. 4 No. 4A, 2013, pp. 127-150. doi: 10.4236/jmp.2013.44A013 http://www.scirp.org/journal/PaperDownload.aspx?paperID=30830 (2013)
4. Yablon, J. R., Grand Unified SU(8) Gauge Theory Based on Baryons which Are Yang-Mills Magnetic Monopoles, Journal of Modern Physics, Vol. 4 No. 4A, pp. 94-120. doi: 10.4236/jmp.2013.44A011, http://www.scirp.org/journal/PaperDownload.aspx?paperID=30822 (2013)
5. Yablon, J. R., System, Apparatus, Method and Energy Product-By-Process for Resonantly-Catalyzing Nuclear Fusion Energy Release, and the Underlying Scientific Foundation, http://patentscope.wipo.int/search/docservicepdf_pct/id00000025517277.pdf?download (2014)
Thank you,
Jay R. Yablon
A link to Hadronic Journal! Founded by Ruggero Santilli, the inventor of anti-photons!
http://www.pepijnvanerp.nl/2016/02/the-continuing-stupidity-of-ruggero-santilli/
I will not disagree about Santilli being a rather unique personality. But he did not write the paper; I did. So judge the paper for good or ill on its own scientific merits, please.
I believe the comment was just pointing out that the Hadronic Journal is not considered a reputably academic journal, and therefore anything published in it would not be considered to be a “paper” in this context; no more than a link to a blog post. I agree that the commenter is trying to be derogatory, but it is fair to point out to others reading this thread that hadronic journal is well-known to be a vanity journal that publishes essentially anything.
SCIRP is on Beall’s list: https://scholarlyoa.com/?s=scirp
You’ve acknowledged being falsely represented by JC as an MIT doctoral researcher. Your undergrad is also not in physics. I only mention these things because casual readers may be taken in by the misrepresentation.
JC has also claimed that Retraction Watch and PubPeer are “in Gill’s pocket”.
My undergrad degree was in Computer Science at MIT. I always had a passionate interest in physics, but chose CS because it was a more economically-practical option. I will not recount my path from there to building a business in which I obtained over 150 US patents see http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fnetahtml%2FPTO%2Fsearch-adv.htm&r=0&p=1&f=S&l=50&Query=lrep%2Fyablon&d=PTXT for my clients, and continuing my physics work as an independent researcher using the freedom gained from my business. I have been able to do this work free of political constraints, concern for advancement, diversion to fundraising, and all of the other overt and subtle coercive plagues of academe out of which I opted in the 1970s. I candidly admit that is a double-edged sword because my opted-out position also works against me for obvious reasons which I freely acknowledge and so will save you the trouble of having to recount. And if you have any other questions about my background that you would like to ask me about, please do so, and I will gladly divulge whatever you wish, publicly or privately.
If I had it to do over knowing what I know now I would have avoided SCIRP, but what is done is done, and that does not make my own work bogus whether or not there might be other bogus works on SCIRP (which did in fact solicit me). Since you are a physics person, Donald, I would truly appreciate your looking at my papers based on their content and not their venue of publication. Then if there are disagreements you have on the scientific merits, let’s have at it. That is what true science, conducted with integrity as scientists seeking natural truth and not as political animals seeking advantage, is all about. That is indeed what Retraction Watch seeks to promote — ethical science based solely on merit and not politics or mudslinging or deception. Indeed — and I say this two hours before the second Presidential debate in the US — we in the scientific community need to be role models for people elsewhere in our world who struggle to seek and present objective truth and apply scientific method and change our minds when presented with evidence contradictory to our biases and preconceptions. And we need to set good examples for people who cannot talk civilly to one another because of their biases.
Finally, let me comment on all the Bell stuff with JC and RG etc., for which the recent AOP retraction is the core subject of this thread. I have communicated with both JC and RG privately from time to time and have no beef with either. I have stayed out of the science of the Bell debates because as I see it JC is trying to correct what he thinks is a blunder by Bell, and RG and others think that Bell made no blunder, and my own research interest is in breaking ground into areas not well-plowed, not in “blunder correction” (or debating whether a blunder was made). At one point last year given that I maintain lines of communication with both JC and RG I had thought to try to “mediate” their differences, but concluded that there is too much personal and scientific distance between them to hope for a fruitful result. I have told Joy, and probably also told Richard, that I feel at times like I am watching two kindergartners squabble, and that it is really does not reflect well, politically, for either of them to continue down that road. And while I do not want to say anything that might be held against me by AOP because I hope to shortly submit to them a paper on geometrodynamic electrodynamics that I have been developing for the past 10 months (disclosure of self-interest), I do think that more transparency about what happened in this retraction case is warranted. We are scientists. We disclose all evidence and we consider all evidence. To date I have not seen that type of full disclosure occur in this case.
Of pertinence, on Donald Graft’s page, 17 published papers since 1970:
http://rationalqm.us/papers/Papers.html
JC could profit immensely from this advice. You should talk to him.
I think it is disgraceful that you malign Richard Gill in this way. JC is the only one tossing around viscious ad hominems, referring for example to Dr Gill as a “third-rate statistician”, claiming that PubPeer and RW are “in Gill’s pocket”, that AoP is corrupt, etc.
I am not maligning anybody, and “disgraceful” is a pretty strong word. As my mom always said, “it takes two to tango.” And as any fair person will acknowledge, there are always two sides to every story. (And more if there are more than two people involved.)
I am dismayed to see the way these two individuals have been fighting for several years, which we can all objectively agree has been going on. And because I am a problem solver I am frustrated that I do not see a clear path to solving this. And in all candor, just because of the level of acrimony, I have tried to stay very far out of this.
But if both RG and JC were to tell me that they think I could help to cool things down, then I would try to do so. I might even try to study some of the intricacies of their scientific points of disagreement, even though it is not in the sweet spot of my own interests.
Perhaps you could comment on the following criticism of a passage in arXiv version 5 of Joy’s paper, https://arxiv.org/pdf/1405.2355v5.pdf.
This version contains some new material, in particular, the derivations (57)-(60) and (61)-(64) on page 8. In (57)-(60), the limit is taken as s_1 converges to a of some expression which depends on s_1, a and lambda. Up to (59) everything seems to be OK. In (59) we have a limit of a sum of two terms. Going from (59) to (60) the following seems to have happened. The limit of a sum of two terms is rewritten as a sum of two limits, one for each of the two terms separately. The second of the two limits is evaluated, the result is zero. The first limit is however not evaluated: instead, no limit is taken at all, so that the end result still depends on the dummy variable s_1.
The expression concerned is continuous in s_1, a and lambda and the limit could therefore have been computed by simply evaluating it with s_1 set equal to a. That results in lambda, as already claimed in (54).
So the step from (59) to (60) is non-sense, and the final result moreover contradicts (54).
Richard, As I emailed to both you and Joy privately, I independently did the math in the question you posted about Joy’s going from (59) to (60), and sent that to each of you. I have posted my review of this calculation at https://jayryablon.files.wordpress.com/2016/10/jcrg-2.pdf. I truly hope this is helpful to everyone. Jay
Richard, thank you for your email response to the above. I have included your response, as well as my analysis of your response, in a new posting at https://jayryablon.files.wordpress.com/2016/10/jcrg-3.pdf, which I also sent you by reply email. Best, Jay
On October 10, 2016 at 1:04 am in a post here at RW I offered to try to help mediate the dispute between Joy Christian and Richard Gill, given that I am good terms with both. Richard accepted this in a post at October 10, 2016 at 9:32 am, and Joy has also agreed. So since then I have been trying to winnow down the dispute as clearly as possible in my own mind, and to provide my analysis of the salient points. While I understand Joy’s outrage about the reaction, in accordance with Donald Graft’s recommendation at October 9, 2016 at 11:41 pm I have advised Joy to stay away from personal attacks here at RW and focus on the science, and he is cool with that. I hope that will enable any posts he makes which stay on the science and not on personalities will be cleared onto RW, because, after all, it is his paper that is the topic of this thread. So, in the waning hours before I begin Yom Kippur observance and turn into a pumpkin for a couple of days, I would like to briefly summarize what I have learned so far from my communications with both Joy and Richard, which have been conducted privately and separately by email with each. I will try to do this in a clear and concise way.
In reference to Joy’s paper at https://arxiv.org/pdf/1405.2355v5.pdf, Richard first maintained in his RW post of October 10, 2016 at 9:32 am that Joy’s “step from (59) to (60) is non-sense, and the final result moreover contradicts (54).” As a consequence of the derivation I provided at https://jayryablon.files.wordpress.com/2016/10/jcrg-2.pdf, Richard agreed in a private email to me that in the steps from Joy’s (59) to (60) for Alice and likewise from (63) to (64) for Bob are valid, if we set a=a_1 and b=s_2 when taking the limits in equations (57) to (64). So we are past that.
Richard has next argued that if s_1–> a and s_2 –> b when taking the limits, then because of the further constraint s_1=s_2 that Joy introduces at (65) to conserve angular momentum for the doublet that emerges from the singlet, then this implies a=b, which looks on the surface to be a trivial consequence of combining the foregoing relations in this sentence. If a=b in fact, then Joy’s work would fail, because that would force Alice and Bob to align their detectors which defeats the premise of the EPR experiment that the two parties choose their detector directions independently without conspiracy.
Now, while I agree with Richard that setting s_1=s_2 does serve also to constrain a and b in retain to one another, I find that the constraint imposed in fact is that the magnitudes must obey |a|=|b|. But there is no requirement to have a=b. In other words, the constraint s_1=s_2 in (65) only requires that a and b have the same magnitude which is a weaker condition, but it does not require the stronger condition that they have the same direction. I have laid out my deduction of this at https://jayryablon.files.wordpress.com/2016/10/jcrg-3.pdf for anybody to review in detail if they wish. Because a and b are unit vectors such that |a|=|b|=1, this condition is already built in to the whole model. And even if these were not unit vectors, |a|=|b| is merely a normalization requirement, nothing more. All the salient issues in EPR are about direction, not magnitude.
My discussions with Richard and Joy are continuing, and I will report on them here at RW as I am able to gain further clarification. However, to the extent that this retraction may have been based on a belief that a=b is a consequence of angular momentum conservation, it is my opinion that the retraction is scientifically-flawed, because in fact the only consequence of conserving angular momentum after the singlet becomes a doublet is that the magnitudes must obey |a|=|b|. Not a=b.
I have cleaned up Jay’s analysis so that there is no “backtracking” (which could be subject to objection):
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=271&p=6850#p6850
Let me point out something on which Jay and I apparently agree: from definitions (54) and (55), it follows that A(a, lambda^k) = lambda^k and B(b, lambda^k) = – lambda^k.
Even Fred Diether, one of Joy’s staunchest supporters, seems to agree: http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=271&p=6850#p6848
Joy himself seems to agree too: http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=271&p=6850#p6850 (he calls it “anti-correlation”)
The measurement outcomes are equal and opposite (and do not depend on the measurement settings).
Why compute the correlation by a roundabout route if it is now already clear that it equals minus one?
Correlation does not equal -1. It equals E(a, b) = -a.b: https://arxiv.org/abs/1405.2355
Yes, the -1 result only shows the fact that A and B are anti-correlated. You have to do the correlation calculation in a way that maintains the S^3 properties of the model. The -1 result is R^3 not S^3.
Perhaps I should elaborate.
We live in Einstein’s Universe. One of the solutions of Einstein’s theory of spacetime for the physical 3D space is a 3-sphere. My local model is based on the assumption that we live in this 3-sphere, S^3, not in the flat Euclidean space, R^3, as is usually assumed. Therefore the EPRB correlations we observe in Nature are correlations among the points of this 3-sphere, not among the points of a Euclidean R^3. Now the functions A(a, lambda) and B(b, lambda) defined by my equations (54) and (55) represent points of this Einsteinian 3-sphere. What is then being calculated in equations (67) to (75) are correlations among the points of this 3-sphere. lambda on the other hand is the orientation of this 3-sphere. Therefore it makes no sense to calculate correlations between the two values of lambda, which in any case is just -1. It is a gross misrepresentation of the physics being considered to write A(a, lambda) = +lambda and B(b, lambda) = -lambda. That is like writing Cat = Dog. It means nothing.
Yep, what it all boils down to is a rejection of your S^3 postulate by the Bell believers. Which is OK if they want to reject the postulate. But they won’t find any errors or fatal flaws of the S^3 model. However since the S^3 model so accurately describes the EPR-Bohm scenario in a local-realistic way and agrees completely with the quantum mechanical prediction and since it also a solution of Einstein’s theory of spacetime, chances are very good that your postulate is probably correct. Something else that it means is that space has unique spinor properties.
I am glad that Joy is talking about the 3-sphere; I want to explore that here more deeply when I have more time.
But let me go to his latter statement. To be fair, I would not use “gross misrepresentation” because in fact Richard is correct that A(a, lambda) = +lambda and B(b, lambda) = -lambda are true equations, mathematically. But what I would say is that they do not tell anywhere near the whole story. Think about it: At one level, everything regarding EPR and Bell is about the numbers +/- 1. It has to be, because these theories are about correlating space-like-separated events that have binary values. So it should not be surprising to see a +/- lambda showing up in the Alice and Bob equations when they are whittled down to their bare skeletons. There is no reason for Richard to critique those equations any more than there is reason for Joy to take exception to Richard pointing out those equations.
But what Joy is effectively doing is a lot of fancy dressing up of the number +/- 1 times lambda, which is perfectly fair to do. All the time in math and science, we take the number 1 and turn it into some very complex and rich expression which is equal to 1. And then we can manipulate the rich version of 1 and learn real things from that. For example, 1=u_alpha u^alpha for the four-velocity u^alpha=dx^alpha/dtau is an integral constant of the geodesic equation of gravitational motion — a lot of information to pack into the number 1. Put that into a variation the right way as Einstein did in section 9 of his 1915 paper, and you find out about such things as perihelion precession, satellite orbits, and what happens when I drop something. One would never say, oh, u_alpha u^alpha is equal to 1, so that cannot mean much. On the other hand, 1=3-2 tells us nada about gravitational motion and really does not mean much. It is all about what you do with the number 1.
So here, as I point out in (60) of https://jayryablon.files.wordpress.com/2016/10/jcrg-2.pdf, lambda^k = sgn(s_i^k dot a) is just a fancy way of dressing up the +lambda for Alice, and likewise can be done for Bob. But when we use +lambda in this form of a sign function, we get A(a,lambda^k)= sgn(s_i^k dot a) and likewise for Bob, which is Bell’s result, and it is also Joy’s result. All of this is a perfectly valid way IMHO to dress up the numbers +/-1 times lambda and get some information of real interest from them.
Jay
Jay, I would have to say that there is no “fancy dressing”. It is the behavior of the 3-sphere topology of space coupled with the particle spin that produces the correlation. After all, it is just clicks in a detector and not really any number other than the number of clicks that you count. But to follow up what I was saying earlier about the rejection of Joy’s postulate, it doesn’t matter whether the postulate is rejected or not because it is still a valid counter-example to Bell’s so called “theorem” since it produces the predictions of quantum mechanics in a local-realistic way.
Hi Fred,
Well, I will admit to being a little colorful to try to make a point. And in fact I have also done a “fancy dressing” 1 as well, for example, at (4.1) and (10.1) of my draft paper on electrodynamics electrodynamics at https://jayryablon.files.wordpress.com/2016/10/lorentz-force-geodesics-brief-4-1.pdf.
But think about it: Every equation ever written can be rewritten in the form:
“something = 0.”
just move everything onto one side and you have a zero. (And to be really clear the zero may have all sort of finite or infinite structure to it; think tensors, and think SU(N), and think Hilbert spaces, and think Heisenberg matrices.) On October 12, 2016 at 8:01 am here at RW, Richard said “why compute the correlation by a roundabout route if it is now already clear that it equals minus one?” IMHO that would be analogous to saying the “why compute anything about the ‘something’ if it is now already clear that it equals 0?” This sits on the slippery slope of degenerating into an argument that mathematical calculation serves no purpose.
All equations of consequence reduce to very simplified skeletal equations. Those equations gain power when they are unpacked and an applied in their more complex and rich forms. That is all I am saying.
Jay
I wanted to take a few minutes at the start of what will be a very busy weekend and coming week to update everyone on the communications I exchanged with Richard on October 11 and 12. I have copied them over verbatim into a two-page PDF document linked here: https://jayryablon.files.wordpress.com/2016/10/rg-jry-10-11-to-10-12.pdf. I converted all of the ascii equations that were in our emails, into visual equations, so you can better read them and I can better analyze and work with them. When I next have some time, I will provide my analysis of Richard’s 10-11 and 10-12 emails in the above link, beyond what I already said in my 10-11 email in this link. I have also had some communication with Joy, primarily about the meaning of the three sphere, which I hope to discuss at length when I catch some breaks next week.
Again, my purpose in all of this is to pinpoint the points of disagreement and / or misunderstanding between Richard and Joy and their respective “schools of thought,” and see if I can at least get them to agree about what they disagree over in a way that can be “refereed” by others reading this thread as well. Hopefully, this may illuminate the scientific merit (or lack thereof), of the AOP retraction.
Good weekend to all. Jay
Gill is still ignoring the fact that s1 = s2 = s is true at creation of the particle pair and it is not necessarily true at detection. s1 = s2 can only be true at detection if the experimenter sets a = b.
Thank you. You seem to be saying that in the step from (69) to (70) in https://arxiv.org/pdf/1405.2355v5.pdf, Christian is assuming that a = b. Because he is using a relationship which is true at creation of the particle pair but not necessarily true at detection. So no longer necessarily true in the limit as s_1 converges to a and s_2 converges to b.
No one is saying that. Not I, not Jay, not Fred.
Fred, maybe I am missing something, but the angular momentum of the singlet –> doublet “system” should remain conserved unless and until there is an interaction with something else outside that “system,” in which case the “something else” becomes part of the “system.” Once you “detect” one of the particles in the doublet, the detection itself will thereafter change things for that particle by virtue of the detection, which many people refer to as “collapsing” the wavefunction. But what the Alice and Bob detectors should detect at the moment of detection, before any other intervening interactions, and no matter how the detectors themselves are oriented, are oppositely-oriented spins. Am I missing something here?
Yes, you are missing the S^3 topology. Plus in an EPR-Bohm scenario, there are polarizers before the detectors. It is all encoded in Joy’s S^3 model.
You may be missing a couple of things. First, none of this has anything to do with “collapsing” the wavefunction. We are not doing quantum mechanics here. The spin-0 angular momentum remains conserved in the 3-sphere model until the detection process, which, following Bell, is encoded by the function A(a, lambda), where lambda specifies the initial state of the spin and “a”, Alice’s freely chosen direction of measurement, specifies the final state of the spin. The vector “s” is not a hidden variable in my model (it is not being summed over in the correlation calculation). So s_1 remains equal to s_2 until the detection process, which, as I mentioned, is encoded in the function A(a, lambda). This function, according to Bell, can be whatever you like, as long as it depends only on a freely chosen direction “a” and the initial state lambda. You are thinking too physically here, beyond the requirements by Bell.
That is a better way to put it.
The following is my Formal Response to the Withdrawal of my article from Annals of Physics. I have emailed it to the Editors of the journal, as well as to its publisher, Elsevier:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=285&p=6885#p6885
Hi Richard:
Joy Christian is attempting in his paper at https://arxiv.org/pdf/1405.2355v5 to show how to reproduce Bell’s theorem consistently with local realism, so long as the spacetime metric includes a closed curved space S^3, in accordance with the Friedmann-Robertson-Walker metric which is Joy’s equation 9. After having had some time to study Joy’s paper and the questions you have raised publicly and privately since we began our communication following your above RW post on October 10, 2016 at 9:32 am, and as I have previously stated at RW, my objective is to try to pinpoint exactly where you believe Joy’s paper is in error, and to perhaps resolve the long-ongoing dispute between the two of you and your respective “schools of thought.” IMHO, it would not only be good for you and Joy, but good for the scientific enterprise, to finally settle a long-standing scientific dispute, one way or the other.
Accordingly, in a 4-page document linked at https://jayryablon.files.wordpress.com/2016/10/jc-analysis-3.pdf, I have sought to carefully lay out my own independent analysis of how Joy tries to connect to Bell in a local realistic manner based on S^3, I would ask for you to please comment on where my analysis fails, if it does, or to agree that it does not fail in which case you would have agreed that Joy’s work is correct and that its retraction was scientifically in error. Toward that end, I have included very specific questions at various places below, and numbered them, so that in reply all you need to do is use the question numbers for your answers.
Thanks,
Jay
Here is an interesting link http://www.scottaaronson.com/blog/?p=1028 with some discussion of Christian’s previous work. Apparently he has form for threatening legal action.
That “previous work” of mine is now published, which proves Aaronson wrong:
http://link.springer.com/article/10.1007/s10773-014-2412-2
More details about that proposed experiment can be found on my blog:
http://libertesphilosophica.info/blog/experimental-metaphysics/
If there is anything that RW has shown us is that anything that has been published can just as easily be unpublished, for a wide range of reasons. The published status is no longer safe.
Richard D. Gill did indeed try extremely hard to get the paper I have just linked retracted. But on that occasion he was not successful, thanks to a very noble Editor-in-Chief, the late Prof. David Finkelstein (RIP): https://en.wikipedia.org/wiki/David_Finkelstein
David Finkelstein urged me to submit my critique to IJTP. It was accepted and published as http://link.springer.com/article/10.1007/s10773-015-2657-4
The above “critique” by Gill he has linked is based on elementary mathematical mistakes, and has been comprehensively debunked by me and others. For example, in it Gill forgets to sum over Einstein’s summation index in the very equation he is supposed to be questioning. See, for example, my reply below to his “critique”, and a reply by Peer 2 on PubPeer:
https://arxiv.org/abs/1501.03393
https://pubpeer.com/publications/4C65BF0EDD5500B6D460273C68E70E
I have had several email communications with both Richard and Joy during this past week, and would like to pinpoint what I think is their key point of disagreement. For this discussion I will refer to v6 at https://arxiv.org/pdf/1405.2355v6.pdf which Joy posted on October 17.
From what I have been able to gather, the disagreement between Joy and Richard and their respective “schools of thought” is focused on Joy’s (67) through (75) (which are similarly numbered in both v5 and v6). These find that the expected value for the correlation:
E(a,b) = – a dot b = – ||a|| ||b|| cos (theta). (Joy 1)
Above, a=(a_x, a_y, a_z) and b=(b_x, b_y, b_z) are three-vectors which respectively represent the orientation of the Alice and Bob detectors in three-dimensional physical space with coordinates (x, y, z). Also above, I have included the definition of the dot product to help highlight what I think is the key issue dividing Joy and Richard.
Richard, however, contends in contrast to (Joy 1), that this expected value is ALWAYS given by:
E(a,b) = – 1 (Richard 1)
Now, if we use the magnitude normalization ||a|| = ||b|| = 1 to which both gentlemen I believe would accede, then (Joy 1) simplifies to:
E(a,b) = – a dot b = – cos (theta) (Joy 2)
Of course, (Joy 2) and (Richard 1) will give the same result in the special case where cos (theta) = 1, i.e., theta = 0, which is to say, in the special case where the Alice and Bob detectors are aligned in the same spatial direction. (For completeness, note that we may also have theta = 2pi times a positive or negative integer, which is still the same orientation.)
At bottom, what I believe Richard is effectively saying is that the mathematics Joy has used to get to (Joy 1) and (Joy 2), and in particular, Joy’s simultaneous application of the limits s_1 -> a and s_2 -> b, has the effect of forcing Alice and Bob to align their detectors in the same direction whereby they must have a=b, which violates the EPR conditions that each must be able to freely choose their detector directions.
In contrast, Joy is saying that no constraint is imposed by taking these limits. And at (85) to (91) and the surrounding discussion paragraph recently added to v6, Joy is saying that the only constraint is that the magnitudes ||a||^2 = ||b||^2 must be the same, but that the directions may be freely chosen and do not have to be aligned.
Joy’s argument here happens to be based on a calculation which I developed and posted at https://jayryablon.files.wordpress.com/2016/10/jcrg-3.pdf in my RW post of October 11, 2016 at 1:08 pm. While this is my calculation, which I believe is a correct as a calculation qua calculation, I will maintain an agnostic position about how it interpretively applies to Joy’s overall result. Specifically, if a=b as Richard maintains, then ||a||^2 = ||b||^2 is implied as well by forward logic (strong logic constraint). However, if ||a||^2 = ||b||^2, then the reverse logic does not necessarily apply, that is, we can, but do not need to have, a=b, which Joy maintains (weak logic constraint). The logic arrow only works one way, and Joy and Richard seem to be disagreeing about whether the stronger or the weaker logic apples. That is the question about which I will stay agnostic for now, while waiting to see arguments for each position.
So, if I may cast all of this into the simplest picture possible (which is just another view of the logic arguments in the last paragraph), let me take the cosine in the (Joy) equations above and write this at the level of 8th grade algebra as y = cos x. What Joy is saying is “I have obtained a result y = cos x which reproduces the Bell correlation using local realism when the spacetime metric contains a curved Friedmann-Robertson-Walker space S^3.” What Richard is saying “sure, you have y = cos x, but the mathematics which got you to that result forces you to have x=0 (or 2 pi times an integer) and thus forces Alice and Bob to align their detectors in the same space direction.” And Joy in his new paragraph surrounding (85) to (91) is attempting to explain why this is not so. Again: Richard says a=b is true which implies ||a||^2 = ||b||^2 is true, while Joy says that only ||a||^2 = ||b||^2 is true, which does not imply that a=b.
So, on to the bottom lines: QUESTION 1 to each of Joy and Richard: do you both at least agree that the foregoing is the crux of your disagreement?
And, if it is, then I would ask QUESTION 2 to Richard, and to anybody else who wants to chime in: is the explanation Joy has added in association with (85) to (91) of his v6, convincing? And please articulate reasons why or why not.
Finally: In going from (67) to (75) to obtain the correlation (Joy 1), Joy says in both v5 and v6 that (72) follows from (71) “by removing the superfluous limit operations.” From everything else discussed above, the root of where Richard and Joy do not agree appears to stem from these limit operations. This is because at some level, Richard sees these limits as dispositive because they force a=b, and Joy sees them as superfluous and only requiring the weaker constraint ||a||^2 = ||b||^2.
So let me pose a final QUESTION 3 as a helpful suggestion to Joy: Please elaborate WHY these limits are superfluous? And if they really are superfluous, is there perhaps some alternative mathematical calculation path you can show to derive (Joy 2) without having to use these superfluous limit operations at all?
Thank you to both Richard and Joy for continuing to bear with me on all of this.
Jay R. Yablon
The real issue for me is not mathematics, but my central hypothesis, which says that the EPR-Bohm correlations, E(a, b) = -a.b, are correlations among the scalar points of a quaternionic 3-sphere (S^3). Once we accept this hypothesis, then the mathematics is trivial. Accepting the hypothesis means, however, that we must accept that the functions A and B defined in my equations (54) and (55) are actually scalar points of a quaternionic 3-sphere, with values +/-1. Once we accept that, then the mathematical steps must follow my equations in the paper, all the way to the end, without exception. Of course there are several other ways to derive the correlations -a.b among the scalar points of a quaternionic 3-sphere without using the limits I have used in this paper, using different definitions of A and B, but with the same result E(a, b) = -a.b, as I have already done in several other papers and a whole book:
https://arxiv.org/find/all/1/au:+Christian_Joy/0/1/0/all/0/1
Now the claim “E(a, b) = always -1” rejects the 3-sphere hypothesis. In fact, there are only three possible ways to get the correlations other than E(a, b) = -a.b. These are: (1) by rejecting the geometry and topology of the quaternionic 3-sphere; (2) by violating the conservation of the spin-0 angular momentum; and (3) by doing both (1) and (2). Thus insistence of “E(a, b) = always -1”, no matter how it is justified, amounts to rejecting the fact that there is a Mobius like twist in the geometry of the quaternionic 3-sphere. This is not something new I am saying. This goes back at least to my discussion of the Hopf fibration of S^3 in one of my papers written in 2011 (if not all the way back to my first paper on the subject written in 2007). This issue is now clarified with your result of ||a||^2 = ||b||^2 versus a = b, as discussed in the new material I have presented in the version 6 of my paper, after eq. (80), towards the end.
So, to answer your question, yes there are ways to derive the correlation E(a, b) = -a.b without the use of the limits, as I have done before. But that by no means suggests that what I have done in the current paper employing limits is somehow wrong. It is just a nifty way of obtaining the same inevitable result: -a.b. By now I have derived the same result in some 10 different ways! Far from my use of the limits being problematic, to me it is one of the most beautiful ways of appreciating the intrinsic coherence of the geometry and topology of the quaternionic 3-sphere. The limits become superfluous because of the three-dimensional rotational symmetry of the 3-sphere, which (as you know, Jay) leads to the conservation of spin angular momentum. Physics is the key to understand my model, not mathematics (which is trivial),
Hi Jay,
Quite frankly, the limits on the functions A and B aren’t even required for eqs. (68) to (71). They could just be taken off. The A and B limits are really only required for eqs. (54) and (55) to demonstrate that the outcomes for A and B are indeed +/- 1.
As I see it, Christian’s own definitions (54) and (55) force E(a, b) = -1 for all a and for all b. No amount of argument changes the direct mathematical consequences of explicit mathematical assumptions. So this is where I have to quit the discussion.
Thanks for your attempts at mediation.
On the contrary, my definitions (54) and (55) force E(a, b) = -a.b for any a and for any b.
But I agree with Gill that “no amount of argument changes the direct mathematical consequences of explicit mathematical assumptions.” My explicit mathematical assumptions inevitably force E(a, b) = -a.b for any freely chosen a and b by Alice and Bob.
QUESTION 4: Why is JC still engaging in uncivil and inflammatory rhetoric when you claim he has promised to desist from it?
“Aaronson also blocked many of my rebuttals from his sordid blog because they would have been too uncomfortable for him. They would have exposed his dogmatism, ignorance, and hypocrisy. Instead of honestly engaging with my rebuttals openly, he systematically blocked them and resorted to hiding behind insults and treachery.”
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=286&p=6903#p6903
Who can fault Scott Aaronson for ignoring posts made in such a style?
Regarding JC’s “challenge” be aware that Fred Diether refuses to approve posts that respond to the challenge. Here we have a classic case of projection or, if you prefer, the pot calling the kettle black.
It’s simple why anonymous posts were disapproved regarding Joy’s challenge. They all violated the conditions that Joy specified in one form or another. Don, you are a member of the forum so you are not subject to pre-moderation by posting anonymously. If you have a valid challenge then sign in and post it.
FD thus confirms that he is preventing the community from deciding for themselves on the validity of the responses to the “challenge”. It is disingenuous in the extreme to delete all the responses so that JC can claim that there are no responses and that therefore he must be correct.
Regarding my membership at the forum, I long ago asked that FD delete my account (because when I signed up I was not aware of its unscientific nature). FD refused to do that. I again ask FD here to delete my account.
That is not true obviously. Only anonymous guest comments were rejected as they are subject to moderation and there is no point in allowing challenges that obviously violated the conditions of the challenge. But people can become a member of the forum anonymously also then post a challenge without moderation.
Sorry, but when you sign up to become a member, you agree that it is forever. Only spammers are banned but their accounts are never deleted. You are the one that went into some kind of rage when everyone was being nice to you so we don’t understand your problem.
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=210&p=5952#p5950
But all of this is quite off-topic to what Jay has posted.
Permit me to paraphrase what I think Joy is saying in his post of October 24, 2016 at 1:27 pm, without getting into the weeds. Again, I will use Joy’s latest draft at https://arxiv.org/pdf/1405.2355v6.pdf.
To start, everyone studying this discussion is or should be familiar with the non-commuting Pauli spin matrices sigma_i with i=1,2,3 corresponding to the x, y, z space coordinates, which are embodiments of the quaternions for which Hamilton first famously inscribed i^2=j^2=k^2=ijk=-1 in the Brougham Bridge in 1843, in the first of what have now been numerous demonstrations including Yang-Mills gauge theory and Heisenberg matrix mechanics that non-commuting objects play a fundamental role in describing nature. It is well-known that any of these spin matrices when multiplied side by side is related to itself and the other two matrices in this group, which is closed under multiplication, by (below, delta is the Kronecker delta, eta is the antisymmetric Levi-Civita tensor, and repeated indexes are summed):
sigma_i sigma_j = delta_ij +i eta_ijk sigma_k (1)
This captures what happens when one of these matrices is multiplied by itself, as well as what happens when one of these matrices is multiplied by a different one of these matrices. When these sigma matrices are in turn used in the Dirac gamma matrices forming the most basic and physically-relevant Clifford algebra, were are able to obtain the electron magnetic moments via the Gordon Decomposition of the vertex factor in the current four-vector, using the Dirac bilinear covariants for magnetization and polarization. And when we enumerate all the loops in the vertex, we obtain the observed anomalies, in perhaps the most-precisely validated theoretical / experimental concurrence that exists in the world today.
So if we give the same commutativity relation (1) to angular momentum via sigma -> L, but with an opposite sign convention and including a hidden variable lambda, this is Joy’s equation (48), which readily goes over into his (49). And Joy correctly points out that a three-dimensional rotational symmetry implies that angular momentum is conserved, which is but one aspect of Noether’s theorem. This is also, it seems to me, why Joy keeps referring to a quaternionic space. And we certainly know form spinor algebra, that spinors reverse sign after a 360 degree rotation, which is visualized by so-called orientation-entanglement, and which is lost when one uses O(3) rather than SU(2). That is why Joy hammers so much on this being S^3, not R^3.
Next, one of the geometric curiosities intrinsic to any three-dimensional object, is that such an object has a given parity, which can be turned inside out and made into an opposite version of itself, such as our right hand and our left hand. Joy uses the barber shop pole as an example. But we cannot turn our right hand into our left hand by any type of rotation. Rather, we must flip its parity, i.e., set x -> -x along all three space axes. So any three-sphere of the Friedmann-Robertson-Walker (FRW) type, because it is three dimensional, may have a right-handed parity, or a left-handed parity. This parity, which Joy has consistently referred to as “orientation” (a misnomer in my humble opinion), is what Joy is assigning to the hidden variable. In my words: the parity of the 3-D FRW space is Joy’s hidden variable.
So with this backdrop, if I am understanding correctly, then what Joy is essentially saying is that he uses the limits s_1 -> a and s_2 -> b to obtain the relation E(a,b) = – 1 that Richard has repeatedly emphasized, but really, E(a,b) = – a dot b for the special case where a=b. And I think Richard and Joy would agree up to here. But Richard will pin Joy to this special case only.
Then, precisely because Joy is using a three-space S^3 with the rotational symmetry governed by the extension of (1) to angular momentum with a hidden variable which is the parity of S^3, Joy says that he may use the rotational symmetry of the three-sphere to go from the special case of a=b, to the general case of any relative orientation between a and b. And this is what he sees as “nifty” about using the s_1 -> a and s_2 -> b limits and then applying spherical symmetry.
Joy, am I understanding this correctly? And Richard, does this make sense to you?
Jay
Hi Jay,
Excellent summary. I only have a couple of very minor and rather pedantic quibbles with what you have written. One is that I would deliberately stay away from the matrices like Pauli or Dirac matrices that are so second nature to us in physics. The whole point of using geometric or Clifford algebra directly and not its matrix representation is to stress that the former are in fact capturing the properties of the physical space itself. So this is my first pedantic but important point. It is a mistake, from my point of view, to fall back to matrices. I am happy to do that when I am doing particle physics with Fred, but not in my work on the Bell stuff, which is based on the intrinsic geometry and topology of the physical space itself.
My second point is also quite pedantic. The world “orientation” is the geometers’ and topologists’ nomenclature for “parity.” The distinction is pedantic but quite important when one is dealing with the general relativity stuff like manifolds and vector spaces, and it is built-in in the literature on “geometric algebra” (which was the term preferred by Clifford because of its significance for geometry).
This somewhat unfamiliar nomenclature I have been using for the past nine years — “orientation” rather than “parity”, “multi-vectors (bi-vectors or tri-vectors)” rather than “matrices”, and “geometric algebra” rather than “Clifford algebra” — has been one of the main difficulties in communicating with those familiar with the physics jargon but not familiar with the one used by topologists and geometers.
You did not respond to my point that there is a simple direct proof that Christian’s assumptions imply that E(a, b) = -1 for all a and b: namely via the equalities A(a, lambda) = – B(b, lambda) = lambda = +/-1 for all a and b, see definitions (54) and (55). No amount of nifty (but correct) mathematical tricks can ever get a different result.
My assumption by no means implies “E(a, b) = -1 for all a and b.” My definitions (54) and (55) for the measurement functions A and B necessarily lead to E(a, b) = -a.b, as evident from my calculations that follow from the definitions (54) and (55). As I have stressed here before, it is impossible to derive the correlations other than E(a, b) = -a.b without violating the conservation of the initial spin-0 angular momentum. In particular, the insistence “E(a, b) = -1 for all a and b” violates the conservation of the initial spin-0 angular momentum, and therefore violates the rotational symmetry of the three-dimensional physical space. As we know, the correlations E(a, b) = -a.b are observed in the real physical world where angular momentum is conserved and the rotational symmetry of the physical space is respected.
Then let me do so here and now.
First, I agree, and have agreed in the past, that A(a, lambda) = – B(b, lambda) = lambda = +/-1. I think Joy and Fred have also agreed. And I have also agreed with some of the nifty and correct mathematical tricks that get us over to Bell’s correlations. But your disagreement with Joy centers around his using a symmetry argument to generalize from this specific result over to the general correlation of Bell. He says he can get there with a symmetry argument and you say he cannot. That is the flashpoint.
Accordingly, I will prepare and post a more extended discourse about this, but at the bottom line, given the high threshold of acceptance that Bell’s results have in the physics community, I will recommend to Joy that he PROVE the symmetry argument, and not just MAKE the symmetry argument, and do so in this very paper and not merely by reference to previous papers.
Jay
Jay asked me privately whether I can prove that conservation of spin momentum is violated if we insist on E(a, b) = -1 for all a and b. The answer is, Yes. If we insist on E(a, b) = -1 for all a and b, then my eq. (68) implies that – (s_1 . s_1) (s_2 . s_2) = -1. Which implies ||s_1||^2 ||s_2||^2 = 1. Which implies ||s_1|| = 1 / ||s_2||. Which violates of the conservation of spin momentum. If we increase the magnitude of s_1, then the magnitude of s_2 decreases, and vice versa. Whereas the conservation of spin momentum requires that s_1 = s_2, and thus ||s_1|| = ||s_2||.
I just want to confirm what Mr. Graft has said. Moderator Diether did refuse to post a politely worded comment of mine related to Christian’s challenge.
I appealed in the hopes that the other active moderator, who had always impressed me as probably a fair-minded person, might be repelled by such unjustified censorship. I later learned that Diether never notified the other moderator of the appeal. He rejected it unilaterally.
This so-called “forum” is not a forum at all. I would never attempt to post there again. I only did so (as “guest1202”, not as a registered user) because having no knowledge of the readership of the “forum”, I was curious what would be their response to what I think is the generally accepted content of Bell’s theorem.
It is not what Christian represents. What he sees as an elementary error in Bell’s proof assumes a position which is not that of Bell, nor of the thousands of modern physicists who have checked the proof. Christian essentially challenges the reader to affirm this false position. No wonder that no one replies!
Regarding the handling of Christian’s paper by Annals of Physics, I agree that it shows both editorial incompetence and bad faith. They accepted the paper after seven months of supposedly careful review. They should have known that Bell’s theorem has been famous for over half a century, and that any refutation of it would be headline news. Regardless of the correctness of Christian’s paper, Annals’ handling of it was reprehensible. No author should be treated in this casually arrogant way.
“Guest1202”
P.S. I am a mathematician with a Ph.D. and a modest research record. In normal forums, I post under my real name, but I did not do so in sci.physics.foundations because I did not want to become associated in any way with that group.
Dear Guest 1202:
I recognize you as having anonymously made a number of very insightful posts which critiqued my own work and helped me to adjust my thinking in a good way, especially as I discussed in the post at http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=233&p=6052&hilit=guest1202#p6052. I credit your input for motivating my work these past ten months that led me to the near-submission-ready paper at https://jayryablon.files.wordpress.com/2016/10/lorentz-force-geodesics-brief-4-2.pdf. I deeply regret that you have stayed away from SPF if for no other reason then the selfish one that I found your comments to be very helpful to me. I hope you will reconsider, or at the very least, will consider having some private discussion with me at [email protected].
Jay
When FD refused to delete my account I purposely tried to get banned, just as Richard Gill was banned.
DG did not purposely try to get banned as he did not spam the forum. He only asked to be banned during his rage. He later tried posting to the forum anonymously several times. But most of those post were rejected in moderation because they were so plainly wrong. We are not sure why DG wanted to post anonymously as he was and still is a member of the forum and his posts wouldn’t have been disapproved. I guess he was just afraid of being shot down publicly.
RG was banned for spamming the forum with many overly repetitious posts after he was warned to stop doing it twice.
The fair number of posts tonight with personal squabbling and finger pointing explain better than I ever could why I sat on the sidelines of all the Bell wars for the last several years, and only got involved when I was dragged in by a potshot that was taken at me here, after which I decided that it would not be wise to sit this out any longer. But I will not be part of the squabbling. That must end, and these Bell wars must be concluded one way or the other on the merits, on behalf of the scientific enterprise. The biggest disservice to science that AOP committed here — and it is regretful given that journal’s storied history — is that proceeding with publishing Joy’s paper could have provided the opportunity to bring a conclusive end to the Bell wars either by Joy’s work gaining definitive acceptance or being definitively shot down. Instead, the war has now only intensified and much of it has roosted here at Retraction Watch.
Given that, I will only discuss the science and the math, and I again ask everybody else to please do the same. I am participating in the discussions here rather than elsewhere, because RW is independently moderated and so may provide the best neutral terrain on which to discuss and sort out the scientific issues with all views represented and none censored. Our only master and our only loyalty in this enterprise, must be to the natural universe we all inhabit.
And please, everybody, keep in mind that we are rarely as wonderful or brilliant as we imagine ourselves to be, nor are others often as bad or stupid as we imagine them to be. Keep an open mind and listen, especially to the people who see things differently from you. They are proxies for the experiments that also often see things differently than we do and force us to change our minds. That is not a weakness for a scientist; it is a strength. But keep the personal stuff out of it. If you want that, just go watch the news from the US elections; this is not how scientists should behave.
Jay
PS: This sort of thing is also why I opted to pursue my own scientific work outside of academia. Back in my MIT days when I was seriously considering getting onto the professorial treadmill, I already had picked up on a lot of the academic mudslinging and decided I would rather pursue science work on my own and develop independent means of support enabling me to do so. In many ways that has been a more difficult path. But in all the ways that count on substance, it is the only path I could have followed. This is because it is much harder to learn then unlearn and fend off political pressures as would be required “inside,” than to learn correctly in the first place with finding natural truth being the only objective as I am able to do “outside.”
I don’t think there is a “Bell war”. There is only Joy Christian, having his paper retracted before it was even printed. Nor is there any kind of censorship here, since his paper is freely available on arXiv for anyone to read. He has been publishing his theory there for almost ten years, with various papers.
I beg to differ. Removing a published paper without telling the author for over two months, even after multiple requests for clarification, and without providing any scientific evidence for the removal of the paper, is a clear form of censorship.
The paper was already public on arXiv, and will presumably be so forever. It’s not censored.
The paper is now permanently stigmatised, due to no fault of its author. It has clearly been censored.
No fault of your own, really? The paper is stigmatised because the author keeps trying to publish something that is demonstrably incorrect. The author continues to argue with people on the internet rather improving the artictle. I don’t think a journal accidentally accepting something briefly is comparable to the stigma surrounding this nonsense already. Why do you think people are electing to comment anonymously?
I invite you to demonstrate (anonymously if you prefer), what is “incorrect” in my withdrawn paper. But I admit that what I have presented in my paper is politically incorrect.
What would the A and B functions be for QM to produce +/- 1 outcomes? They would certainly have to be something like Joy’s A and B functions. So does that mean that quantum mechanics also predicts -1 as a result? No! The -a.b prediction from quantum mechanics for an EPR-Bohm scenario is derived by a probabilistic method and is NOT simply A*B. Bottom line is that the predictions for a theory are calculated in a different way from how experimental results are calculated. So it is quite a mystery as to why you are rejecting that.
Christian defines E(a, b) in the first equality of (67) of https://arxiv.org/pdf/1405.2355v6.pdf Now substitute A(a, lambda) and B(b, lambda) with their definitions, evaluating the two limits, as Christian already correctly did in (54) and (55). That’s all.
When we substitute the functions A(a, lambda) and B(b, lambda), defined in my equations (54) and (55), into the definition (67) of E(a, b), and then evaluate the two limits, making sure that nothing unphysical (such as a violation of the conservation of spin angular momentum) is unwittingly (or on purpose) smuggled-in, then E(a, b) works out to be E(a, b) = -a.b. That’s all.
I would be curious if Richard or anybody on the opposite side of this debate has a view on what Joy says he has proven above re spin momentum conservation? Jay
I was under the impression that s_1 and s_2 were meant to be unit vectors, throughout this part of the paper. See (66) and the sentence preceding it. So their magnitude is constant, anyway.
In (54) and (55) they are apparently dummy variables (also known as bound variables: https://en.wikipedia.org/wiki/Free_variables_and_bound_variables). When we compute the limit of a function as one of its arguments approaches some value, the result does not depend on the name which we give to the argument. The result depends on the function, and on the point at which we compute the limit.
As Christian himself points out below equation (75), the limits in (68) and (69) result in
-D(a) L(a, lambda) L(b, lambda) D(b)
From (56) it follows that this equals – L(a, lambda)^2 L(b, lambda^2) = -1
It goes wrong at the step from (69) to (70). Here Christian is using a property of s_1 and s_2 derived from the physical interpretation which he likes to give them (the property that s_1 = s_2), while in fact he is working here with two mathematical variables whose names are irrelevant. They are placeholders. Might as well be called t_1 and t_2, or anything else. We study a function of t_1 and t_2 in the limit as t_1 approaches a and t_2 approaches b. If you want to impose t_1 = t_2 you’ll need to restrict attention to the case a = b.
Essentially, he briefly assumes a = b so that L(a, lambda) L(b, lambda) = -1, giving
-D(a) L(a, lambda) L(b, lambda) D(b) = D(a) D(b) = L(a, lambda) L(b, lambda)
Then he forgets that he has assumed a = b and continues as if his result is generally true.
Later, yet more goes wrong (he uses identity (49) which contradicts (50) and (51)). Though perhaps the symbol for cross product is not well-defined; probably the cross product should depend on lambda since lambda is supposed to be the random orientation of S^3. Does this mean that whether we use the right-hand or the left-hand rule to compute axb should depend on lambda? If so the notation needs to be more elaborate. And this would spoil the transition from (73) to (74).
The issues raised by Gill in his long comments above have all been thoroughly addressed by me and others, many times over, during the past several years, and they have all been long settled. See, for example, my answers in the link below.
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=271#p6808
More importantly, his comments do not address the question asked by Jay in his post, or to me privately. What Jay asked was whether it is possible to prove that conservation of spin angular momentum is violated if we insist on E(a, b) = -1 for all a and b as Gill does. The answer is, Yes. If we insist on E(a, b) = -1 for ALL a and b, then E(a, b) = -1 must also hold just in case “a” happens to be equal to s_1 and “b” happens to be equal to s_2. Now we simply substitute a = s_1 and b = s_2 into eq. (68), without ANY prior assumptions about the vectors s_1 and s_2. Then eq. (68) becomes – (s_1 . s_1) (s_2 . s_2) = -1 after some very elementary mathematical steps (which I will be happy to spell out if necessary). But that then implies ||s_1||^2 ||s_2||^2 = 1. Which in turn implies ||s_1|| = 1 / ||s_2||. Which evidently violates of the conservation of spin momentum. If we increase the magnitude of s_1, then the magnitude of s_2 decreases, and vice versa. Whereas the conservation of spin angular momentum requires that s_1 = s_2, and thus ||s_1|| = ||s_2||. Thus insisting on E(a, b) = -1 for all a and b as Gill does is equivalent to smuggling-in (if only unwittingly) the violation of the conservation of the spin angular momentum. QED.
By the way, nowhere in my paper, or in the above proof, have I assumed, or have I been forced to assume, that a = b must hold, unless of course Alice and Bob freely choose to set a = b, which they are certainly allowed to do.
The average of the crossproduct in (74) vanishes anyways without lambda. a x b = c.
As a non-physicist, I have to admit that I am really lost in what these scientists are discussing. Who can independently verify the accuracy of what is being said by the conflicted parties in this discussion thread?
If you trust computer code then here is your answer.
http://challengingbell.blogspot.com/2015/05/further-numerical-validation-of-joy.html
Any questions, please ask.
You needn’t be a physicist in order to detect bad math. Look for unclear notation, e.g., implicit arguments or use of the same symbol for different meanings; unsound or poorly justified reasoning; contradictory assumptions; &c. If the presentation is bad the article should not be published even if the conclusions were correct.
Having read the RW discussion these past couple of days, it is very plain to me that the central flashpoint between Joy and Richard is that Joy uses the limits s_1 -> a and s_2 ->b together with s_1=s_s==s to conserve angular momentum, in order to derive the nonlinear correlation E(a, b) = -a.b predicted by quantum mechanics and observed in experiments. Richard says that this is tantamount to setting a=b and restricting Alice and Bob to aligning their detectors in the same direction in violation of the EPR premises. Joy says that he can use the spherical symmetry of S^3 to generalize back to independent a and b such that ab. Richard says that this symmetry argument is lipstick on a pig and no Joy can’t generalize a and b. Joy says no it is not and yes he can. Richard says no. Joy says yes. And then we are on a yes, no, yes, no treadmill. But this all traces back to whether Joy is or is not imposing the constraint a=b. We need to break that up.
I do not have enough information to know whether in the end Joy is right or Richard is right. But I have seen enough to have concluded that it is not enough for Joy to MAKE this symmetry argument. I believe that Joy needs to rigorously, mathematically PROVE this symmetry argument. And I am making a public call on Joy to do so, with some suggestions about how to do so.
At the same time, I know that Joy is very hesitant to do this, because he believes that his opposition has shown a historical pattern of “moving the goalposts” any time he revises a paper to account for their critiques. And I know that Joy believes he has already done this before, and only used the limits-plus-symmetry argument to help overcome earlier objections to his more general development that does not use limits-plus-symmetry. This is why Joy sometimes publicly vents his frustration more than he would if he was a computer rather than a human. So I am also making a public call on Richard and his side of the debate to clearly commit to what they would regard as an acceptable threshold to concede that Joy has proven his results, with some suggestions about exactly what they need to publicly commit to, in advance of Joy taking up my recommendations here.
As to Joy, I would suggest that in the main paper, Joy derive the end result E(a,b) = – a dot b without ever relying on taking the limits s_1 -> a and s_2 -> b. Perhaps this really is a “nifty” shortcut. But if it is, PROVE THAT IT IS. How do I suggest Joy organize this proof? If Joy can indeed derive E(a,b) = – a dot b on a local realistic basis without anything that smacks of setting a=b and then arguing he can generalize back to ab because of rotational symmetry, then Joy will have overcome Richard’s central objection that all Joy has derived is E(a,b) = – a dot b = -1 for the special case of the two unit vectors representing detectors pointing in the same direction with a=b thus a dot b = 1 in violation of the EPR premises. Then, I suggest that Joy take the s_1 -> a and s_2 -> b limits line of development, put it into an appendix, and start the appendix with something along the lines of: “Having shown how E(a,b) = – a dot b can be derived for all vectors ab on a local realistic foundation, in this appendix we now show a nifty shortcut for obtaining this same result, making use of rotational symmetry arguments based on the S^3 of the FRW metric.” Then show the nifty shortcut proof. But don’t just MAKE the rotational symmetry argument to get to the general case. PROVE E(a,b) = – a dot b in the general case alone, and then use that proof to prove the limits-plus-symmetry argument that you are making in the present draft of the paper as a nifty, condensed viewpoint which flows from that.
As to Richard et al., my clear reading of everything you have written is that all roads lead back to your believing that Joy’s results only hold if a=b, in which case E(a,b) = – a dot b = -1 when a and b are normalized to unit vectors. Conversely, I take this to mean that IF Joy was able to mathematically obtain the quantum correlation E(a,b) = – a dot b in the general case where a and b are clearly not constrained to one another, you would agree that he has proven that it is possible to obtain the non-linear quantum mechanical correlation using a local realistic theory with hidden variables, and will have disproved Bell’s contention that it is impossible to do so. So: am I correct that your contention that Joy is constraining a=b – and various other objections which flow from this – are in fact the ONLY objections you have? If the answer is yes, then everything is clear, and Joy has his work clearly defined. If the answer is no, then please advise what other objections you also have, which other objections do not trace directly or indirectly back to the a=b objection as their source, but are independent of the a=b objection.
Thanks,
Jay
There are a couple of places where “ab” appears in the above. This is a typo, and this should read “ab,” that is, “a not equal to b.”
OK, the RW intake program does not accept a between the a and the b. So “ab” should read “a not equal b.”
And for some reason, the less than and greater than signs or not coming in either. So, ab in my original post means “a no equal to b.”
The left and right arrow head symbols are html code so you can’t use them on a web page for math.
Jay, that has already been done.
http://challengingbell.blogspot.com/2015/05/further-numerical-validation-of-joy.html
There are no limits on the A and B functions in the computer code that validates Joy’s model.
The burden of proof is on Gill. He claims that I assume a = b. He has not proved this. I will wait until he produces a proof of his claim. If he cannot produce a proof, then his claim is empty. If he produces a proof, then I will either refute it, or consider revising my paper.
Proof of a = b.
From (54) and (55), A(a, lambda) = -B(b, lambda) = +/-1
Substituting in (67), we find E(a, b) = -1
But according to (75), E(a, b) = – a dot b
a and b are unit vectors. So a dot b = 1 implies that a = b
Disproof of a = b:
(54) and (55) define the analytical functions A(a, lambda) and B(b, lambda) explicitly, each with random values +1 of -1, with 50/50 chance.
Substitutions of (54) and (55) into (67), together with s_1 = s_2, immediately gives (72).
(72) then gives E(a, b) = -a dot b, as in (75), once the hidden variable lambda is summed over for large n. QED.
Side note: As already proved above, setting E(a, b) = -1 for all a and b violates the conservation of spin-0 angular momentum, which implies and holds if and only if s_1 = s_2.
In addition to wrong physics, note also the logical error in Gill “Proof of a = b.” He first derives E(a, b) = -1 (incorrectly, as I showed). But then jumps to (75) without any attempt of deriving it, to conclude that E(a, b) = -a dot b.
Where did (75) come from in his derivation of his “Proof of a = b” ?
Jay, can you see the logical gap in Gill’s “proof” ?
For the purposes of my “proof”, I assume that Christian’s paper is completely correct. So I am free to use (75).
Christian also provides us with A(a, lambda) = -B(b, lambda) = lambda = +/-1 in (54) and (55), from which it follows that E(a, b) = -1.
If there is any problem with my conclusions, then Christian’s assumptions and/or derivations must be to blame.
My paper is indeed completely correct. But no one is allowed to use two completely different assumptions in the same proof. One cannot have s_1 = s_2 and not have s_1 = s_2 in the same proof. That is what Gill has done in his proof, but he does not seem to recognise that.
E(a, b) = -1 follows if and only if s_1 is not equal to s_2, thus violating the conservation of spin angular momentum.
E(a, b) = -a dot b follows in general if s_1 = s_2 is assumed, as we must do to conserve the spin angular momentum.
Gill mixes up two different assumptions to derive two different correlations, E(a, b) = -1 and E(a, b) = -a dot b, and then compares them to “prove” a = b.
Let me spell this out so that everyone can see the elementary logical flaw in Gill’s “proof.” Let me produce two separate arguments in the manner of Gill to make this clear.
(1) First, let s_1 = s_2. Then my definitions (54) and (55) for A and B, once substituted into (67), immediately gives (72). From (72) then (75) follows giving E(a, b) = -a dot b. This is what is derived in my paper. The key in my derivation is this: Given my definitions and assumptions, (67) immediately and directly gives (72). This step is not understood by Gill. Given my definitions and assumptions, E(a, b) = -1 can only hold in a very special case in which Alice and Bob specifically set a = b. Otherwise E(a, b) = -a dot b necessarily follows.
(2) Now let us drop the assumption of s_1 = s_2, thus violating the conservation of spin angular momentum. In that case my definitions (54) and (55) for A and B, once substituted into (67), do indeed give E(a, b) = -1. The rest of the equations in that case do not add anything, and we end up with a new equation, (75-b), which also gives E(a, b) = -1.
Gill has badly mixed up the cases (1) and (2) above to “prove” a = b. He is not free to compare apples with oranges in this manner to “prove” anything, let alone a = b.
I expect that Richard will reply in some way to the above and I do not want at this moment to intercede in his and Joy’s discussions as principal actors. But from reading what Joy wrote at October 27, 2016 at 7:40 am and at October 26, 2016 at 11:20 pm, I am wondering if what Joy is saying here can be rigorously recast as a “proof by contradiction.” Specifically, such proofs take the logical form of “assume x=true. Do some direct calculation based on that assumption. Obtain the result x=false.” There are several points where we might start this logic chain: a) assume E(a,b)=1 always is true, then show it is false. b) assume a=b always is true, then show it is false. c) assume s_1=s_2 always is true, then show it is false.
If one or more such contradictions serve to disprove Richard’s logic, and are not simply showing an internal contradiction in Joy’s logic, then perhaps that is a way to go. Though I have a feeling that this might also shift the discussion into one of whose logic was disproved, Richard’s or Joy’s.
It does amaze me, however, how something so seemingly-simple, is so darned opaque. Which is why I will continue to recommend to Joy that he utilize in this paper, a proof of non-linear quantum correlations that do not in any way rely upon the “limits plus symmetry” argument, but rather, which PROVE that argument. Joy advises that https://arxiv.org/abs/1211.0784 which is effectively the preprint of his paper http://link.springer.com/article/10.1007%2Fs10773-014-2412-2, which he says contains such a proof without relying on these limits that have led to such opacity in the a=b discussion here. Perhaps if Joy is still reluctant to amend his present paper in this way, we can at least discuss that alternative “no-limit” approach here at RW?
Anyway, just some suggestions…
Jay
If Christian can fix the problems with his paper he’ll revolutionise physics.
In fact the main obstacle to rescuing his work is Bell’s theorem, which says that what he is trying to do is impossible. (Bell’s theorem has stood for fifty years. A proof of the theorem adequate to cover Christian’s framework is very easy.)
To answer your question to me, Jay, unfortunately the a = b question is not the only problem with the present version of his theory. I only emphasised it because the part of the paper where it arises is quite self-contained and contains extremely obvious self-contradictions and errors, involving only elementary algebra and calculus. So any mathematically inclined reader can decide for themselves.
Please note that Gill has not provided a proof for his claim that I assume a = b as I asked him to do after Jay’s suggestion. Instead, Gill now makes several unproven and unprovable statements. He says
“If Christian can fix the problems with his paper he’ll revolutionise physics.”
There are no problems with my paper to be fixed. There is not even a comma misplaced in my paper. As I have already noted above, all the issues raised by Gill and others over the past nine years have been comprehensively addressed and disposed of by me and others. I have provided detailed mathematical refutations of each of the claims made by Gill against my model. Each time he has shifted the goalpost and raised the bar, I have been able to still make the goal and hop over the bar.
Gill now claims that “in fact the main obstacle to rescuing his work is Bell’s theorem, which says that what he is trying to do is impossible. (Bell’s theorem has stood for fifty years. A proof of the theorem adequate to cover Christian’s framework is very easy.)”
In fact, Bell’s theorem remains unproven. What is more, I have comprehensively refuted the unproven claims by Bell independently of my local model presented in the current paper. You can find one of my refutations of Bell’s theorem in the Appendix D of this paper:
https://arxiv.org/abs/1501.03393
This refutation of Bell’s theorem has nothing to do with my physical local model of the EPR-Bohm correlations being discussed here. In addition to the above refutation, some of the participants on this debate are well aware of my open challenge to Bell’s theorem, posted at Fred’s forum, which remains uncontested even today. Thus Bell’s theorem is both refuted as well as remains unproven, and my challenge to Bell’s theorem remains uncontested. The bottom line is that Bell’s so-called theorem is based on some naïve physical assumptions, and thus it is not applicable in the real physical world, as all the experiments done to date in this context have repeatedly demonstrated.
Gill further claims that “…unfortunately the a = b question is not the only problem with the present version of his theory. I only emphasised it because the part of the paper where it arises is quite self-contained and contains extremely obvious self-contradictions and errors, involving only elementary algebra and calculus. So any mathematically inclined reader can decide for themselves.”
This statement is quite objectionable given the fact that all of the claims against my model made by Gill and others have been systematically and comprehensively refuted by me and others. There are no contradictions or errors in my paper, period. It is also interesting to note that if there were “extremely obvious self-contradictions and errors, involving only elementary algebra and calculus”, then the distinguished editors and referees of Annals of Physics were unable to spot a single one of them during their seven-months long rigorous peer review of my paper. In fact Gill’s claims above are simply false, as anyone proficient in Clifford algebra and general relativity can readily see. It is also important to note that Gill is not a physicist. Nor has he ever published a peer-reviewed paper in Clifford algebra or general relativity on which my paper is based. I therefore urge the readers to take his opinions about my paper with a pinch of slat.
Does this dispute on RW make really sense? Joy Christian feels confident that he is right, whereas others feel confident that he is wrong. As Joy Christian’s papers are freely available on arXiv.org, everyone who is interested in quantum physics can thus make up his or her own mind. Time will finally tell whether we can hold fast to the concept of ‘realism’ or not.
It is pretty easy to show that Bell was wrong even without Joy’s model. However, the question that Joy’s model raises more importantly is; should space be modelled with 3-sphere topology or is it flat R^3? Recent macroscopic experiments are leaning towards 3-sphere topology. Joy has proposed a mechanical macroscopic experiment if successful would settle the question forever. If not successful, it rules out 3-sphere topology macroscopically but not microscopically. It should be done.
If it is indeed pretty easy to show that Bell was wrong, you only need to have a little patience. In this case, it’s merely a matter of time when the „disproof of Bell’s theorem“ will find its way into future textbooks on quantum physics.
” If not successful, it rules out 3-sphere topology macroscopically but not microscopically.”
Fred, I am glad you raised this, because I am myself not really clear what you really have in mind here when you say “microscopically.” We are all presumably familiar with the macroscopic FRW curved 3-space that could apply to the whole universe. But please explain what you and Joy have in mind , when you refer to a “microscopic” three-sphere. And presumably, huge numbers of such three spheres in any given region of spacetime. How are we to think about this in terms of molecules or atoms or or leptons or protons or neutrons or photons? Or, if more pertinent, in terms of Alice and Bob. Are these variants of wormholes or Planck-scale black holes or Hawking radiation or some such thing? The more concrete and visual, the better. Give us a picture of the spacetime geometry.
If this is not clear to me, then there is a fair chance it is also not clear to some other readers. And because Joy appears to hang much of his hat on the rotational symmetry of the three-sphere from one spacetime event to next including events with spacelike separations from one another, perhaps this is part of what is blocking an understanding between him and his theory’s critics.
Thanks, Jay
Hi Jay,
It just means that 3-sphere topology could still be relevant for quantum particles but not macroscopic large objects if the mechanical macroscopic experiment is not successful. But since some recent macroscopic experiments using EM waves beat Bell, Joy may be very right about it.
Hi Jay, By 3-sphere we are not referring to little marbles at every point of space or spacetime. We are referring to the physical geometry and topology of the 3D space itself. A 3-sphere is a spatial part of one of the spacetime solutions of Einstein’s field equations of general relativity. What it means in practical terms for our problem is that the physical space is not modelled for us as the 3D vector space R^3, but as a 3D quaternionic sphere S^3. From this perspective it is misguided to be hung up on the 3D vectors like a, b, and s that Gill is hung up on. They are not the essential parts of the 3-sphere at all, but are dual to the bivectors L and D that constitute the 3-sphere. Of course locally experimenters like Alice and Bob would still experience the same old R^3 made up of ordinary vectors, but the global properties of S^3 are dramatically different from those of R^3. Unlike R^3, which is flat, S^3 is curved, with constant spatial curvature. Intuitively this is a bit like how globally the Earth is round, but locally it is nearly flat. Of course I am not talking about the external curvature in the case of S^3. I am talking about its intrinsic curvature, quantified by the Riemann curvature tensor. Therefore the distinction between micro and macro is actually non-existent for us, unless of course we are concerned with Planck scale, which we are not. We are only concerned about what Alice and Bob can do in their labs on Earth, not what they can do at the centre of a black hole. I think what Fred has in mind is the usual classical versus quantum distinction.
“I am talking about its intrinsic curvature, quantified by the Riemann curvature tensor.”
OK, Joy, let’s go from there. A Riemann curvature is locally measurable as a tidal force via the geodesic deviation, and is real and cannot be transformed away by any means. If there is a curvature then there is a gravitational field and vectors (which I know you say we should not be hung up on) will parallel transport with path-dependency. So, tell me / us about gravitation and / or the measurable tidal force curvature and / or parallel transport and how one ought to think about those in the context of your theory. Jay
Ah… I am glad you asked. This is discussed in great detail in my paper you linked earlier that has been published in the International Journal for Theoretical Physics. See, for example, the Figure 4 in the IJTP paper you have linked: https://arxiv.org/abs/1211.0784 . The difference between the linear versus the cosine curve you see has to do with the geodesic deviation.
As I follow the comments on this post, I cannot help of two things:
First, while working on one of my first publications, one of the reviewers kept making changes in what I was doing. In the end, this reviewer said that paper should be rejected because it was a poorly written first draft. All I could think was that it was a poorly first draft that he, the reviewer, had written.
Second, from the gist of the comments, it would seem that a new manuscript is being created through the process of open review. I look forward to the final product.
🙂
Hear hear! That would be a wonderful validation of the mission of Retraction Watch, and it would hopefully conclude a very divisive scientific debate one way or the other. Jay
Thanks Richard, I’d like to explore this a bit further:
I am new to Bell, which maybe puts me at a disadvantage relative to you and Joy and some other folks here, but also gives me fresh eyes and perhaps a naiveté that can be helpful. My first inclination was to say “tell me what else is wrong that is not tied at all to the a=b disagreement, and be very specific as to where these other problems lurk.” That is only fair to Joy, because I have asked him to consider a major revision that does not use the “limits plus symmetry” argument and he should know what additional hurdles he will face after he does that. So I’d like to see if we can agree on the “goalposts” in advance of any major revisions that Joy might undertake if he is persuaded to do so, and not then “move” them. And if you can reply in the specific terms I was inclined to ask about above, certainly, please do so.
But there may be a more global way to approach to this, so let me try that too.
Bell seems to have a number of “theorems.” So for clarity, if my understanding is correct, then the Bell theorem that is most important in this discussion is the one which says that it is impossible to derive the nonlinear quantum correlation
E(a,b) = – a dot b = -||a|| ||b|| cos (theta) = – cos (theta) (quantum)
for unit vectors ||a|| = ||b|| = 1, for any and all a and b orientations, except by a nonlocal theory. Rather, according to Bell, any local realistic theory can do no better than to obtain the linear classical correlation
E(a,b) = -1 + 2 theta / pi (classical)
These two correlations are respectively shown by the blue and red curves at the link https://en.wikipedia.org/wiki/Bell%27s_theorem#/media/File:Bell.svg.
So it seems to me that for Joy (or anybody else) to disprove this particular Bell theorem, three things are needed:
1) a demonstration that their theory leads to E(a,b) = – a dot b.
2) a demonstration that it does so for any and all orientations of a and b, and not just for the special case of a=b where the two detectors are aligned in which case E(a,b) = -1 based on either the quantum or the classical correlation.
3) a demonstration that the theory which leads to 1 and 2 is local realistic.
Now the questions, which I will label for easy replies:
A) Do Richard and Joy both agree with the above statements? And if I have stated this wrongly in any significant way, please advise.
If we are OK so far, then let’s proceed. Richard, I believe you have agreed that Joy has derived E(a,b) = – a dot b, but you believe that in the process he restrains Alice and Bob to align their detectors in the same direction in violation of the EPR premises, whereby a=b and so the quantum correlation becomes E(a,b) = -1. B) Is this correct?
C) If so, then IF (and I have said “if”) Joy were to provide a convincing proof that the quantum correlation could be derived from his theory for any and all a and b orientations, then in this hypothetical case, would he have not already satisfied 1 and 2 above? D) And it were the case that Joy could satisfy 1 and 2 above, would not the one remaining mandate be #3 for Joy to also show that his theory uses local realism all along the way? E) And finally, is it your contention that Joy’s theory is local realistic but that he does not derive E(a,b) = – a dot b other than for the special case where a=b? F) Or is it your contention that his theory also has problems with its locality?
Jay
I am sorry, Jay. But before we can consider anything else Gill must produce a proof of at least one of his claims without using a proposition and its negation in the same proof. So far he has attempted to prove only one of his claims, the a = b claim, but has ended up using a proposition and its negation in the same proof, as I pointed out above. Where is an irrefutable proof of a = b?
I received a private email from Richard, who is trying not to post any more on RW, and does not want to respond to my latest questions A, B, C, D, E in public. But he did provide me with the following private reply which I am forwarding along to inform this discussion.
Jay
From Richard Gill via email:
“There is not a short answer since there are two quite distinct models in the paper we are discussing. They seem to be hardly connected to one another at all. The first model reproduces the quantum correlations, but is non-local – it is Pearle’s detection loophole model. Christian learnt it from me. See https://arxiv.org/abs/1505.04431
The second is the crazy model A(a, lambda) = -B(b, lambda) = lambda = +/-1 which is clearly local but which clearly does not reproduce the quantum correlations.
…
I suggest study of one of the simpler proofs of Bell’s theorem, if you want to understand what Christian is up against. His most recent analysis of Bell’s theorem is that it is illegal (a) because it uses probability theory and (b) because it uses unphysical manipulations, such as writing the difference of two integrals as the integral of the difference of the integrands. You can’t even define what is a local hidden variables model without using the language of probability theory. And since when are mathematicians forbidden from mathematically exploring a given mathematical model? Wherever it comes from?”
Are you posting RG’s private emails with permission from him?
Yes. I think Richard would confirm that he and I have an understanding that we have each respected. As I think would Joy. And I have not posted anything of a private or personal nature from him or Joy, only those materials which are of general objective scientific or mathematical interest.
Let me respond to each of the points made by Gill:
Gill: “…there are two quite distinct models in the paper we are discussing.”
This is incorrect. There is only one model — the 3-sphere model: https://arxiv.org/abs/1405.2355 . To be sure, in my paper there are two different representations of the 3-sphere considered, each with some advantages and some disadvantages. Both representations are valuable, and they complement each other quite nicely.
Gill: “They seem to be hardly connected to one another at all.”
This is incorrect. How can there be no connection between two representations of one and the same manifold, namely a 3-sphere? This is why I am sometimes forced to stress that Gill has never published a single peer-reviewed paper in Clifford algebra (which, by the way, he learned from me) or general relativity to understand the physical model presented in my paper.
Gill: “The first model reproduces the quantum correlations, but is non-local – it is Pearle’s detection loophole model.”
This is incorrect, on both counts. Both representations of the 3-sphere model are manifestly local. Gill here makes an empty statement without a proof. And contrary to his claim, the first representation is by no means Pearle’s detection loophole model. His claim makes me suspect whether he has actually read my paper at all. To be sure, I use some of the formal mathematical constructs used by Pearl, but there is no exploitation of a “detection loophole”, or any other “loophole” for that matter, in my local model. There is a one-to-one accounting in the model between the initial (or complete) state (e, s) and the measurement outcomes A and B. So Gill once again makes a false claim without providing a proof for his claim.
Gill: “Christian learnt [Pearle’s model] from me.”
This is incorrect. I did not learn Pearle’s model from Gill. I learnt it from my former Ph.D. adviser, Prof. Abner Shimony, in the mid 1980’s, and few years later from Philip Pearl himself (whom I know personally). And while I am at it, let me also mention that I learnt about Bell’s theorem also form Abner Shimony (the “S” in the Bell-CHSH inequality) while I was his student in the mid 1980’s, and a few years later from John. S. Bell himself (with whom I was also well acquainted, thanks to my mentor Abner Shimony).
Gill: “The second is the crazy model A(a, lambda) = -B(b, lambda) = lambda = +/-1 which is clearly local but which clearly does not reproduce the quantum correlations.”
This is incorrect, on several counts. As anyone can see Gill is confusing the measurement outcomes A and B with the hidden variable lambda when he writes A = +lambda and B = -lambda. Look carefully. That is what he has written, just as he has done many times in the past. This shows that he has not really understood my model. And it is quite extraordinary that he confuses the measurement outcomes A and B with the hidden variable lambda. Also, contrary to his claim, my model evidently reproduces the quantum correlations, as anyone can see by simply studying my paper carefully and investigating the details of the analytical as well as numerical evidence presented therein. But Gill has got one thing right –- my model is indeed manifestly local, as anyone knowledgeable in the subject can readily see.
Gill: “I suggest study of one of the simpler proofs of Bell’s theorem, if you want to understand what Christian is up against.”
I recommend that too. And then, if you think you have understood Bell’s argument, I suggest that you take up my challenge to Bell’s theorem and prove me wrong. Here is my challenge, which is open to anyone, and can be taken anywhere on the Internet, not only at Fred’s forum: http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=275#p6681
Gill: “You can’t even define what is a local hidden variables model without using the language of probability theory.”
This is incorrect. John S. Bell, in his famous paper of 1964, used only expectation values, not probabilities, to define a local hidden variables framework, and then produced an explicit analytical local model of his own, without using any unnecessary notion from the probability theory.
Finally, I will not quit participating at RW. I will be here to answer any questions, or respond to any reasonable criticism that I have not addressed already over the past nine years.
I am amazed this is still being discussed at length when the foundation of Christian’s work is based on a simple math error he to this day refuses to admit despite having it pointed out in excruciating detail by many people. Here is the deal that ANYONE understanding what “an algebra” is can process and understand: you can’t directly add algebraic element coefficients from one algebra to algebraic element coefficients of another algebra with a different basis. He uses two orientations for the bivector basis, each have distinct basis elements so coefficients attached to one can’t be directly added to the other until one basis is first mapped to the other. For the geometric product of two bivectors A and B, the coefficients on the bivector part of the result do indeed change signs when the orientation of the algebra is changed. His problem appears when he fails to recognize the map between the bases is another negation, one he does not account for in his work, allowing him to incorrectly dispatch the bivector partial results and leave the desired averaged -a.b partial results from the geometric products summed over many runs of “fair coin” orientation choices each run. This is one of the elementary algebra errors Richard Gill is talking about.
As for this paper specifically it has been pointed out to Christian that for A and B members of S^3, A+B is assured not to be a member of S^3, so the triangle inequality used in his derivation is non-sequitur. Still present. I could go on but what is the point?
It was found by me that Lockyer is double mapping the multiplication table.
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=226#p5859
I am surprised to this day that he still hasn’t figured that out. Joy’s GA math is 100 percent correct. The computer program GAViewer confirms it.
http://challengingbell.blogspot.com/2015/05/further-numerical-validation-of-joy.html
Indeed there is no point. For none of what Rick Lockyer has written has anything to do with my paper, as has been pointed out to him by me and others (especially by Fred Diether, in great mathematical detail) on many occasion. We are discussing here my paper, not Rick Lockyer’s or anyone else’s misrepresentation of it. I too am amazed that even after it has been pointed out to him so many times he continues to misrepresent my paper in this manner, at every opportunity. It is also important to note that, like Richard Gill, Lockyer is not a physicist and has never published a single peer reviewed paper in either physics or math journal.
The author has had several papers on the same theory available online on arXiv for eight years now. I can’t see they have been cited in any other credible publications (exept for some fringe papers where they are only mentioned in passing).
To me that speaks volumes about the sociology and politics within the physics community than about my local-realistic model for the EPR-Bohm correlations (I don’t have a “theory”).
This touches on what seems to me a critical issue, and I agree that a clarification would be helpful. The three-sphere S^3 is locally “the same as” ordinary three-space R^3 that we see around us. (Technically, the two are locally diffeomorphic.)
I assume that what Fred terms “microscopically” is what mathematicians call “locally”. But there is a big difference in intuitive meaning. “Locally” does not carry the implication of “very small” as does “microscopically”.
Drop down one dimension to get an intuitive picture. The two dimensional sphere S^2 can be obtained by adding a “point at infinity” to the Euclidean plane R^2. (If you have studied complex analysis, this is the usual “Riemann sphere” construction.) Thus R^2 can be considered as sitting inside S^2, obtained from S^2 by removing a single point. In the same way, R^3 sits inside S^3, and can be obtained from S^3 by removing a single point.
Any analysis that applies to R^3 will also apply in S^3 if we ignore the one point which was removed to obtain R^3.
Bell’s theorem is proved using analysis in R^3. The exact same analysis works in S^3. Although S^3 is not topologically the same as R^3, nothing in the usual proofs of Bell’s theorem uses properties of R^3 that are not shared by S^3.
I think it is fair to say that Christian’s model is sufficiently subtle and complicated that it requires careful study. But Bell’s theorem has a very simple proof which surely has been checked by thousands of physicists and mathematicians. If the usual proofs of Bell’s theorem are correct, then Christian’s claims must be faulty. Before investing a lot of time in the subtleties of Christian’s model, it seems sensible to first find the error in the simple Bell theorem proofs, if there is an error.
Christian and his followers claim that there *is* an error. I disagree. I think that they have misunderstood the hypotheses of Bell’s theorem.
Whatever the case, I recommend that anyone interested in Christian’s claims first examine the usual proofs of Bell’s theorems and Christian’s objection to them. That has the potential to save a lot of time.
Bell’s theorem has been considered for over half a century as expressing a profound contradiction between the predictions of quantum mechanics and what most people believe about “reality”. It has been mentioned as deserving of a Nobel prize (as I imagine that everyone would agree were its proof not so simple). If there is something wrong with its usual proofs, that would be headline news, independently of whether someone has actually found a counterexample (as Christian’s model would be if correct).
Doesn’t it make sense to first gather the low-hanging fruit (that the usual proofs of Bell’s theorem are incorrect, assuming they are as Christian and his followers claim) before investing weeks of hard work learning Clifford algebra, etc., to determine the validity of Christian’s model?
Good points, especially the last one. It is also good to know that at least some participants in this discussion have the background to understand the topological issues that are important in my model (such as the one-point compactification to get S^3 from R^3 by adding a single point at infinity). However, that alone is not enough to understand the strong correlations. What is needed in addition is a parallelization of the 3-sphere by quaternions. All of these issues are discussed in my relevant papers and book.
Despite what may appear at first sight, I have no interest in Bell’s theorem per se. My interest mainly is in understanding the physics and mathematics underlying the strong correlations we observe in Nature. But sociologically Bell’s theorem is obviously quite important, because it is believed-in by most physicists. But it is also true that it has never been universally accepted. In my opinion both Einstein and Pauli would have found it laughable. In my opinion it is fundamentally misguided. I have tried to explain in very simple terms what I consider to be wrong with it; for example, in the Appendix D of this paper: https://arxiv.org/abs/1501.03393 . I also have several other arguments against Bell’s theorem, but this one is perhaps easiest to understand because it is completely independent of my model. And of course there is also my open challenge, which may be instructive.
It seem to me, Joy, that the most important statement in your appendix D is the following: “But this rule of thumb is not valid in the above case, because (a,b), (a,b′), (a′,b), and (a′,b′) are mutually exclusive pairs of measurement directions, corresponding to four incompatible experiments.”
Is this not a variant of saying that dependent probabilities are not the same as independent probabilities? And if so, and if that is really the problem here, given as Dr. Parrott points out that “Bell’s theorem has been considered for over half a century” by multitudes of physicists, how could such an elementary error have been missed by everyone? Jay
You are quite right. My sentence you have quoted is indeed the key problem. One can frame it in terms of dependent versus independent probabilities, but “dependent probabilities are not the same as independent probabilities” is not what anyone has missed in my view. The real problem with Bell’s argument has to do with how that rule of thumb, namely how replacing a sum of four averages with an average of four products — however innocent that may seem mathematically, leads to nothing less than absurdity as far as the physical experiments in question are concerned. While the sum of four averages is an innocent sum of four physically meaningful experiments, its replacement — the average of four products involving four mutually exclusive pairs of directions (a,b), (a,b′ ), (a′,b), and (a′,b′ ) — does not correspond to any physically meaningful experiment at all, let alone to an EPR-Bohm type experiment involving two simultaneous spin measurements.
Let me explain this in terms of realism. The key here is this question: What if we had measured the spin of a particle about the direction b′ instead of the direction b? This is like asking: What if I visit Miami instead of New York next Monday? It sounds like a perfectly innocent possibility, unless I ask: What if I visit Miami AND New York next Monday at exactly 1:00 PM local times? Then you would surely say that I have lost my mind. But that is exactly what we are taking as a Real possibility when we consider the average of four products involving four mutually exclusive pairs of directions (a,b), (a,b′ ), (a′,b), and (a′,b′ ). It involves physical measurements of the two spins about the directions b and b′ (as well as about a and a′ ) at the same time! However innocent that may seem mathematically, it is an utter absurdity physically. We simply can’t be in Miami and New York exactly at the same time! Alice and Bob simply do not have the ability to measure spins of their respective particles about the directions a and a′ and b and b′ at exactly the same time.
I have exchanged a few more email with Richard Gill and he has confirmed that we are cool with my communicating his scientific and mathematical responses here at RW.
In reply to Richard saying “I suggest study of one of the simpler proofs of Bell’s theorem, if you want to understand what Christian is up against,” I asked him to send me such a proof. His reply with a “simpler proof” is below, for all to consider.
From Richard Gill:
“Given functions A(a, lambda) and B(b, lambda) taking values +/-1 and a probability distribution over lambda, rho(lambda) d lambda
Consider two values for a and two for b: a1 and a2, b1 and b2.
Define Ai = A(ai, lambda), Bj = B(bj, lambda)
These four are now four random variables taking the values +/- 1.
So there are just 16 possible outcomes for the quadruple (A1, A2, B1, B2)
Important observation: A1B1 – A1B2 – A2B1 – A2B2 takes only the values +/- 2
(please check this yourself! either run through all 16 possibilities by hand, or come up with a smart trick to prove it more easily)
Therefore E(A1B1 – A1B2 – A2B1 – A2B2) lies between -2 and +2
Thus E(A1B1) – E(A1B2) – E(A2B1) – E(A2B2) lies between -2 and +2
This is called the CHSH inequality.
Application to EPR-B – find some measurement settings such that the correlation E(a, b) = – a dot b violates this inequality.
Christian’s objection to this proof is that the quantity “A1B1 – A1B2 – A2B1 – A2B2” is physically meaningless
My response to this: so what?
We may not be able to perform any experiment and observe it directly, but according to our mathematical assumptions it does mathematically exist, even if we measure nothing at all. It’s just some function of lambda and lambda is drawn at random according to some probability distribution.”
Looks like you or Gill have some signs wrong there. The standard CHSH inequality has plus signs for 3 of the term and a minus sign for one of them:
https://en.wikipedia.org/wiki/CHSH_inequality
Multiply by -1 to get “the standard” CHSH inequality: three plus signs and one minus.
Fair enough to multiply by -1 since the upper and lower limits are -2 and +2. It’s worth clarifying that for people starting out with Bell-like inequalities.
At the risk of a brief digression to illustrate a point, let me tell a true story:
When my daughter Paula was in middle school, I attended a “meet the teacher” day along with many other parents. One of the teachers we met was her math teacher. This teacher told us that her current set of lessons were about taking averages. And then she wrote an example on the blackboard for the parents. Because the “greater than” and “less than” signs do not get through on RW, I will use square brackets to denote expected value, so that [x] here will mean “the expected value of x.”
The teacher wrote (I am picking the numbers out of a hat) that:
[5+3] = 8/2 = 4
She then wrote that:
[8 + 10 + 12 +14] = 44/4 = 11
She then announced to us that she could write the average of all these numbers as:
[(5+3) + (8 + 10 + 12 +14)] = [5+3] + [8 + 10 + 12 +14] = 15,
or really, because she was adding two expectations, as 7.5 = 15/2.
But of course that is wrong, and the correct answer for the average is:
[(5+3) + (8 + 10 + 12 +14)] = 52/5 = 10.4
The teacher’s fallacy, was in equating: (“ne” means “not equal to”):
15 = [5+3] + [8 + 10 + 12 +14] ne [(5+3) + (8 + 10 + 12 +14)] = 10.4
which is to say, you cannot just take a sum of expected values and turn it into the expected value of a sum. The parents all looked at each other with rolled eyes, and when I told my daughter about this, she said, “yeah, she doesn’t know what she is doing, and the students are having to always correct her.”
Of course, in this story, one will easily see that the problem is that the 5+3 contains two numbers, and the 8 + 10 + 12 +14 has four numbers, and yet the teacher was erroneously giving equal weight to a two number set and a four number set. But the moral of the story is that the rule of thumb that “the sum of averages is equal to the average of sums” is not universal, and that one must look closely and carefully at the very specific problem at hand before reflexively but wrongly applying this rule.
So back to Bell etc. It is good that Dr. Parrott has steered us away from Joy’s model for a little while, and to “the low-hanging fruit (that the usual proofs of Bell’s theorem are incorrect, assuming they are as Christian and his followers claim).”
Joy in his October 28, 2016 at 8:33 pm post points right to his (D3) to (D6) and the related discussion on less than half a page of https://arxiv.org/abs/1501.03393 which is easily digestible material, as to why Bell made an error and why the range of the recited binary expectation values runs from -4 to +4 and not from -2 to +2. Richard in what I relayed on October 29, 2016 at 11:43 am, makes an argument on the very same point that this range runs from -2 to +2. So now, with apples being compared to apples, have put a fine point on the root of the Christian versus Gill disagreement over Bell, and that of their respective minions, without having to look at Joy’s model at all right now.
With the cautionary tale from my daughter’s middle school about looking closely at the expected values which are being summed by Bell and whether the expectation value of the sum is or is not equal to the sum of the expectation value in this specific situation, the core of the present dispute over Bell, independent of Joy’s model, is clearly and succinctly drawn.
I suggest that the participants in this discussion try to thrash out this single question one way or another before doing anything more. Did Bell combine these expectations properly in the given situation, or did he not? If Bell did in fact make a faux pas by incorrectly converting over a sum of expectation values to an expectation value of sums, then as Dr. Parrot says, “that would be headline news, independently of whether someone has actually found a counterexample (as Christian’s model would be if correct).”
To all: stay focused, do not wander off on tangents, and let’s just answer this one simple question. Page 8 of https://arxiv.org/abs/1501.03393: Is Joy right or wrong? And in either case, why?
Jay
Joy is 100 percent correct. Besides what Joy points out, it is easy to see that quantum mechanics and the quantum experiments use the four separate averages where the bound is |4| from what Joy presents. Then claim they have “violated” Bell-CHSH’s bound of |2|. Case should be closed.
So, you think that the several thousands of physicists and mathematicians that have gone over Bell’s proof in the past 50 years all made the same trivial mistake as your daugther’s teacher?
Please keep in mind that I am about 2.5 weeks new to all of these Bell discussions, but let me take a stab at my own answer to this. I claim no authoritative mantle for this, I am just trying to think this through independently, for better or worse.
In Joy’s (D3) which absent objection from some partisan I will take to represent a valid result from Bell, we have A_k(a)B_k(b) showing up four times with the four combinations of ab, ab’, a’b and a’b’. Let’s for the moment study A_k(a), and let’s take the first experimental run for which k=1. Using the vector a to establish a “north pole” direction for Alice’s detector, if the spin arrow of the particle being detected points toward anywhere in the northern hemisphere, then Alice will detect A_1(a)=+1. If the spin arrow points toward anywhere in the southern hemisphere, then Alice will register A_1(a)=-1. A binary outcome. By experimental design.
Now let us suppose that on the second run Alice chooses a different direction a’ to establish her north pole. Then A_2(a’)=+1 means that on the second run she got a hit in the new north, and if A_2(a’)=-1 she got a hit in the new south. No problem so far.
But what Joy asks is if Alice does actually use a, not a’, for the k=1 first run and thus measures A_1(a)=+1, then what is the meaning of A_1(a’)? And by “meaning” I really do mean “meaning,” in the deepest sense of the word.
Certainly, Joy is correct that A_1(a’) is the detector orientation not chosen in that first run. Or metaphysically, it is the “road not taken.” So all we can ask about is whether, had Alice hypothetically chosen A_1(a’), this choice would have turned in a +1 or a -1 result? But there is no experiment that can ever be done to know that for sure, any more than human life is riddled with the often unanswerable heartbreaking mystery of “what if I had made a left not right before the accident?” Or, “what if JFK’s motorcade in 1963 had taken a different turn in Dallas?” Or “what if Versailles had not been put in place after the First World War.” Or the seeming miraculous outcome of having taken a good path when a bad path could have been taken. Like the time you or a loved one lived when it might have been otherwise, but for a stroke of luck or a great doctor. The point being that this question drives deeply into all of the quantum debates about causation, time, things that might have been, determinism, and so on. And this ought not be a surprise: we are talking here about quantum physics and acausal entanglement verses realism and locality. These have been vigorously debated not only in their own terms, but perhaps even more so because of the riddles about life’s deep mysteries with which they all appear to be connected. So it is asking for trouble for anybody to assert that questions about the meaning of A_1(a’) – the path not taken – have a clear, pat, no-brainer answer, if in fact A_1(a) was Alice’s decision on the first run.
So the question whether A_1(a’) “woulda, coulda, shoulda” been a +1 or a -1 is unknowable, because that is not what was detected. And this brings us straight into the ontological question whether after Alice chooses A_1(a), it become impossible to make any statement WHICH CAN BE EXPERIMENTALLY CONFIRMED, about what A_1(a’) would have been had that been chosen instead. My interpretation of Joy (and he can set me straight of I am misguided) is that he claims that this very problem is why Bell went wrong: because he was summing a “did happen” outcome with “might have happened but we can never know” outcome. I do not know if anyone can assert without equivocation that this question has to be answered one way or the other. But, we should look to the guidance we have from what physics already teaches.
Here, were I to offer my own tentative opinion, I would look to quantum theory, and would hoist it on its own petard. Quantum theorists have been telling us since Copenhagen that it is impossible to talk about where an electron was or what path it traveled prior to being detected: “The photon landed here, but which slit did it go through? Unknowable.” Feynman path integrals computed in space with double->triple->quadruple->infinite slits and one->two->3->infinity numbers of diffraction grates only consider probabilities for certain paths to have been followed, but make no claim to be able to know, even in principle, what path actually was taken. All we know is where the electron was detected. And once it is detected, this very same petard says that the wavefunction has been “collapsed” and by the very act of observing the particle we have interceded in the particle’s life and destroyed all of the other “might have beens” about the particle. Make no mistake, a fundamental contention of quantum theory is that you can only talk sensibly about what you can in principle measure.
So when Alice makes the choice on the first run to use a and eschews using a’, by a hoist on quantum theory’s own petard she has collapsed a wavefunction and made it impossible and unknowable to make any statements at all about what “might have been” had Alice elected a’ rather than a. Therefore, it seems that Joy does well to have in essence asked the question whether we can actually sum events which did occur and were observed, with events that did not occur and were never observed and will remain unknowable and according to quantum theory cannot be talked about with any definiteness, at least by any human being in this world.
This is a far cry from the mistaken way my daughter’s math teacher computed averages. (And to HR’s post that came in while I was writing this, really? Come on!) But the question Joy has raised here is what we are permitted to do in our mathematics, when we are considering the paths we did travel against the paths we never took. I can assert that if I had stayed at MIT and pursued a professorial track (which a mentor who died wanted me to do and which I probably would have done if he had lived, but also I would not have met my wife because I met her via his death), I would have joined the EE department and sooner or later moved over to the physics department and been working on physics problems from a different place with different information and different relationship and different life imperatives. But if you, dear reader, as a scientist properly-trained to be skeptical are saying to yourself “yeah, right Jay, dream on, but we’ll never know,” than you kind of have to say the same thing about whether A_1(a’) can properly be used in calculating an observable result such as the Alice and Bob correlations, when Alice decided on the first run to measure A_1(a), and not A_1(a’) which will never be known.
That may not dispose of the question whether Joy is right or wrong on page 8 of https://arxiv.org/abs/1501.03393. But I think it does frame some of the questions that we need to be asking about (what I think all would agree? was) Bell’s step from Joy’s (D3) to (D6).
Or at least, those are my two cents.
Jay
Christian thinks that it is not allowed to write
E(A1B1 – A1B2 – A2B1 – A2B2) = E(A1B1) – E(A1B2) – E(A2B1) – E(A2B2)
because (he says) the expectation value on the left refers to an experiment where experimenters simultaneously measure spins in directions a1 and a2 on one particle, and on b1 and b2 on the other; while the four expectation values on the right refer (he says) to four separate experiments in which one particle’s spin in measured in just one direction (four combinations). The first experiment is not meaningful and therefore (he says) the mathematical expression is meaningless.
I think that his interpretation is unwarranted. We have assumed local hidden variables. So we are given a mathematical model in which there are given two functions A and B, and a probability distribution rho. As a mathematician, I compute from these ingredients just anything I like. If I find anything useful, I will let my friend the physicist know what comes up. ie what are the necessary *mathematical* consequences of *his* mathematical assumptions.
For instance, we could imagine simulating the hidden variables model on a computer. Secretly I will let my computer compute A(a1, lambda) and A(a2, lambda) at the same time, even though it only outputs just one of them: the one which the experimenter has chosen, through the experimenters choice of the setting. The experimenter gets to see a perfect simulation of a real experiment. But in secret the computer has been “observing” also the outcomes of the measurements for the settings which the experimenter didn’t choose, each time.
The experimenter will be perfectly happy with my simulation of his “real” experiment. And at the same time, I will have actually computed (by simulation) E(A1B1 – A1B2 – A2B1 – A2B2), which of course will be equal to E(A1B1) – E(A1B2) – E(A2B1) – E(A2B2).
Hidden variables may be hidden to experimenters, but they are not hidden to God.
But why such a convoluted argument? Why not derive the bounds on E(A1B1) – E(A1B2) – E(A2B1) – E(A2B2) directly? If the bounds are true, then direct derivation should be possible.
I just want to second what Dr. Gill says above, and put it differently, hopefully a little more concisely though necessarily less completely.
A *hypothesis* of Bell’s theorem is that *it is possible in principle* to measure A1B1, A1B2, A2B1, A2B2 *simultaneously*. This is the “realistic” part of “local realistic”. It is *not* a hypothesis that *all* experimenters can do this. But maybe God can. Gill’s hypothetical computer can.
The *conclusion* of Bell’s theorem is that quantum correlations are impossible (assuming this hypothesis). Put contrapositively, if we believe quantum mechanics, the *hypothesis* is *false”.
The objection of Christian and his followers to the usual proofs of Bell’s theorem (like the one given by Gill) amounts to a misunderstanding of its hypotheses and conclusion. They seem to claim that because the above *hypothesis* to Bell’s theorem is probably physically false (as just about everyone agrees, including me), Bell’s theorem itself must be false. Whether a logical implication (like Bell’s theorem) is true or false has nothing to do with whether its hypothesis is true or false.
Even if everyone agrees with both Christian and me that the hypothesis of Bell’s theorem is false, that does not make Bell’s theorem itself false. Following is an example in a simpler setting:
THEOREM: In any algebra, if 1 = -1, then 1 x 1 = (-1) x (-1) , where “x” stands for multiplication.
I hope we can all agree that the theorem is true. However, its hypothesis (that 1 = -1)
is false if we are in a grade school universe in which all we know about are real numbers.
(There are algebraic systems like the integers mod 2 for which 1 = -1.) That the theorem’s hypothesis is false from the viewpoint of a sixth grader is irrelevant to the truth of the theorem.
Stephen Parrott wrote: “A *hypothesis* of Bell’s theorem is that *it is possible in principle* to measure A1B1, A1B2, A2B1, A2B2 *simultaneously*. This is the “realistic” part of “local realistic”. It is *not* a hypothesis that *all* experimenters can do this. But maybe God can. Gill’s hypothetical computer can.”
But this hypothesis has nothing to do with realism. It is anti-realistic, in the sense that it amounts to being able to be in New York and Miami at exactly the same time. I don’t know about “God”, but I am confident that Gill’s hypothetical computer cannot be in New York and Miami at exactly the same time. Let me explain why I call this hypothesis anti-realistic:
Let me first define some objects and possible events that I hope everyone will agree are manifestly real (i.e., they do not compromise Einstein’s Local Realism in anyway):
(1) New York City is a manifestly Real place.
(2) Miami is a manifestly Real place.
(3) You are a manifestly Real person.
(4) You can be in New York City on 4th of July 2017, at 1:00 PM. A manifestly possible, Real event.
(5) You can be in Miami on 4th of July 2017, at 1:00 PM. A manifestly possible, Real event.
(6) You can be in New York City AND in Miami on 4th of July 2017, at 1:00 PM. An impossibility, in any possible world. I don’t believe even “God” can make this possible.
But the last impossibility is precisely what is claimed by Bell and his followers to be possible when they consider the average of impossible events like:
E( a1, b1, a2, b2 ) = Average of [ A(a1) B(b1) + A(a1) B(b2) + A(a2) B(b1) – A(a2)B(b2) ].
These events simply cannot occur in ANY possible world. They are absurdities, like the item (6) above.
Consequently, anything derived from considering such absurdities, such as the upper bound of 2 on the Bell-CHSH-type inequality, is also an absurdity. It has nothing whatsoever to do with the notion of Realism, or locality, or causality, or anything in physics in general.
On the other hand, note that it is perfectly legitimate to make counterfactual statements like:
(7) You can be in New York City OR in Miami on 4th of July 2017, at 1:00 PM. A manifestly possible, Real event.
But if the Bell-followers replace AND of (6) with OR of (7) in this manner, then the upper bound on their Bell-CHSH inequality is 4, not 2. And the upper bound of 4 has never been “violated” in any experiment (indeed, nothing can violate it). So the hypothesis you mention has nothing to do with realism. It is simply an absurdity, in any possible physical world.
Those readers who are following this ongoing dispute on Bell’s inequality and who are interested in the extraordinary implications of Bell’s theorem should read N. D. Mermin’s “Bringing home the atomic world: Quantum mysteries for anybody” (Am. J. Phys, 1981). Maybe, they will then understand what this dispute is actually about and will be able to make their own conclusions.
The implications of quantum entanglement are really “shocking” when one tries to bring them in line with our classical concept of “realism”. As long as physicists hold on to think about quantum phenomena within the framework of – so to speak – classical ideas and notions, such disputes will never end. Be that as it may. Or one simply accepts the “weirdness” of quantum phenomena by dismissing some conceptions of “physical reality” which we are – as “everyday life” suggests these to us – intuitively convinced of.
For Bell (and for Einstein for that matter) the dispute was never about “realism”, but about local causality. This is quite clear from Bell’s writings, especially from his last paper: “La nouvelle cuisine.” He was very much an Einsteinian in this respect. For both Einstein and Bell it made no sense to question the “physical reality”, or “realism”, per se. For them the issue was then about the apparent non-locality implicit in the notion of quantum entanglement.
On my part, quantum entanglement is nothing but a 20th century phlogiston. It will not be with us in the next century. The sooner we overcome it the better:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=237#p6160
As Einstein said: “I think that a particle must have a separate reality independent of the measurements.”
The Bell inequalities derive from George Boole’s “Conditions of Possible Experience” and BT is as much about the applicability of classical logic as anything else. Why should it be surprising that classical (Boolean) logic — which Bell experiments involving quanta seemingly violate in favor of QM — might not be universally applicable simply because it serves us so well in our macroscopic environment?
Then what is the appropriate inequality for quantum experiments?
You mean since the Bell inequalities are classical logic and thus arguably not applicable in the quantum domain, how can they logically or meaningfully be employed to frame experiments involving quantum phenomena? The answer is that what the experiments do is establish a contradiction within the parameters of classical logic itself and indicate that however you slice it you’ve found a domain in which purely classical thinking is inadequate. (See Itamar Pitowsky for all of this.) Non-locality isn’t necessarily proven and locality thus proven (see Chris Timpson on the Simulation Fallacy) but reality, to the extent it’s congruent with Boole’s “Conditions”, isn’t given a pat on the back either.
You do science with the tools you’ve got. BT’s contention that EPR is wrong — BT’s flat assertion that no hidden variable theory can replicate all the predictions of QM — is proven by experimental results interpreted by classical logic. If the Quantum Supremacy project works out there’ll be empirical as well as theoretical proof of Bell Entanglement and the cake is iced.
Should read: “Non-locality isn’t necessarily proven and locality thus disproven” instead of “locality thus proven”.
You are then comparing apples to oranges. The fact of the matter is that QM and the quantum experiments actually use a different inequality and then point out falsely that they have violated the Bell inequalities.
Sorry, the “fact of the matter” in regard to what are being compared is that you’ve got it wrong. Anyway, are you referring to the Tsirelson’s Bound controversy, the supposedly “incorrect” CHSH derivation or something more esoteric? The acoustics aren’t the best in here all of a sudden.
Edited from previous pending:
Diether, thanks so much for linking the incorrectly titled thread “Lockyer’s math errors” from you “blog”. There were no math errors on my part.
I would like all readers to take the link and observe Christian’s entry on December 8, 2015. Here we have a clear demonstration of the sign error in his own written equations. Note the two equations in the white box of his entry. The left hand sides of both are equal by the rule of multiplication by real numbers: for any possible algebra with algebraic elements A and B with multiplication rule *, we always have (A)*(B) = (-A)*(-B). And yet the right hand sides differ by sign on the cross product he will be dispatching by adding the two. A very clear demonstration by the man himself of the sign error he committed.
These two equations are of course represent the two orientations for the GA product which he picks between on a fair coin basis expecting the pesky cross product terms (GA bivector) to statistically evenly add and subtract themselves to insignificance. Problem is they don’t, which Christian half shows us. Just like I proved in the referenced thread, when cast in a form they actually can be added, they are equal as accurately indicated on his left hand sides.
This erroneous math leads to the incorrect passage from eq. (74) to (75) in the referenced paper, which does occur after the clear misstep between (69) and (70). The former appears throughout all of Christian’s work. Bottom line is S^3 orientation is a non-starter, he needs to stop pushing it and take a different tack.
No amount of bad math can make good physics
You are exactly right, Rick.
In the present paper https://arxiv.org/pdf/1405.2355v6.pdf the sign mixup can be seen in equations (49), (50) and (51). According to (50) and (51), L(a, lambda)L(b, lambda) does not depend on lambda (in fact, using the fact that I commutes with everything and I^2 = -1, we find L(a, lambda)L(b, lambda) = a b. But according to (49) it apparently depends on lambda. Ah, but that is supposing that the cross product does not depend on “orientiation”. In order to rescue this mistake we must have two cross products, one for lambda = +1 and one for lambda = -1. Call them x and xx.
Then a x b is the usual cross product, while a xx b = – a x b = b x a
Unfortunately fixing the mistake here, causes irreparable damage later on. There is no escape.
Sorry, Lockyer is completely wrong as is are you. You forgot to include lambda. For the cross product you will have +(a x b) when lambda is +1 and -(a x b) = b x a when lambda is -1. The bad math belongs to you and Lockyer.
I agree that a x b (lambda = +1) = – a x b (lambda = -1). Otherwise (49) is inconsistent with (50) and (51). But we also have, from (50) and (51), L(c, lambda) = lambda I c. Hence L(a x b, lambda = +1) = L(a x b, lambda = -1). So the desired cancellation in (73) won’t occur.
I did not forget to include lambda. The problem is that it turns up twice, while Christian wants it only to turn up once.
There is no sign mistake or a contradiction of the kind Gill thinks there is in my equations (49), (50) and (51). The cross product in my paper is universally defined by the right-hand rule (which is the standard convention). The mistake is actually made by Gill in his post. He has written an equation in a manner of mine, but the orientation of the 3-sphere on the LHS or his equation does not match with the orientation of the 3-sphere on the RHS of his equation. Moreover, he ignores the fact that orientation is a relative concept.
Actually Gill’s mistakes have been repeatedly pointed out to him, many times over, at least since 2012. See, for example, the following two papers of mine:
https://arxiv.org/abs/1203.2529
https://arxiv.org/abs/1501.03393
At one point Gill even admitted his mistakes to me:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=226#p5872
I did not admit any mistake; but I did admit that there was a way to re-interpret your formulas so that one particular step in the argument was rescued. It’s your alternative proposal (78): transpose the geometric product in line (72) when lambda = -1. That was certainly a clever way to fix the cross-product cancellation problem in (72) to (75), and I recall it was proposed by your computer programmer in order to fix a problem with the GAViewer verification of (72) to (75). GAViewer found the mistake (your mistake) there too!
Unfortunately, changing a fundamental definition in one line of a long paper likely has repercussions elsewhere. So far I did not see any revision of the paper to take account of the fact that geometric products are understood to be transposed when lambda = -1. Such a change would have enormous repercussions throughout the part of the paper we are discussing now.
It doesn’t really matter. The average of the cross product vanishes anyways; a x b = c. But to be completely mathematically and physically correct, the left-handed orientation must be translated to the right-handed orientation before summing. It quite simple physically. When you look at a left-handed orientation from a right-handed frame of reference, the order is reversed.
Maybe, the following quote from a paper by Gröblacher et. al („An experimental test of non-local realism“, Nature 446, 871-875 (19 April 2007)) clarifies what I mean:
„Most working scientists hold fast to the concept of ‘realism’—a viewpoint according to which an external reality exists independent of observation. But quantum physics has shattered some of our cornerstone beliefs. According to Bell’s theorem, any theory that is based on the joint assumption of realism and locality (meaning that local events cannot be affected by actions in space-like separated regions) is at variance with certain quantum predictions. Experiments with entangled pairs of particles have amply confirmed these quantum predictions, thus rendering local realistic theories untenable. Maintaining realism as a fundamental concept would therefore necessitate the introduction of ‘spooky’ actions that defy locality. Here we show by both theory and experiment that a broad and rather reasonable class of such non-local realistic theories is incompatible with experimentally observable quantum correlations. In the experiment, we measure previously untested correlations between two entangled photons, and show that these correlations violate an inequality proposed by Leggett for non-local realistic theories. Our result suggests that giving up the concept of locality is not sufficient to be consistent with quantum experiments, unless certain intuitive features of realism are abandoned.“
As I recall that’s a paper from Zeilinger’s group about Leggett-Garg and it ought to be noted that Bohmian mechanics (which of course “preserves reality” with nonlocal HVs) is specifically excluded from consideration for a variety of technical reasons. The Gisin group takes Bohmianism on with experiments involving moving reference frames and establishes that hidden variables (Bohm variety) would need to operate >4 magnitudes faster than lightspeed.
As a sort-of aside, I want to share my own experience re: grandiose claims in science.
I remember when I realised the magnitude of my contribution, I was so elated. I naively assumed that those well-versed in the areas-of-interest would immediately “see” why. While the result was extremely convincing, and more importantly: methodologically rigorous and correct, the explanation behind it still required work.
Prior to embarking on the journey of getting this work reviewed (to improve the explanation/presentation), I will never forget the initial concern of the individual who ended up shepherding me through this arduous process.
Initially they were so concerned about my state of mind that they asked for my (thesis) supervisor’s contact information and were upfront about potentially recommending psychiatric assessment. After experiencing some (in retrospect, much-needed) quiet disappointment, I gave them this information and also provided a bit of explanation, which afforded me some lattitude (I was bugging them near-daily at the time :P).
While their concern initially seemed offensive, I never said that because I knew the likelihood of a “big result” (comparable to what Christian is claiming) was so small that it could not be (at that point) immediately embraced, and thus the only place their comments could come from was a genuine concern for my well-being.
Lo and behold, a few months later, when I had submitted this work to a prominent journal in the area-of-interest, it was sent for review to the top scholars that I had asked for, thanks in large part to the aforementioned shepherd (whom I now consider a mentor). This rare situation, of presenting work to area-of-interest’s top scholars whilst being nearly half their age, was such an honour that it put things into perspective.
Not only that, but I hope that I’ve made my aforementioned shepherd-turn-mentor proud in carrying the torch exactly where they had left off two decades prior, both in terms of mathematical rigour and explanatory power of the data whose observations our methodology sought to improve.
I am so thankful to this aforementioned shepherd-turn-mentor because we have never met in person, but I felt it was clear we shared much in common when looking at how we approached the problem-in-question.
It is interesting Mr. Christian never had the luxury of being pulled aside by a scholar of similar calibre to the one who ended up being my main influence, especially when considering Mr. Christian’s alleged institutional affiliations. Given my experience of having someone shepherd me, going as far as “putting their neck out” for me when they have never met me (!!), I find it hard to believe that Christian’s result is as-good as he is claiming because I think experts in the area would get behind him (he has been doing this 10 years, from what I read above).
I know that if I had expressed my contribution in the way Mr Christian has (“Disproving Bell’s theorem”, a grandiose title for work that is supposed to be rigorous), the aforementioned shepherd-turn-mentor would not have invested their intellectual merit behind me.
Being combative a niche website forum, then PubPeer, and now RW comments, does not seem to be the best way to build consensus around a result (if it’s correct).
Just some things to think about, Mr Christian.
In one of his posts above Gill derives the Bell-CHSH inequality as follows:
“Given functions A(a, lambda) and B(b, lambda) taking values +/-1 and a probability distribution over lambda, rho(lambda) d lambda
Consider two values for a and two for b: a1 and a2, b1 and b2.
Define Ai = A(ai, lambda), Bj = B(bj, lambda)
These four are now four random variables taking the values +/- 1.
So there are just 16 possible outcomes for the quadruple (A1, A2, B1, B2)
Important observation: A1B1 – A1B2 – A2B1 – A2B2 takes only the values +/- 2
(please check this yourself! either run through all 16 possibilities by hand, or come up with a smart trick to prove it more easily)
Therefore E(A1B1 – A1B2 – A2B1 – A2B2) lies between -2 and +2 … (1)
Thus E(A1B1) – E(A1B2) – E(A2B1) – E(A2B2) lies between -2 and +2 … (2)”
Now until inequalities (1) everything seems to fine. But inequalities (2) are evidently not the same as those in (1). Unlike the bounds on a single average as in (1), they involve bounds on four separate averages. But let us go along with Gill and accept, for the sake of argument, that inequalities (2) are the same as inequalities (1). If so, and since (2) is a mathematical claim after all, it should be quite easy to derive the bounds of -2 and +2 on the sum of the four averages in (2) without any reference to the inequalities in (1), which involve an average over the unphysical quantities (A1B1 – A1B2 – A2B1 – A2B2). So I wonder why Gill doesn’t directly derive the bounds of -2 and +2 on the four separate averages considered in (2)?
Because he can’t derive (2)’ without using (1), and there is no reason that he should be able to.
It is a *hypothesis* of Bell’s theorem that the “unphysical quantities (A1B1 – A1B2 – A2B1 – A2B2)” can be simultaneously observed. Within quantum mechanics, they cannot be simultaneously observed, but in a “local realistic” theory they can be.
A *hypothesis* of Bell’s theorem is that these quantities can be simultaneously observed, and its *conclusion* is that the quantum mechanical correlations are impossible.
The (logically equivalent) contrapositive of this is that if the quantum correlations are possible, then the universe cannot be “local realistic” This is the most intuitive way to state Bell’s theorem
Of course Gill’s can’t derive the bound of 2 without considering (1). That was the point of my rhetorical question. I just posted the following reply to you in my post above:
Stephen Parrott wrote: “A *hypothesis* of Bell’s theorem is that *it is possible in principle* to measure A1B1, A1B2, A2B1, A2B2 *simultaneously*. This is the “realistic” part of “local realistic”. It is *not* a hypothesis that *all* experimenters can do this. But maybe God can. Gill’s hypothetical computer can.”
But this hypothesis has nothing to do with realism. It is anti-realistic, in the sense that it amounts to being able to be in New York and Miami at exactly the same time. I don’t know about “God”, but I am confident that Gill’s hypothetical computer cannot be in New York and Miami at exactly the same time. Let me explain why I call this hypothesis anti-realistic:
Let me first define some objects and possible events that I hope everyone will agree are manifestly real (i.e., they do not compromise Einstein’s Local Realism in anyway):
(1) New York City is a manifestly Real place.
(2) Miami is a manifestly Real place.
(3) You are a manifestly Real person.
(4) You can be in New York City on 4th of July 2017, at 1:00 PM. A manifestly possible, Real event.
(5) You can be in Miami on 4th of July 2017, at 1:00 PM. A manifestly possible, Real event.
(6) You can be in New York City AND in Miami on 4th of July 2017, at 1:00 PM. An impossibility, in any possible world. I don’t believe even “God” can make this possible.
But the last impossibility is precisely what is claimed by Bell and his followers to be possible when they consider the average of impossible events like:
E( a1, b1, a2, b2 ) = Average of [ A(a1) B(b1) + A(a1) B(b2) + A(a2) B(b1) – A(a2)B(b2) ].
These events simply cannot occur in ANY possible world. They are absurdities, like the item (6) above.
Consequently, anything derived from considering such absurdities, such as the upper bound of 2 on the Bell-CHSH-type inequality, is also an absurdity. It has nothing whatsoever to do with the notion of Realism, or locality, or causality, or anything in physics in general.
On the other hand, note that it is perfectly legitimate to make counterfactual statements like:
(7) You can be in New York City OR in Miami on 4th of July 2017, at 1:00 PM. A manifestly possible, Real event.
But if the Bell-followers replace AND of (6) with OR of (7) in this manner, then the upper bound on their Bell-CHSH inequality is 4, not 2. And the upper bound of 4 has never been “violated” in any experiment (indeed, nothing can violate it). So the hypothesis you mention has nothing to do with realism. It is simply an absurdity, in any possible physical world.
Well in an EPR-Bohm scenario it is also impossible for a local-realistic theory to observe or even predict the four quantities simultaneously since it is a quantum scenario. And that is Bell mistake number one. Second Bell mistake; “shifting the goalposts” by allowing QM and the experiments to use the four averages separately where the bound is |4| instead of |2| and then claiming they “violated” the inequality with the bound of |2|. It is pure mathematical nonsense.
Christian wonders why I go via (1) to get my proof of (2). Isn’t there a direct proof?
Yes there is also a simple geometric proof. The probability distribution of (A1, A2, B1, B2) can be thought of as a vector of 16 non-negative real numbers adding up to one. The quantity E(A1B1) – E(A1B2) – E(A2B1) – E(A2B2) is therefore a linear function defined on the 15-simplex, https://en.wikipedia.org/wiki/Simplex It’s maximum and minimum values are therefore attained at the extreme points of the simplex. Earlier in my proof we agreed that at the extreme points, the points where one of the 16 probabilities equals 1 and the other 15 are all zero, E(A1B1) – E(A1B2) – E(A2B1) – E(A2B2) = +/-2.
You might like to draw a picture to visualise this argument.
Stephen Parrott says “It is a *hypothesis* of Bell’s theorem that the “unphysical quantities (A1B1 – A1B2 – A2B1 – A2B2)” can be simultaneously observed”. I don’t think the hypothesis of local hidden variables has any implication about what can be simultaneously observed by a real experimenter. The hypothesis says that when the experimenter measures spin in the direction a, he is merely observing the value A(a, lambda) of some function of a and of some physical quantity lambda. It doesn’t say that he, or anybody else, can actually observe lambda.
That “simple geometric proof” simply hides the absurdity of the kind I have clearly brought out in my post. It simply obfuscates the fact that an unphysical possibility of simultaneously measuring an impossible event such as A(a1) B(b1) + A(a1) B(b2) + A(a2) B(b1) – A(a2)B(b2) is a necessary assumption to derive the bounds of -2 and +2 on the Bell-CHSH correlator. But once we start admitting such absurdities in our logic, then we can derive anything we like.
To say it in other words:
Local realism is precisely the world view that even measurements that are not executed have definite values, regardless of what happens at the same time in another place. Statistically, that means:
E(A1B1 – A1B2 – A2B1 – A2B2) = E(A1B1) – E(A1B2) – E(A2B1) – E(A2B2)
Whether I am able to measure (A1B1 – A1B2 – A2B1 – A2B2) or not, who cares? From the viewpoint of an experimental physicist, this would be annoying a little bit, but that’s all.
And local realism, including counterfactual reason, obtains here in the macroscopic world, at any rate when you’re dealing with tangible, directly observable physical objects. For instance just try violating the Wigner-d’Espagnat inequality (derived ultimately from Boole’s COPE) using as your experimental set the attributes of canines in an enclosed dog park or the tomes on a stationary shelf of books. You’ll die first. But that’s arguably not the case in the quantum realm when particle spins are the characteristics being investigated. Hence the hysteria.
NB — Bell discusses the application of Wigner-d’Espagnat to statistics in his iconic “Bertlmann’s Socks and the Nature of Reality”.
The last event, namely the simultaneous determination of those four quantities certainly *can* occur in Gill’s computer. It is not a logical impossibility. Indeed, it is what anyone unfamiliar with quantum mechanics would *expect*, on the basis of everyday experience with
the classical world.
If we can’t agree on that, then I doubt that we shall ever agree, so there seems little point to discuss it further. But others may be convinced, which is the point of this last reply.
I can’t see any valid analogy with an impossibility of being in two different places at the same time.
The impossible event I denoted as (6) may occur in a computer, but it most certainly cannot occur in any possible physical world, classical or quantum. It is simply an absurdity. therefore the stringent bounds of -2 and +2 are also absurdities, having no relevance for physics.
In your proposed macroscopic experiment, is it possible to compute the results of more than one measurement angle on the same macroscopic fragment from the photographic images?
Yes.
And since I know what you are getting at, let me save time and answer your next question.
Even in my proposed experiment in which it is possible to compute the results along all the directions such as a1, a2, b1, and b2 just as it is possible in Gill’s computer, the quantities such as A(a1) B(b1) + A(a1) B(b2) + A(a2) B(b1) – A(a2)B(b2) are still physically meaningless, because those are not what we are supposed to be computing for the correlations E(a1, b1), E(a1, b2), etc. It is very important to understand that for rotations or spins in the actual physical space satisfying the symmetries of the group SU(2), the averages such as E(A1B1 – A1B2 – A2B1 – A2B2) and E(A1B1) – E(A1B2) – E(A2B1) – E(A2B2) are NOT the same thing.
So by “yes”, I take it that we can compute the results of different measurement angles on the same fragment at the same time. But in your final pargraph in that answer, you say we are not supposed to do those computations, because they are physically meaningless. So what are the experimenters expected to compute in your macroscopic experiment?
HR, you already know the answer to your question since I have given it to you many times in the past. There is no such thing as “at the same time” in my proposed experiment, because the actual spin directions are supposed to be measured in it, not their components along some prespecified measurement directions. The measurement directions such as a1, b2, etc. are not supposed to be specified during the actual experiment. The experimenters are then supposed to compute the actual binary correlation functions, E(a1, b1), E(a2, b2), etc., each computed completely *separately*, after the entire run of the experiment is completed and recorded on some device. You will find more details about my proposed experiment here:
http://libertesphilosophica.info/blog/experimental-metaphysics/
There is a proof of Bell’s theorem by Steve Gull (one of the pioneers of geometric algebra), http://www.mrao.cam.ac.uk/~steve/maxent2009/images/bell.pdf which uses Fourier analysis. If you know some Fourier theory, you might like it. It’s formulated as an impossible project: “Write a computer program which is to run on two independent Personal Computers which mimics the QM predictions for the EPR setup. There is to be no communication between the computers after the time of program load.”
He goes on to emphasize: “This is a mathematical project. There are no physical assumptions”.
He then presents a rather neat one page “sketch proof of impossibility”.
My point is that at the heart of Bell’s theorem are some mathematical facts which aren’t changed by consideration of what can and cannot be done in physics laboratories. Pointing to one line of the mathematics and saying “but that expression does not correspond to anything we can do in the laboratory” does not invalidate the mathematics. Once you have committed to any mathematical model, for instance a local hidden variables model, you are bound by the mathematical consequences of your mathematical assumptions.
This is hardly a “proof” of Bell’s “theorem.” It is a sketch of a possible idea of a proof at best.
To begin with, the “proof” declares, with emphasis, that “This is a mathematical project. There are no physical assumptions.” I couldn’t care less about a mathematical project in this context. I am concerned about physically realizable EPR-Bohm type experiments. What is more, I see no derivation of the bounds on the CHSH correlator at all in this supposed “proof.” Finally, there actually exists an explicit, clear-cut local-realistic model, trivially derived and verified in several independent event-by-event computer simulations: http://arxiv.org/abs/1405.2355 . Therefore the above “proof” cannot possibly have any physical significance.
In reference to my earlier post on October 29, 2016 at 5:11 pm, I am going to start an argument . . . with myself. In that post I took the ontological view that A_1(a’) in the circumstance where Alice chooses to measure A_1(a) on the first run, is a “path not taken” which can never be known. Relatedly, Dr. Parrott has stated that the view one takes of A_1(a’) is actually a hypothesis, and that the question is not whether Bell is right or wrong, but rather, what Bell tells us if we make one hypothesis versus a different hypothesis about the meaning of A_1(a’) which is the experiment Alice chose not to do on the first run. And he gave a simple but clear example on October 29, 2016 at 9:53 pm. In short, Bell becomes a “machine” which tells us what happens depending upon the input hypothesis we employ. Again, by way of notation, I will use [x] for the expected value of x, because the greater than (gt) and less than (lt) signs do not show up on RW posts.
Let me start with (D3) of Joy’s https://arxiv.org/abs/1501.03393, namely:
-4 le [A_k(a)B_k(b)] + [A_k(a)B_k(b’)] + [A_k(a’)B_k(b)] – [A_k(a’)B_k(b’)] le +4 (1)
which is an intermediate result of Bell which Joy thinks is really a final result because Bell cannot turn this sum of averages into an average of sums because once Alice chooses a and Bob chooses b, the a’ and b’ choices present an incompatible experiment. I will come back to whether Joy is right or wrong about this after doing some analysis below. I have called a’ and b’ the “roads not taken” and Joy has used an example about being in New York and Miami at the same time and Dr. Parrott has said essentially that we shouldn’t even talk about this because Bell makes no judgement about the treatment of the roads not taken other than as hypotheses which are input to the theorem to obtain consequences.
By way of arguing with myself, I am now thinking that notion of a’ and b’ being an incompatible and unknowable road not taken may be overstated by me and by Joy, up to a point. And also, I want to consider that perhaps the treatment of these roads not taken does not have to be a hypothesis in the manner of what Dr. Parrott has said, but can be a true albeit probabilistic statement about actual agreed-upon physical reality. Further, I am starting to think that the correct approach really was stated in my initial reaction on October 29, 2016 at 12:03 am that this should be approached via the fact that “dependent probabilities are not the same as independent probabilities,” though I should have called these “conditional” and “unconditional” probabilities so there is no confusion as to what I mean. So let’s begin a simple progression of analysis starting (1) above — which is really a Bell equation — so I can illustrate this more precisely.
Let me take (1) above and write this for the first run of the experiment, k=1 as:
-4 le [A_1(a)B_1(b)] + [A_1(a)B_1(b’)] + [A_1(a’)B_1(b)] – [A_1(a’)B_1(b’)] le +4 (2)
We do not know yet whether Alice has chosen a or a’ for this run, nor Bob chosen b or b’. But now, let’s have Alice choose a and Bob choose b so that a’ and b’ are the “roads not taken.” We might say that we know nothing about the a’ and b’ options because these were never chosen for this run, but I think this may be overstated and we need to be careful and deliberate.
Specifically, let us posit that in this first run, upon choosing a and b for their detector alignments, Alice and Bob each found a value +1, that is, they detected A_1(a)=B_1(b)=+1. Because we now know about the first run result with certainty – both as regards how the detectors were aligned as well as what was detected by each person – we can drop the expectation brackets from the [A_1(a)B_1(b)] term only, and write (2) above as:
-4 le A_1(a)B_1(b) + [A_1(a)B_1(b’)] + [A_1(a’)B_1(b)] – [A_1(a’)B_1(b’)] le +4 (3)
I have only removed the expectations from the first term, because that is the term which tells us how Alice and Bob actually decided to align their detectors in the first run. And because we have posited that they each detected A_1(a)=B_1(b)=+1 in this first run, we may write A_1(a)B_1(b)=1, and so simplify (3) into
-4 le 1 + [A_1(a)B_1(b’)] + [A_1(a’)B_1(b)] – [A_1(a’)B_1(b’)] le +4 (4)
Now, this contains the actual result that was found in the first run based on the actual detector settings chosen, as well as three terms for experiments that were not performed, namely, with a’, b’, and both a’ and b’. But these terms still mean something, and it seems prudent to closely study what these other three terms do mean. This is where using conditional probabilities, which was initial reaction to how to approach this problem, comes into play.
Although Alice used a not a’ and Bob used b not b’, we still have some useful information about a’ and b’. Specifically, we can know the angles theta_a between a and a’, and theta_b between b and b’. We can capture this, for example, in the dot products a dot a’ and b dot b’. And if these are unit vectors, then a dot a’ = cos theta_a and b dot b’ = cos theta_b.
So, given that we now know that Alice chose a and Bob chose b and that A_1(a)=B_1(b)=+1 is what each detected in this first run, suppose now that a differs from a’ and b from b’ by a very small angle, say, 1 degree. Then because A_1(a)=B_1(b)=+1 were detected, although we cannot know for certain what “would have been” detected had a’ and b’ been used, we can and do know that there is a very large likelihood (close to but slightly less than a probability of 1) that we would have found A_1(a’)=B_1(b’)=+1 had those alternative alignments been chosen. Conversely, if a and a’ differ by 179 degrees, and likewise b from b’, we know that there is a high probability close to but slightly less than 1 that we would have found A_1(a’)=B_1(b’)=-1. Not a certainty, but a very high probability. Conversely cast, the probability is close to but very slightly greater than zero that Alice and Bob would have found A_1(a’)=B_1(b’)=+1. So while we cannot know for certain what would have happened had we performed these alternate experiments, we can use the information we do have to talk about the probabilities of what “would have happened.”
All of this can be sensibly discussed using the language and mathematics of conditional probabilities. Once Alice chooses a and Bob chooses b in the first run, we can now state these as known conditions. We may write the known conditions that Alice chose a as |a and that Bob chose b as |b. We may say that |a means “given the choice of a” and that |b means “given the choice of b. Therefore, we may go back to (4) and place these conditions into the three terms for the experiments that were not performed, now writing (4) as:
-4 le 1 + [A_1(a)B_1(b’|b)] + [A_1(a’|a)B_1(b)] – [A_1(a’|a)B_1(b’|b)] le +4 (5)
But we also have posited that A_1(a)=+1 and B_1(b)=+1 for the experiments that were performed, so this simple integer number +1 can be factored out from the expectation values and (5) can be written as:
-4 le 1 + [B_1(b’|b)] + [A_1(a’|a)] – [A_1(a’|a)B_1(b’|b)] le +4 (6)
Now, [B_1(b’|b)] contains the expected value for what Bob would have measured had he used b’, given that he actually chose to use b. So long as b is not equal to b’ and is not equal to –b’ (180 degrees different), we will have -1 lt [B_1(b’|b)] lt +1. That is, [B_1(b’|b)] will not be an integer, but will be a non-integer number between -1 and +1. Likewise [A_1(a’|a)] will also be an non-integer number between -1 and +1. The final term [A_1(a’|a)B_1(b’|b)] is the expected value of a product, will mathematically differ from the product of the separate expectations [B_1(b’|b)] and [B_1(b’|b)] by the covariance, see, e.g., https://en.wikipedia.org/wiki/Covariance, which I will not try to calculate here. In all of these terms in (6) for the experiments that were not performed, the key driving numbers regarding the these expected values will be a dot a’ = cos theta_a and b dot b’ = cos theta_b.
In (6) above, in contrast to what Dr. Parrott has said that were are making hypotheses about the meaning of experiments which were not performed even to the point of he and Richard talking about what God knows (to which I have no objection in principle), we have used the humanly-available information to deduce as much as we can about these experiments not performed, in a definite, calculable, albeit probabilistic way. So long as a and a’ do not point in exactly the same or exactly opposite directions, and likewise for b and b’, the number in (6) will now be greater than -4 and less than +4, with the equalities removed (le becomes lt). That is, (6) now becomes:
-4 lt 1 + [B_1(b’|b)] + [A_1(a’|a)] – [A_1(a’|a)B_1(b’|b)] lt +4 (7)
so long as a is not parallel or antiparallel to a’ or likewise for b and b’.
Finally, this brings us back to Joy’s contention that Bell makes a mistake when he equates the sum of the averages to the average of a sum. Given (7) above, this question now becomes whether (7) above can or cannot be further turned into the relation:
-4 lt 1 + [B_1(b’|b) + A_1(a’|a) – A_1(a’|a)B_1(b’|b)] lt +4 (8)
And specifically, factoring out superfluous mater from (7) and (8), the question is whether (?=? denotes that we are asking, not asserting, whether these are equal):
[B_1(b’|b)] + [A_1(a’|a)] – [A_1(a’|a)B_1(b’|b)]
?=? (9)
[B_1(b’|b) + A_1(a’|a) – A_1(a’|a)B_1(b’|b)].
While I can be persuaded otherwise, my initial thought is that these are NOT equal, which would make Joy correct, and which would mean that Bell did make an actual mistake when he equated a sum of averages to the average a sum, and that Joy’s Appendix D in https://arxiv.org/abs/1501.03393 actually does contain headline news. This is because we can talk about [B_1(b’|b)] and [A_1(a’|a)] and [A_1(a’|a)B_1(b’|b)] separately, as definitive albeit probabilistic values for certain results that might have been observed had we used a’ and b’, given that we actually used a and b in this first run. This is knowable and calculable. But we cannot remove the expectation brackets and then add B_1(b’|b) and A_1(a’|a) and A_1(a’|a)B_1(b’|b) together, because these are unknowable as exact, definite numbers. They are knowable only probabilistically, and INDIVIDUALLY, as expectation values.
Again, I am new to Bell, and perhaps this line of thinking has already been developed before and shot down. Or maybe it hasn’t. I put this out there simply as my two cents about all of these discussions, talking as Dr. Parrot suggested, only about Bell’s theorem itself, and not about Joy’s model.
The bottom line is that if (9) is an equality then Bell made no error and Joy is wrong. If (9) is not an equity, then Bell did make an error and Joy is correct.
Jay
It has never been shot down and that is why the debate rages on. But it is easy to shoot down the Bell reasoning for the simple fact that QM and the experiments all use the four averages separately where the bound is |4| and then claim they have “violated” Bell-CHSH where the bound is |2|. That fact is real easy to see from what Joy has presented.
Once your notation has been fixed, (9) is true, as far as probability theory is concerned.
You consider just one “trial” (one particle pair) and you take expectation values conditional both on which pair of directions Alice and Bob chose to measure and on which pair of outcomes they then obtained. Your notation does not express this very well.
But anyway, conditional expectation is also additive, just as ordinary expectation.
E(X + Y | Z = z) = E(X | Z = z) + E(Y | Z = z).
Here, Z is a vector of random variables, and z a vector of realised values.
It is sufficient for the macroscopic experiment to just calculate E(a1, b1). If that produces the negative cosine curve -a.b or even close to it, then Joy is right. Bell-CHSH is not even needed.
And by “a1,b1”, what particular values do you have in mind?
Please read the paper: https://arxiv.org/abs/1211.0784
Did that. And it is clear that in that paper a1,b1, could represent any values, which you now claim would be “unphysical” .
I claim no such thing. Please read the paper again.
I just did a one page calculation now posted at https://jayryablon.files.wordpress.com/2016/10/bell-limits.pdf regarding the CHSH limits, which sidesteps the question Joy raises about Bell’s purported “illegal operation.” I instead use the covariance of the product of Alice and Bob’s observations, and take that covariance to be zero by virtue of Alice and Bob’s independence. I believe all of these operations are legal, and I am not anywhere turning a sum of averages into an average of sums. Nonetheless, the net result still appears to yield the outer CHSH limits of -2 to +2, and not of -4 to +4, so long as the covariance between Alice and Bob is zero, which I believe it has to be by the EPR premises. Jay
In the very first line of my attachment, I meant to write (D3) not (D4).
Well, a perfectly valid local hidden variable model would be one where Alice measures +1 no matter what, and Bob measures -1 no matter what. The covariance is not 0, unfortunately.
“The covariance is not 0, unfortunately.”
Here is the same calculation for non-zero covariance: https://jayryablon.files.wordpress.com/2016/10/non-zero-covariance.pdf.
To start with, where do you get (5) from? “We independently know that:”
Jay, your argument is fatally flawed. In the experiments each of the numbers A and B are always observed to average to zero. So the expression in your eq. (3) is identically zero.
Yes, Joy, but for a given run, (3) need not be zero. Is that correct?
What does an average for a given run mean?
For one run, it is a result (-1 or +1) for the experiment that was actually done, and an expectation value for the three experiments that were not run. Along the lines of what I posted on October 30, 2016 at 12:34 pm. I believe that is part of your appendix D argument, namely, that there are four incompatible experiments, one of which was done and has a known result, and three of which were not done and so can only be discussed probabilistically, if not by God, then at least by humans.
Yes, his assumption that the covariance must be zero is much stronger than what is imposed by a LHV.
HR, forget the hidden variable theory. The experimental fact is that A and B always average to zero. This means that the expression inside the average in Jay’s eq. (3) is identically zero.
And, since Jay was not aware of this, he is unwittingly factoring zeros out of zeros in his eq. (5). That is like dividing by zeros. Sorry, Jay.
Yes, but Bell’s theorem (which I assume Jay tries to prove) covers all hypothetical local hidden variable theories (including obviously wrong ones), many of them where A and B does not average to zero, no matter what we observe in experiments. Point being that the covariance does not need to be zero in a local hidden variable theory (in fact, that would amount to postulating that all correlations must be zero, so the whole theorem would be moot).
I may need to pull back this line of approach, or at least I am probably “thinking out loud” more than I ought. Sorry. What this may actually lead to is a proof by contradiction as to Joy’s view that Bell should not have equated the sum of the averages to the average of the sums, making use of non-zero covariances. But I will work it through and sleep on it to see if it fans out before I do anything else.
Jay, nice tries, but you are missing two crucial ingredients of the story.
One, Bell’s CHSH inequality (the one with bounds -2, +2) is about theoretical expectation values, not about experimental averages. The four experimental averages are only close to the corresponding four theoretical expectation values if the number of observations in each of the four sub-experiments is large. You are not using this assumption in your derivations.
Two, the CHSH inequality depends on the assumption of local hidden variables. You are not using this assumption in your derivations.
The second assumption (local hidden variables) tells us that the *theoretical* correlation between measurements at settings a and b is
E(a, b) = integral over lambda of A(a, lambda) B(b, lambda) rho(lambda) d lambda
Exercise: prove CHSH for these *theoretical* quantities.
Corollary: if all four sample sizes are all large *and* local hidden variables is true, the *experimental* correlations will *approximately* satisfy the CHSH inequality.
Note that it is important to assume that the probability distribution of the hidden variable, rho(lambda), does not depend on the settings used by the experimenter. Otherwise we can’t so easily say something E(a, b) + E(a, b’) + E(a’, b) – E(a’, b’).
Hint: the proof strategy involves combining the four integrals under one integral sign.
PS there are four subexperiments each with its own number, N(a, b) say, of trials; and of course the hidden variables lambda^k which nature generates for each of the four subexperiments will of course be completely different from one sample to another. Better to write lambda^k(a, b).
The important point is that we assume that they form four large, independent, random samples from the *same* population with probability distribution rho.
Well I am aware now. 🙂 So if you set all the zeros to zero, we can get an equation for the covriance that runs from -4 to +4. Wondering if that helps?
That is correct, but also equivalent to what I have in my Appendix D. At the moment I don’t see why considering only covariance would help in any way.
OK, so Richard and Joy (and anyone else) look at this calculation https://jayryablon.files.wordpress.com/2016/10/proof-by-contradiction.pdf, to advise whether this helps shed any light on the expected value of sum = sum of expected value disagreement. I fully understand and sympathize with Joy’s point about “A physically meaningful quantity = A physically meaningless quantity.” But I am trying to see of there is a straight mathematics proof, or at least a different way to view Joy’s physics argument.
My worry is about physics, as you have noted. But mathematically I have not seen a proof in the standard literature (such as the Clauser-Shimony report etc.) of either of your equalities (1) or (5). In the standard literature equality (1) is simply taken for granted. But there is a vast literature on Bell’s theorem, so there might be a proof of (1) out there somewhere, within this context. Needless to say, the burden of proof of proving either (1) or (5), or both, is on those who believe in Bell’s theorem. Ignoring my physics worry (and logic worry), at the moment I can’t think of any purely mathematical reason why (1) may not true. (5), on the other hand, may not be true in general even mathematically.
Here is a simple proof of why your equality (5) cannot be true in general. The quantity on the LHS of (5) is bounded by -4 and +4, since each of the four covariance terms in it are bounded by -1 and +1. The quantity on the RHS of (5), however, is bounded by -2 and +2, which is easy to prove, as I have done in the Appendix D of my paper you have cited. Therefore the equality (5) cannot possibly be true in general, even mathematically.
Regardless of whether Jay succeeds in his efforts, my concerns are only about physics.
It is physically meaningful to write the sum of four averages in the manner of CHSH:
E(A1B1) + E(A1B2) + E(A2B1) – E(A2B2).
In other words, the above quantity is a physically meaningful quantity.
On the other hand, the following single average is physically meaningless, because it involves an average of four incompatible experiments performed about mutually exclusive directions:
E(A1B1 + A1B2 + A2B1 – A2B2).
Now Bell-CHSH inequality is derived by equating the above two expressions:
E(A1B1) + E(A1B2) + E(A2B1) – E(A2B2) = E(A1B1 + A1B2 + A2B1 – A2B2).
Even if someone can rigorously prove that the above equality holds mathematically, I would not be impressed at all. Because physically it is equivalent to the following absurdity:
A physically meaningful quantity = A physically meaningless quantity.
As long as the above absurdity is used in the derivation of any Bell-CHSH type inequality, Bell’s theorem has no relevance for physics.
Thanks Joy. I updated the file I posted late last night to obtain an equality that contains only the covariance expressions, with all expectation values removed. That is linked at https://jayryablon.files.wordpress.com/2016/10/proof-by-contradiction-updated.pdf. The equalities (5) (as before) and (9) are now the two alternative representations of (1) that we are evaluating for truth or falsity. I suspect your answer regarding (9) will be the same as your answer regarding (5), namely that the bottom line of (9) is bounded by -2 and +2 while the top line is bounded by -4 and +4? And to Richard, do you agree or disagree (and why) with Joy’s reply above regarding (5)?
Jay
Thanks, Jay. Your equalities (1), (5), and (9) are all equivalent. And my dispoof also applies to them equally. Since the averages of A and B are vanishing, we just have E(A, B) = cov(A, B). Therefore we might as well concentrate on (1) [but we are free to apply my observation below to (5) and (9) as well]. I have noted previously that physically equality (1) leads to absurdity:
E(A1B1) + E(A1B2) + E(A2B1) – E(A2B2) = E(A1B1 + A1B2 + A2B1 – A2B2) …… (1)
implies
a physically meaningful quantity = a physically meaningless quantity.
But now I claim that the above equality is false even mathematically. Here is a simple proof: The LHS of (1) is bounded by -4 and +4, since each of the four terms in it are bounded by -1 and +1. The quantity on the RHS of (1), however, is bounded by -2 and +2, which is easy to prove, as I have done in the Appendix D of my paper you have cited. Therefore equality (1) cannot be true in general even purely mathematically. That is to say, it is both physically absurd and mathematically false. Therefore Bell’s theorem is a seriously erroneous claim.
Jay, of course I disagree with Christian’s reply. The logic is very obviously incorrect.
BTW, you missed my posting of October 30, 2016 at 6:43 pm
It would have been better to take a decent proof of Bell’s theorem as your starting point. To get back on track, apply the law of large numbers right at the start, and replace those limits of averages by … their limits. Expectation values. Integrals over lambda. After that it is all plain sailing.
It is important to note that Gill has not provided any proof, or even a hint of a reference to a proof, of the following equality assumed by Bell in the proof of his “theorem”:
E(A1B1) + E(A1B2) + E(A2B1) – E(A2B2) = E(A1B1 + A1B2 + A2B1 – A2B2).
I have been trying to “mediate” the long-standing Bell disagreement between Richard and Joy for about three weeks now. I have observed that their stark differences on substance mirror the fact that Richard’s fundamental center of gravity is as a mathematician studying a physics problem, while Joy is a physicist using mathematics (as all serious physicists must do) to study natural reality. And Bell Theorem sits right between mathematics and physics: the theorems themselves are largely mathematics and statistical, yet they intrude directly into physical explorations about what is and is not reality.
Richard tends to take a view that the “mathematics is the mathematics and you cannot use physics considerations to override the mathematics.” This is perhaps exemplified by his statement on October 30, 2016 at 8:33 am that:
“My point is that at the heart of Bell’s theorem are some mathematical facts which aren’t changed by consideration of what can and cannot be done in physics laboratories. Pointing to one line of the mathematics and saying ‘but that expression does not correspond to anything we can do in the laboratory’ does not invalidate the mathematics. Once you have committed to any mathematical model, for instance a local hidden variables model, you are bound by the mathematical consequences of your mathematical assumptions.”
On the other hand, Joy is always asking whether something makes physical sense, for example, on October 30, 2016 at 7:15 pm when he says:
“Even if someone can rigorously prove that the above equality holds mathematically, I would not be impressed at all. Because physically it is equivalent to the following absurdity:
A physically meaningful quantity = A physically meaningless quantity.
As long as the above absurdity is used in the derivation of any Bell-CHSH type inequality, Bell’s theorem has no relevance for physics.”
I see both sides of this, and I often work from both sides of this myself. If it is possible, I always prefer a rigorous mathematical proof to a physical “argument,” if such proof can be obtained. That is why, for example, after a couple of false starts yesterday, I developed https://jayryablon.files.wordpress.com/2016/10/proof-by-contradiction-updated.pdf to recast the disagreement over what Joy calls “A physically meaningful quantity = A physically meaningless quantity” of (1) in the above link into a form which might, perhaps, be adjudicated on mathematical grounds alone. And with the covariance functions, this could be adjudicated on statistical grounds, which is right where Richard lives. So if Richard should agree that equations (5) or (9) in this linked file above are in fact invalid, and that they are proxies for (1), then that would I think force him to change his mind about Bell’s sum of expectations = expectation of sums equation (1) on mathematical grounds alone.
But because my money is on Richard not conceding that (5) or (9) are invalid, let me point out a mixed physical and mathematical argument that in my view of things, and possibly in the view of many others, could carry the day, and get us all beyond this dispute about Bell’s handling of CHSH:
First, I think that Joy needs to refine his statement “A physically meaningful quantity = A physically meaningless quantity” and rewrite this as:
*A physically observable quantity = A physically unobservable quantity*
In other words, if Joy can demonstrate that (1) in https://jayryablon.files.wordpress.com/2016/10/proof-by-contradiction-updated.pdf which is what Bell uses in his processing of CHSH in fact sets a physical observable equal to something that is physically unobservable, then I would take the view that the physics carries the day and agree with Joy that this shows the mathematical result to be absurd. How might one do this? Let me use two examples, one from a recent post by Richard, and one from gauge theory which I have closely studied in my own research.
First example: Richard recently pointed out on October 30, 2016 at 6:43 pm, that when a theory has a hidden variable lambda, any *observable* of the theory, while it might be computed using the hidden variable, may not actually contain that hidden variable. So, in Richard’s:
E(a, b) = integral over lambda of A(a, lambda) B(b, lambda) rho(lambda) d lambda
the expectation value is an observable. But it is derived from A(a, lambda) and B(b, lambda), so one might think that we are setting an observable to a function of a hidden variable (by definition unobservable) which cannot be done. And with this example, Richard makes clear he agrees this cannot be done. But lambda is the dummy variable of integration, so it necessarily drops out once the integral is taken. So although there is a physically unobservable quantity on the RHS above, the integral is an observable because the unobservable lambda is integrated out, and we end up with:
A physically observable quantity = A physically observable quantity (with the unobservable quantity integrated out)
And so this is valid, physically and mathematically. The physics result is *invariant* with respect the the quantity lambda which is unobservable.
Second example: gauge theory also uses an unobservable lambda which is the phase angle in the unitary “gauge” factor exp(i lambda) which multiplies wavefunctions, and redirects their orientation in complex phase space without changing their magnitude. Here, we develop observable physics from an unobservable parameter lambda by requiring the theory to be *invariant* with respect to the value of lambda. So by sticking to this requirement for gauge symmetry, we ensure that none of the observable quantities will be equal to something that depends on the gauge parameter, or which is not invariant under a gauge transformation. For example, if you look at (4.5) of my draft paper at https://jayryablon.files.wordpress.com/2016/10/lorentz-force-geodesics-brief-4-2.pdf, you will see that when I derive the Lorentz force law from a variation, there is included a term with A_\sigma A^\sigma which is not part of the usual Lorentz force law and which is not gauge symmetric, related to an observable du^\beta / d\tau which is gauge symmetric. Mathematically, we might say “so what?” But physically, this is not allowed, because the equation is:
A gauge symmetric quantity = A quantity that is not gauge symmetric
Consequently, I am required to pursue additional development to derive and support the geodesic gauge condition (5.6) that allows me to then reduce (4.5) to (5.8). Now the result is of the form:
A gauge symmetric quantity = A gauge symmetric quantity
and all is good.
So what are the morals of these stories?
Joy, I think when you discuss Bell’s (1) in https://jayryablon.files.wordpress.com/2016/10/proof-by-contradiction-updated.pdf which is (D4) in your https://arxiv.org/pdf/1501.03393v6.pdf, you should very carefully focus on *physical observables*. If (D3) contains expressions which are all physical observables (which IMHO it does) in the sense that the term *“observables”* is commonly accepted, and if you can convincingly demonstrate that (D6) contains an expression which is *a physical unobservable* in accordance with generally-accepted principles about *”observables”*, then at least in my view, that would be a convincing resolution in your favor. And in that situation, I would say that the physics argument would trump the math argument, because no physicist will accept mathematics which leads to the result of:
A physically observable quantity = A physically unobservable quantity
If that is what Bell really did, then Bell would have to be reversed insofar as he did that.
As to burdens of proof (which I have seen Joy try assert in his favor), in principle Bell should have proved that he could equate the sum of expectations to the expectation of the sums, and likely did not do that because he applied a rule of thumb and it didn’t dawn on him. In the meantime, these results have become widely accepted and ingrained, and perhaps nobody else noticed this either until Joy pointed it out some 50 years later. IMHO, this history shifts the burden to those who would reverse Bell. And, Joy, even if you do not like it or don’t think it fair, you are advocating for an historic reversal of a bedrock theory of physics which you seek to relegate to a phlogiston. However the history led to the present state of affairs, if I were speaking in “legal” metaphors alone, and as deeply sympathetic as I personally am to the Einsteinian view of local realism and “no dice,” I would say that the statute of limitations expired in Bell’s favor years ago, so that anyone seeking a reversal of Bell today (not 50 years ago) carries the full burden of proof to show that he made an error.
Jay
Hi Jay,
I agree with the sentiments in your last paragraph about burden of proof. But I have more than provided all sorts of detailed proofs and disproofs for the past nine years in the form of some 16 papers and a book containing detailed analytical arguments and several computer simulations, not to mention thousands of blog posts and comments, many of them not only refuting Bell-type arguments but also debunking some fallacious criticisms of my physically and mathematically impeccable local-realistic model for the EPR-Bohm correlations. Most recently, just above, I have provided a proof of why Bell’s argument is seriously flawed.
As for my observation that Bell’s unjustified use of the equality
E(A1B1) + E(A1B2) + E(A2B1) – E(A2B2) = E(A1B1 + A1B2 + A2B1 – A2B2)
in the proof of his so-called “theorem” is equivalent to the absurdity
“A physically meaningful quantity = A physically meaningless quantity”,
I am most certainly NOT talking about observable versus unobservable physical quantities. I am talking about physically possible quantities such as “an event of my being in New York OR Miami at a given time” versus physically impossible quantities such as “an event of my being in New York AND Miami at exactly the same time.” This contrast has nothing to do with what can or cannot be observed in principle. The event such “my being in New York AND Miami at exactly same time” cannot possibly occur in any possible physical world. It cannot be made possible even by the God of Spinoza. So I am most definitely rejecting your recommendation above in this regard. What I have pointed out above is a much bigger blunder by Bell than it would have been had he only confused between observable and unobservable quantities.
A one-page document at https://jayryablon.files.wordpress.com/2016/10/bell-limits-21.pdf summarizes the argument I believe Joy has made here. Joy, is that so? If so, I would ask Richard to please reply here as to whether option 1 or option 2 or something else (and if so, what) is the conclusion we are permitted reach. Thanks, Jay
Yes, that is essentially my argument. It is easy to prove that “top” lies between -4 and +4. It is also easy to prove that “middle” lies between -2 and +2. Therefore “top” = “middle” equality is simply false, and we cannot conclude that “top” also lies between -2 and +2. The equality between “top” and “middle” simply does not hold. But Bell erroneously equated “top” with “middle” in the proof of his famous “theorem.”
If you generate some random numerical values and compute the averages first according to expression “top” and then according to expression “middle”, do you find the results to be equal? If so, would it be natural to rule out Option 2?
If “top” can range only from -2 and +2, then it should be easy to prove that analytically without referring to the “middle”, which is an entirely different expression from the “top.”
Am I missing something, or is the discussion here about whether the expectation value is linear? That is, if E(A+B) = E(A) + E(B)?
Because that is just basic mathematics, I can prove it in two lines if you want me to. It is also stated in Wikipedia.
It may help if you read the Appendix D of this paper: https://arxiv.org/abs/1501.03393
I would be even more helpful if you answered my question. Are you claiming that the expectation value is not linear?
No. I am not claiming that. The source you have linked says under the equality you mentioned: “The second result is valid even if X is not statistically independent of Y.”
Yes, it is valid even when they are not statistically independent. They are not statistically independent, but the equality is valid. I don’t see your point.
My point is that in the case under consideration E(X + Y) and E(X) + E(Y) do not describe the same physical experiment. In fact E(X + Y) does not even describe any possible physical experiment. Therefore it is essential that we evaluate the bounds on E(X + Y) and E(X) + E(Y) without any reference to each other. These bounds are then not the same, and hence the usual rule of thumb E(X + Y) = E(X) + E(Y) does not apply, as I discussed in my paper.
This is not a “rule of thumb”. E(X+Y) = E(X) + E(Y) is a mathematical identity. It follows directly from the definition of expectation value. Should I prove it for you?
If the two sides of this equation are giving you different results, then either you are making a mistake in your calculation, or the things you are calculating are not expectation values.
The rule of thumb you mention is not valid in the case I have discussed in my paper.
It is always valid. That is the point of being a mathematical identity. If you really think you have found a case where it is not valid, then forget about Bell’s theorem, you are on to something much greater, you are overturning probability theory! You should rewrite not only Wikipedia, but countless textbooks where this basic rule is proven and used.
I think without your actually reading the two pages of the Appendix D of this paper,
https://arxiv.org/abs/1501.03393 ,
you are unlikely to see my point. If you do read the two pages, then let me know which calculation or equation in it is wrong.
Whatever is in that paper is not relevant to the argument. I’m asking a simple mathematical question, about the linearity of the expectation value. There are only two possibilities:
1 – E(A+B) = E(A) + E(B). If this is the case, then Bell’s theorem follows.
2 – E(A+B) not equal E(A) + E(B). If this is the case, then you are arguing against basic probability theory. I’m serious, if you think this is true this is much bigger than Bell’s theorem.
What option do you take?
I like the option I have presented in the Appendix D of the paper I have linked.
Which is?
It is the one I have presented in the Appendix D of my paper I have linked.
Well, the appendix D of the paper you have linked argues both that E(A+B) = E(A) + E(B) and that E(A+B) is not E(A) + E(B). If I were to take it seriously, I should conclude that it is simply logically inconsistent.
So in order not to have such a uncharitable interpretation of your work, I’m asking you directly which alternative you are defending, option 1 or option 2?
Read the fine print in the Wiki link you provided: “for any two random variables X and Y (which need to be defined on the same probability space).”
Sure, but the variables you are talking about are defined in the same probability space. I don’t see what is the problem.
Correct. The all take binary values in {-1, +1}.
Yes, then you would have to go with option 1, whether you get to a -2 to +2 range limitation for “top” by a simulation or analytically. Conversely, if you can get outside the range of -2 to +2 for “top,” then as far as I can tell (and I always keep an open mind to being persuaded otherwise or to somebody pointing out that I made some mistake) you would have to go with option 2 and Bell is cooked and local realism is back on the physics map and Joy will lay claim to one big “I told you so.”
But have you tried to actually generate som random values and see what you get?
Jay may not have tried to generate some random values to see what he gets, but I have:
http://rpubs.com/jjc/84238
It is easy to see from this simulation that the bounds of -2 and +2 are broken in the case of “top” expression.
PS… and Einstein will be smiling from the heavens and ready for playing dice to fall next. 🙂
In the post just following Christian’s quoted above, Richard Gill says “the logic is very obviously incorrect”, but he doesn’t say why it’s incorrect. Since there have been no howls of outrage from the audience, either there is no one seriously following this discussion or the error is not obvious to those who are. Elementary as it is, I thought it might be helpful to explicitly point out the error.
I shall set up a parallel argument which is logically equivalent to Christian’s but is obviously faulty. Christian starts with an equality (1). Instead of that, start with another equality (1)’ :
1.5 = 1.5 (1)’
The LHS (left hand side) of (1)’ is bounded by -4 and 4.
The RHS (right hand side) of (1)` is bounded by -2 and 2.
Therefore, (1)’ cannot be true in general .
Is there anyone participating in this discussion who will publicly affirm the correctness of this logic? Did no one notice?
You have rightly noted that the argument you have set up is faulty. If a logically equivalent analogy you were trying to set up, then a more credible analogy is the following:
X = Y ….. (1)”,
where the variable X is bounded by -4 and +4, the variable Y is bounded by -2 and +2, and the equality is physically absurd (like my being in New York and Miami at the same time). Therefore (1)” cannot be true in general. Evidently there is nothing wrong with this logic.
With all of the great respect that is due to you, Dr. Parrott, I would have to dispute your logic and your math and state that I find them to be incorrect. When you pick a number 1.5 out of a hat, and say that it is between -2 and 2 and also between -4 and 4 also picked out of a hat with *no relation* to 1.5 other than thin air, that is entirely different than if the 1.5 actually had a real *interdependent* relation to the +/-2 and the +/-4. The CHSH bounds of +/-4 about Joy’s (1) are not just picked out of thin air independently and arbitrarily. They are the actual, calculable outside bounds on the expected value expressions inside the limits and *they arise organically and directly from the expressions inside those limits*. They are accepted physics today, and to the extent I have learned the history in my three-week old study of Bell’s theorems, were accepted and used by Bell himself. If the +/-2 and +/-4 had some actual genesis in the number 1.5, I might see this differently. But they don’t. Rather, you turned Joy’s argument into something it isn’t and then shot down your own creation and not anything that Joy argued.
I wrote out Joy’s actual logic and math in https://jayryablon.files.wordpress.com/2016/10/bell-limits-21.pdf and he has confirmed that I have properly represented his view. Years ago you told me that my 40 page tomes were too long. This is all on one page. So please read the actual point that Joy is arguing (and all I have done is try to re-present his argument as clearly as can be so I could asses it also) and let me know if you find any flaws in that actual argument on its own terms, not flaws in an argument that is simplified, incorrectly, into something it is not.
I always stand to be corrected, if you or anyone is able to correct me. If I am convinced that Bell was right I will say he was right. If I am convinced he was wrong I will say he is wrong. I am trying to learn here, and help others learn here. Correct science is my only allegiance. I must assume that as a premier scientist whose views are evidence-based, you adhere to the same philosophy. Thank you.
Jay R. Yablon
The bounds of 2 and 4 are not picked out of the air. The stronger bound is obtained by making stronger assumptions. No contradiction there.
The linearity of expectation, equivalently, the linearity of the integral, is not a “rule of thumb”. It’s a logical necessity. If X and Y are bounded random variables defined on the same probability space then E(X + Y) = E(X) + E(Y).
The assumption of local hidden variables is the assumption that A1, A2, B1, B2 are all random variables defined on the same probability space; and hence so are all four of the products A1B1, A1B2, A2B1, A2B2. The probability space involved here is the set of all possible values of lambda, endowed with the probability measure rho(lambda) d lambda.
In this mathematical world, A1B1 + A1B2 + A2B1 – A2B2 is just yet another random variable also defined on the same probability space. It exists in our mathematical model. Whether or not it has an obvious physical counterpart in the physical situation we are modelling today is irrelevant.
But the stronger assumptions are not only false, physically they lead to absurdities, as I have explained in my posts above.
In the actual experiment, performed in the actual physical world, four separate experiments are performed, and four separate averages E(a1, b1), E(a1, b2), E(a2, b1), and E(a2, b2) are calculated. It is physically impossible to do otherwise. In particular, it is impossible to experimentally observe a quantity such as A1B1 + A1B2 + A2B1 – A2B2. In fact, the quantity A1B1 + A1B2 + A2B1 – A2B2 cannot even exist as an event in any possible physical world, let alone in our world. Therefore the corresponding average E(a1, a2, b1, b2) has no physical meaning, as I have already stressed.
What is more, no one is obliged to perform four experiments and calculate four averages. All one needs to do to verify the prediction “E(a, b) = – a dot b” of quantum mechanics is perform one experiment and calculate one average, say E(a1, b1). It is then quite clear that the average E(a1, a2) lies between -1 and +1, by virtue of the fact that A1 and B1 can only take the values +1 or -1. It is also quite clear that if we perform four such experiments, then each of the four independent averages, E(a1, b1), E(a1, b2), E(a2, b1), and E(a2, b2), similarly lies between -1 and +1. Consequently, their CHSH sum lies between -4 and +4.
The “stronger assumption” in fact is not an assumption at all but an ad hoc replacement of the physically meaningful CHSH sum of four separate averages with a physically meaningless single average of an unphysical quantity A1B1 + A1B2 + A2B1 – A2B2, which cannot possibly be observed in any possible physical world. The latter average then lies between -2 to +2. But who cares if it does. It has nothing whatsoever to do with the physical world we live in.
On further reflection overnight, let me take this all a giant step further. If we were to accept Dr. Parrott’s logic, then Dr. Parrot will have just disproved Bell’s theorem and allowed local realism back into physics and Joy Christian should be thrilled. As should any person in the word who is incarcerated somewhere. But none of those actually happens based on that logic. Permit me to explain:
Let me work with E(A1B1 + A1B2 + A2B1 – A2B2), which to make life simple, I will represent by Dr. Parrott’s number 1.5. Bell’s theorem says that:
-2 le E(A1B1 + A1B2 + A2B1 – A2B2) le +2 (1)
or using the Parrott logic:
-2 le 1.5 le +2 (2)
Equation (1) and its shorthand (2) are Bell’s theorem, and they lead to the consequence that no local realistic theory can reproduce the observed non-linear quantum correlations, precisely because the -2 and +2 outer boundaries prevent that from being possible.
Now, because 1.5 = 1.5, we may also certainly write
E(A1B1 + A1B2 + A2B1 – A2B2) = E(A1B1 + A1B2 + A2B1 – A2B2) (3)
But since 1.5 is also between -4 and +4, it is likewise a true statement that:
-4 le E(A1B1 + A1B2 + A2B1 – A2B2) le +4 (4)
or using the same logic:
-4 le 1.5 le +4 (5)
And just like that, because E(A1B1 + A1B2 + A2B1 – A2B2) are now permitted to go outside the bounds from -2 to +2 all the way from -4 to +4, it becomes possible to obtain quantum correlations using local realism, Bell is cooked, Joy Christian can break out the champagne, and the headline writers can start their stories.
However, none of this is so, because while (1) being true does imply that (4) is also true as a matter of *logic*, the -2 and +2 boundaries are not just a logic statement: they are a statement about a *physical constraint* which nature imposes on the probabilistic quantity E(A1B1 + A1B2 + A2B1 – A2B2). If nature says that E(A1B1 + A1B2 + A2B1 – A2B2) shalt not wander outside of the range from -2 to +2, then one cannot arbitrarily say that E(A1B1 + A1B2 + A2B1 – A2B2) can wander from -4 to +4.
To use a different analogy, if someone locks me inside a 10′ x 10′ room and sets the room inside a football stadium, then it is certainly true that because I am locked inside the room I will remain inside the stadium. But that does not mean I can travel anywhere I wish inside the stadium but outside the room. Outside the room is a forbidden range. Were Dr. Parrott’s logic to be true, then every prisoner the world over should rejoice, because now their confines have been removed and they are free. Logically, the true statement is “inside room -> inside stadium.” The logic inverse would be “inside stadium -> inside room.” But that inverse is false. I can be inside the stadium but not inside the room, but once inside the room I must also be inside the stadium.
So Dr. Parrott has made a false start here — which we all do, and if we think we never do, then we are deluding ourselves and to boot we are not good scientists because we do not take risks. And Joy Christian still needs to keep his champagne on ice, and the headline writers can go back to the US Presidential campaign which has pushed half the people in the US and most people elsewhere in the world to the edge of a nervous breakdown.
All that said, based on Joy’s actual argument which I have re-presented in https://jayryablon.files.wordpress.com/2016/10/bell-limits-21.pdf, I have enumerated two logical options to consider and only one of those, Option 2, ends Bell’s theorem. The other one whittles the CHSH=E(A1B1) + E(A1B2) + E(A2B1) – E(A2B2) boundaries down from +/-4 to +/-2. That is, it makes the CHSH “prison” smaller.
On this point, as a Bell novice, if I had not read all of the material that says the CHSH limits run from -4 to +4, and read that Bell and Gill and Christian and all of the other Bell cognoscenti agree with this, and were I to take seriously the arguments presented by HR and FE and Gill as to the linearity of averages in statistics assuming the same probability space, then I would be inclined to think that CHSH itself lives in a smaller prison, bounded by +/-2 and not +/-4, and I would pick option 1, which I recall HR first suggested yesterday.
So, my question to the Bell cognoscenti, which I am asking not as a challenge but as a student of this esoteric stuff, is this: what would be the *physics* consequences if in fact the size of the CSHS prison was given by:
-2 le E(A1B1) + E(A1B2) + E(A2B1) – E(A2B2) le +2 (6)
and not by the +/-4 bounds?
Truth be told, if I assign each of A1, B1, A2 and B2 the values +/-1 independently (i.e., assign each of these to one of four coin tosses with heads =+1 and tails =-1), than for all 16 combinations of these four variables (A1, B1, A2 and B2), for any single run of four coin tosses, I cannot discern a situation where (6) gets outside the prison from -2 to +2, because of the minus sign before the final term in (6). And because each term has an implicit k subscript for the experiment “run” when (6) is fully written out, this tells me that I may not treat each of the four terms independently, but must have them all cover the same runs of the experiment. That is, if E(A1B1) is the average of the first ten runs k=1…10 of the experiment, I cannot have E(A1B2) come independently from runs k=11…20 and E(A2B1) come from k=21…30 and E(A2B2) come from k=31…40. They all must come from k=1…10. I am not stating, I am asking: As far as I can tell, (6) above a.k.a. (5) in my https://jayryablon.files.wordpress.com/2016/10/bell-limits-21.pdf is the correct CHSH inequality and my option 1 is the correct option.
So I have three questions that I am asking as a student of this stuff, and not as a critic of anybody or anything:
1) What am I missing if anything?
2) Please lay out / explain to me a scenario where (6) can break out of the +/-2 prison in (6) above, mercifully please, without showing me computer code.
3) If my option 1 in https://jayryablon.files.wordpress.com/2016/10/bell-limits-21.pdf were to be correct and if (6) above in fact established the size of the CSHS prison, then what would be the metaphysical physics consequences of this, akin to the barring of local realism by Bell’s theorem?
Jay
Let me ask you this, Jay. And you don’t need to know anything about Bell’s theorem to answer this. Suppose we do only one experiment — because that is all quantum mechanics asks us to do to verify its prediction — and calculate E(a, b), using the observed values of A = +1 or -1 and B = +1 or -1. What do you think the bounds on E(a, b) would be in that case?
N = 10,000 is large enough to see strong correlations -a dot b in a local-realistic simulation. N need not have to be sent to infinity (whatever that word means in physics) for that purpose.
Jay, you ask if you are missing anything. I think you are missing two things. They are both concerned with the fact that there is a limit as N goes to infinity in your definition (Christian’s definition) of correlation.
Firstly, you are missing probability theory. Do you know the law of large numbers? It tells us that those limits of averages which you are studying are equal to corresponding expectation values, which we can write out as integrals. Combine those four integrals under one integration sign …
Secondly, you are missing statistics. In a real experiment, the four observed correlations (averages of products of *finite* number of outcomes) can be almost anything, and it is certainly possible to get an observed value of three correlations minus the fourth equal to +/- 4. However for larger and larger experiments, and if local hidden variables is true, large deviations outside of +/- 2 become less and less likely.
Real experimenters put an error bar on their correlations, or compute standard deviations, and are only excited if CHSH is quite a few standard deviations outside of +/- 2.
This is not an efficient forum in which to answer your question because most readers will not have a copy of your document. Here, I will only give the essence of a reply without referring to your document.
By definition, a (local) “realistic” model assumes that correlations between what Alice and Bob observe can be explained as follows. It *assumes* a probability space whose outcomes are traditionally denoted “lambda”, which is called a “hidden variable”. What Alice observes (either +1 or-1) when her measuring instrument is set to “a” and the hidden variable is lambda will be denoted A(a, lambda). Similarly what Bob observes when he measures “b” is denoted B(b, lambda). The “lambda” outcome for a particular experiment is the same for Alice and Bob.
Thus for fixed *a*, the function which assigns to outcome lambda the number A(a, lambda) is a random variable on the probability space. I shall denote this random variable A(a), and similarly for the other three A(a’), B(b), and B(b’). I emphasize that these are all random variables on the same probability space, as are sums and products of them. This is a *hypothesis* of Bell’s theorem. Nobody claims that it is true in the real world. (The object of Bell’s theorem is to show that it is *not* true in the real world.)
The expectation of a random variable X will be denoted E(X). Bell’s argument starts with a simple algebraic manipulation showing that for any lambda,
A(a, lambda) B(b lambda) + A(a lambda) B(b’, lambda) + A(a’, lambda) B(b, lambda) – A(a’, lambda) B(b’, lambda)
is between -2 and +2. Everyone agrees that this is correct, including you and Dr. Christian. Taking the expectation of this expression and using the fact that the expectation of a sum of random variables is the sum of the expectations, i.e., E(X + Y) = E(X) + E(Y), shows that
E(A(a)B(b)) + E(A(a)B(b’)) + E(A(a’) B(b)) – E(A(a’)B(b’)) (1)
is between -2 and 2. This is a valid operation because all of the random variables are defined on the same probability space. The bounds of -2 and 2 for (1) is the conclusion of Bell’s theorem.
Of course, if (1) is between -2 and 2, it is also between -4 and 4 ! But the bounds of -4 and 4 are not “tight” bounds, in physics jargon. (In mathematics, we say that the bounds of -4 and 4 are not attained.) Christian’s “simple proof” would be correct if he also showed that the bounds of -4 and 4 are attained. But that cannot be proved under the hypothesis that all the random variables are defined on the same probability space. (Actually, the -4,4 bounds can’t even be attained in quantum mechanics; the tight bounds under that assumption are
+- 2 sqrt(2). )
I hesitate to dwell too much on Christian’s mistake, but it is a mistake that is constantly repeated on Christian’s “home forum” SciPhysicsFoundations. It is not a simple slip. It is a mistake which Jay Yablon seems to make in his equation (1) of the document he quotes assuming that his bounds for that equation are assumed “tight” (as the rest of his argument seems to assume). So I shall conclude by summarizing:
Assuming the usual hidden variable model of “local realism”, the “tight” bounds for (1) are -2 and 2. Assuming quantum mechanics but not local realism, they are +- 2 sqrt(2). This shows that quantum mechanics is incompatible with local realism. It does not show that Bell’s theorm (which concludes the -2,2 bounds) is incorrect.
I has no *physics* consequences at all, because Bell’s theorem is not a physical theory, in the sense that it can be falsified by some experiment. It is rather a “meta-result”, it deduces some limits on what correlations a particular class of theories (the local hidden variable ones) can predict. Bell could be sitting in a prison cell his entire life without knowing any results of physical experiments, and still be 100% sure of the correctness of his theorem.
A nice way of stating Bell’s theorem is to imagine three algorithms, one called “source”, the two other called “Alice-wing” and “Bob-wing” respectively. At regular intervals “source” should send some data to “Alice-wing” and “Bob-wing”, and these two should then take an additional random input (called “detector setting”), and together with the data from “source” produce either +1 or -1 as output.
The algorithms could be anything, as long as they adhere to the above rules. Now Bell’s theorem says that there are restrictions on what correlations such a setup can produce. You could just as well regard it as a theorem in Computer Science. For the theorem itself, physical experiments are completely irrelevant.
Interesting. So why do you think physicists have spent hundreds of thousands of dollars on experiments over the past 50 years testing the predictions of Bell’s theorem in contrast to those of quantum mechanics, and each time a new such experiment is performed it gets published in journals like Physical Review Letters and Nature, with renewed speculation of a Nobel Prize? Why were Bell’s papers then published in the physics journals in the first place?
They are not testing the predictions of Bell’s theorem, because Bell’s theorem doesn’t make any predictions about what correlations Nature generates. What they do demonstrate, is that the correlations Nature generates are outside the bounds of what the computer algorithms in my previous reply could generate.
And by the way, nothing Bell related has ever been awarded a Nobel prize.
Bell’s theorem does explicitly predict that Nature would generate only linear correlations.
In the first ever experiment performed to test this prediction of Bell’s theorem in 1972, the lead experimenter John Clauser eagerly expected his experiment to confirm the prediction of Bell’s theorem and revolutionize physics (I learned about this long ago from my mentor Abner Shimony). But instead Clauser’s experiment confirmed the predictions of quantum mechanics and both surprised him and dashed any hopes he had of revolutionizing physics.
It is good that nothing Bell related has been awarded a Nobel Prize, because it is evident from my local-realistic model that Bell’s theorem is false.
The consequence is that quantum mechanics is incompatible with determinism and locality. You can stick with determinism and give up on locality, like the Bohmians, or stick with locality and give up on determinism, which is the standard choice done by most physicists.
Logically speaking, one could also give up on quantum mechanics to stay with determinism and locality, or stay with quantum mechanics while giving up both determinism and locality, but I don’t know anyone that takes these options.
None of these options are necessary, as I have shown: https://arxiv.org/abs/1405.2355
You still haven’t answered me whether you think the expectation value is linear or not. Until you do so, I’ll regard your work as merely logically inconsistent.
No problem.
No, it certainly doesn’t. The theorem mathematically demonstrates that LHV-models can generate only linear correlations. That is something completely different.
I agree with Clauser on this. As you know, I have agreed with you only once, and it was not about physics.
Good idea Joy, let’s walk this through a step at a time, so I can pinpoint whatever I may be misunderstanding, if anything. But let me write this as E(A(a)*B(b)) because A = +1 or -1 and B = +1 or -1 are the actual measurements and they depend upon the detector orientations a, b, and let me assume that a and b are in fact what were chosen and used by Alice and Bob, not a’ or b’. And to simplify I will just use the binary + and -. I am leaving out any mention of a hidden variable, for the moment, because we’ll just look at this for now as a coin toss problem.
There are four possible results from one experiment, given that a and b are the actual orientations used: (A,B)=(+,+), (+,-), (-,+), (-,-). Were I to do this multiple times the *expected value* is zero, but the *actual average* need not be zero, but could range from -1 to +1 with the highest probability result being zero. Just like 7 is the expected value for two dice rolls. But I might do this ten times and end up with an actual average of 8 or 6, and less likely, 9 or 5, and theoretically possible but astronomical small, 2 or 12 each of which has a 1 in 36^10 likelihood. So we must really be clear whether we mean “expected value” which is a theoretical “most likely” average or the “mean,” or “actual average” following real runs which are centered about the average / mean but which may vary from the mean, which variation is captured by a “variance” (my, what a clever name 🙂 ). So in one run A(a)*B(b)=+1 has a 50% likelihood and A(a)*B(b)=-1 has the other 50%. And for multiple runs the average can run from -1 to +1, and will most likely be zero, but if lightening strikes and I beat astronomical odds, in theory the average could be as low as -1 and as high as +1. Nice little “Bell” curve. (Oh, the puns just keep coming. 🙂 )
Now, I expect your next step will be to say, good, then just do that three more times for E(a, b’), E(a’, b) and E(a’, b’), get a -1 to +1 range for each, then add those all up using
E(a, b) + E(a’, b) + E(a, b’) – E(a’, b’) (1)
and notwithstanding the minus sign you will get to a range of -4 to +4.
But where I am stuck whenever I stare at 1, is on the fact that E(a, b’)=E(A(a)*B(b’)) for example, is not independent of E(a, b)=E(A(a)*B(b)), because they each contain A(a). So if A(a)=+1 for a single run, then we must use the same A(a) in E(a, b’)=E(A(a)*B(b’)) that we used in E(a, b)=E(A(a)*B(b)). And if I apply those cross-constraints to all four terms, then the minus sign for the last term seems to limit me to +/-2.
Let me also do it this way: Just calculate
AB+A’B+AB’-A’B’ (2)
for all possible 16 combinations where each of A, B, A’, B’ = +/- 1. If I want to reach +4, then I must have A=1 and B=1 to get +1 from the first term. Then I must have A’=1 to get a 1 from the second term and B’=1 to get a 1 from the third term. Therefore A’B’=1 and the final term must subtract 1, so I only get a 2 not a 4. It is the interdependence between the four terms that creates a narrower “prison” for the sum, and which kept me awake till 3 AM last night. And I just don’t see how you get past that will multiple runs or taking averages either, so long as all four terms are interdependent.
Jay
“But where I am stuck whenever I stare at 1, is on the fact that E(a, b’)=E(A(a)*B(b’)) for example, is not independent of E(a, b)=E(A(a)*B(b)), because they each contain A(a). So if A(a)=+1 for a single run, then we must use the same A(a) in E(a, b’)=E(A(a)*B(b’)) that we used in E(a, b)=E(A(a)*B(b)). And if I apply those cross-constraints to all four terms, then the minus sign for the last term seems to limit me to +/-2.”
Good place to get stuck. So my next question to you is: Why are you using the same index k for both E(a, b’)=E(A(a)*B(b’)) and E(a, b)=E(A(a)*B(b)) [you have not actually shown the index k, but surely you have it (or you should have it) in your mind’s eye]? After all, I could do the experiment E(a, b’)=E(A(a)*B(b’)) today in Oxford, and you could do the experiment E(a, b)=E(A(a)*B(b)) tomorrow in New York. So should we both be using the same index k?
OK, Richard, first see the reply I posted to Joy a few minutes ago, November 1, 2016 at 1:31 pm.
Then one question for now to you and everyone else, and I will study the rest for later:
Is it true that you and Joy and Bell and everybody else here concurs that:
-4<E(a, b) + E(a’, b) + E(a, b’) – E(a’, b’)<+4 (1)
and not +/-2 are the outer territorial boundaries of the CHSH "prison"? Even though of course the statistical probabilities to land inside the -/+ 2 boundaries rather than outside are much higher? Is the ANYONE watching who believes that the ranges from -4 to -2 and from 2 to 4 are IMPOSSIBLE to attain, and not simply much less likely to attain?
If everyone else here agrees that the range in (1) really is from -4 to +4, then I am willing to stipulate that to be so, even if I still don't understand it well enough to see what you all see, and because I do not feel right about holding up this discussion while you all help me figure this out.
But if that is the case, then I have to go to Option 2 in https://jayryablon.files.wordpress.com/2016/10/bell-limits-21.pdf, and conclude that Bell made a mistake and that "top" is not equal to "middle" and that the usual rule about linear addition of averages or expectations in the probability space (which is backed up by Wikipedia, of all places 🙂 ) has some new exception that has to be explained. Or, somebody needs to tell me what is wrong with my math or logic in https://jayryablon.files.wordpress.com/2016/10/bell-limits-21.pdf.
Jay
With a finite number of runs, as you correctly observe, the range from -4 to 4 is *possible* to attain, but if the data is generated by a LHV model, it becomes less and less likely as you increase the number of runs. By choosing the number of runs large enough, the probability that the result will lie outside [-2, 2] can be made as small as you want. It seems you are on a good path here now 🙂
And, by the way, this does not mean Bell was mistaken. It means that the bounds in his theorem applies to expectations, i.e., large runs of the models.
I completely disagree with HR and Gill and all the other adherents of Bell’s theorem. The correct local-realistic (i.e., the 3-sphere model) as well as the experimental and quantum mechanical bounds on CHSH (your expression named “top”) are in fact from -2root2 to +2root2, which have the range smaller than -4 to +4 but larger than -2 to +2. The range of -2root2 to +2root2 is achieved in my 3-sphere model both analytically (at least in three different derivations) and in several different types of numerical simulations, based on geometric algebra and otherwise. This definitely means that Bell’s theorem has long been refuted, at least in my work. I am well aware that my work has not been universally accepted, but that is not because of any scientifically valid reason. Needless to repeat, both quantum mechanics and experiments also agree with the bounds -2root2 to +2root2.
I’m afraid you might have misinterpreted Richard Gill. The bounds -2 and 2 are valid for the case of expectation values, like the ones you have written, i.e., E(a, b) + E(a’, b) + E(a, b’) – E(a’, b’).
What he is saying is that if you don’t take an expectation value, but instead calculates the average for a finite number of runs, then it is true that the bounds are -4 and 4, with probability very high of being between -2 and 2.
If the true value of E(a, b) + E(a’, b) + E(a, b’) – E(a’, b’) is +2, the observed value could just as well be a bit larger as a bit smaller. One can say things like: if the number of runs in each of the four sub-experiments is at least N = 1 million, and local hidden variables is true, then it is close to certain that the experimentally observed value will be between -2.01 and +2.01.
Here, my “margin of error” 0.01 is 10 times 1 divided by square root of N. An error of more than 10 times the standard deviation is *very* unlikely.
OK, Joy, so let me say what I think you are saying about how I should understand all of this, and ask a) whether you agree, and b) whether other folks agree with how you believe I should understand this. Let’s put in the k now to be very explicit, and return to:
E(A(a)*B(b))_k + E(A(a)*B(b’))_k + E(A(a’)*B(b))_k – E(A(a’)*B(b’))_k (1)
And I think about E as an *average* based on actual runs, and not a mathematical expectation value a.k.a. mean — important difference. I believe everyone has agreed that if on the first k=1 run by Jay in New York, Alice and Bob choose a and b for their detector orientations, then the result will be A(a)*B(b)_1 = +1 or -1. And if I do ten runs here in New York, call them k=1:10, all choosing a and b not a’ and b’ then the *average* E(A(a)*B(b))_1:10 will be between -1 and +1, with much higher probabilities at the center of the spread. And because I have not used a’ or b’, the other three terms in (1) somehow get “ignored,” or not, for reasons that seem to be a subject of dispute here. I’ll return to that shortly.
Now it is Joy’s turn at Oxford. But he chooses a and b’. If he does ten more runs all with these orientations, call them k=11:20, then his average E(A(a)*B(b’))_11:20 will also range from -1 to +1 with the mean at zero. Now over to Richard in the Netherlands who conducts k=21:30 all with a’ and b exactly the opposite of Joy (which makes perfect sense 🙂 ) and his average E(A(a’)*B(b))_21:30 is also from -1 to +1 with zero mean. And finally, Dr. Parrot over in Beantown who is wondering why he ever got involved in all this says that I am going with a’ and b’ because I do not want to do anything the rest of you guys are doing. And his E(A(a’)*B(b’))_31:40 also runs from -1 to +1, zero mean. So now, we add all this up, and we get:
E(A(a)*B(b))_1:10 + E(A(a)*B(b’))_11:20 + E(A(a’)*B(b))_21:30 – E(A(a’)*B(b’))_31:40 (2)
which most certainly can run from -4 to +4, though a pig is likely to fly before you ever get something right at -4 or +4. Joy, is that what you are saying? And everyone else, is that what Joy should be saying?
Of course, the four parties might take a deep breath and agree to each do a mix of a, a’, b, b’ experiments, so I can borrow a few of Joy’s vector choices and Richard can trade and so can Stephen, so that each of the four parties is making a mix of choices from one run to the next and then summing their results together. After all, the sum in (2) should be invariant under the order in which the result arrived or who they came from. Then, after everybody grows tired of all this, Joy and Richard and Stephen all agree to let Jay do all of the runs, but Jay still ends up with (2). And then once I get this handoff, I can mix it up and do lots of experiments and make lots of choices from one run to the next, always ending up with something akin to (2), which I will now write with numbers i=1 and p=as large as I’d like and j, k, l, m, n, o anything I choose along the way, as:
E(A(a)*B(b))_i:j + E(A(a)*B(b’))_k:l + E(A(a’)*B(b))_m:n – E(A(a’)*B(b’))_o:p (3)
And this too will range from -4 to +4, but as I do more and more runs I will get much more of the probability situated closer to zero and further from the extremes. And I suspect you guys could even draw distribution graphs to show the narrowing toward the center as you do more and more runs and talk all about the standard deviations etc.
If everyone agrees this far (which may be its own miracle with odds close to that of observing a 3.8 result for 1000 runs), then there is the fact that (2) above “ignores” all of the following:
E(A(a)*B(b))_11:40; E(A(a)*B(b’))_1:10&21:40; E(A(a’)*B(b))_1:20&31:40; E(A(a’)*B(b’))_1:30 (4)
and that (3) likewise has terms that it “ignores” because these experiments were never done. And from what I can tell, there is a lot of disagreement about what to do with those, including what they mean, whether they can be ignored, etc.
So, stopping here for a pause, am I understanding this correctly so far?
Jay
Well, -4 to +4 is never achieved in the real world because of the geometry and topology of the physical space we live in (i.e., because of the geometry and topology of the 3-sphere).
This is not just me saying this. It is an experimental fact that the range of any correlations observed to date has never exceeded from between -2root2 to +2root2. But this is wider than between -2 to +2, and hence it goes beyond the limit set by Bell. We have been talking about -4 to +4 range for simplicity only, even though that range is clearly unphysical.
Having said that, you are now on the right track as far as I am concerned. You now know what prevented you previously from getting out of the prison of -2 to +2 set up by Bell.
You are really close. You were worried about the missing E(A(a)*B(b))_11:40. But let the total number of runs be one million times larger. The average of the first ten million A(a)*B(b) is going to be almost identical to the average of the next thirty million. It doesn’t matter whether or not those measurements were actually made: the values we would have observed if we had done them, still exist – that’s what local hidden variables is saying.
Jay, you are stuck because you are still missing *probability*.
You are still not making use of the law of large numbers, and you are still not making use of the definition of expectation value. Without them, you stay stuck.
Let those sample sizes converge to infinity, apply the law of large numbers (probability theory), use the definition of “local hidden variables” and the definition of expectation value (probability theory).
Result: the expectation value E(a, b) equals the integral over lamdba of A(a, lambda) B(b, lambda) rho(lambda) d lambda.
Now (only now!) add three such integrals and subtract the fourth.
The rest should be plain sailing.
I do not have much else to add, apart from the EPR condition:
The assumption of local hidden variables is the assumption that A1, A2, B1, B2 – which are simultaneous elements of reality – are all random variables defined on the same probability space.
It is important to distinguish between “bounds” and what physicists call “tight” bounds.
(Mathematicians say “bounds which are attained”.) For example, to say that “the variable X is bounded by -4 and 4) does not imply that it could not also be bounded by -2 and 2. To say that “X is tightly bounded by -2 and 2” means that X is bounded by -2 and 2 and also can sometimes be equal to -2 or to 2.
I interpreted your argument as written, but let’s look at it again assuming that you meant “tightly bounded” instead of just “bounded”. In that case, to make your argument valid,
you would have to show that your CHSH sum
E(A1B1) + E(A1B2) + E(A2B1) – E(A2B2) = E(A1B1 + A1B2 + A2B1 – A2B2)
is *tightly* bounded by -4 and 4. But it’s not, under the assumption of local realism (i.e.,
that the response functions A(a, lambda), etc. are all defined on the same probability space).
It also is not tightly bounded by -4 and 4 under the weaker assumptions of quantum mechanics. Do you think you can you show otherwise?
I am going to let that settle in overnight, and I may need to slow down a bit here because I have to write a patent for a client over the next few days. (A nifty electromagnetic device.)
But let me infer where this is heading. Richard, I know you support Bell who rules out hidden variables, so I have to assume that you will register an objection to using anything about results from experiments that might have been done but were not actually done? And if so, then I feel like I just walked through a mirror, because a couple of days ago — when it came to a single run — Joy was arguing, using the words “incompatibility” and “meaningless,” that you cannot take account of an experiment that might have been done but was not done. So I feel like I have heard you guys each arguing both for and against hidden variables in different circumstances!? So are we into scientific philosophy about how one accounts for events that might have occurred but never did, based on the probability of what would have happened if those events had occurred? For example, I did not flip a coin 100 times tonight. But I know that if I did, the chance is 2^100 – 1 out of 2^100 that I did not get heads 100 times. But how does that weave into a history of what actually happened in the real physical world in New York on November 1, 2016?
If we are about to cross a line from pure math and physics calculations and results into how we *interpret* our math and physics results and account for what did happen versus what could have hypothetically happened on a probabilistic basis but didn’t, that is fine, but I want to know (and us all to acknowledge and know) that this is what comes next.
Also, I also saw a post the past hour from Dr. Parrott talking about “bounds” and “tight bounds.” If we add in the above hypothetical results for experiments that did not occur but might have, to those for experiments that did occur, it seems we will alter the probability distributions in some way (or maybe not if there are large numbers?). Do these “road not taken” experiments bear a relation to what Dr. Parrot has interjected tonight? (And by the way “bounds” is a good term to use when we discuss all of this. Better than prisons. 🙂 ) 1 is greater than 0 and less than 2, and you can choose an infinite quantity of other numbers which are less than or greater than 1. But 1 is bounded from above and below by itself, only.
Jay
If you think that this is Joy’s mistake then it *should* also be a mistake in (1) of my document at https://jayryablon.files.wordpress.com/2016/10/bell-limits-21.pdf because my purpose was to re-present Joy’s argument as clearly as possible so it could be evaluated by me and everyone else. One cannot assess somebody else’s work until we step into their viewpoint to understand what they are really trying to say. This was and is presented with the goal of advancing your earlier suggestion which I have fully supported that we focus in on whether Bell made a mistake, before we go over to spending time on Joy’s model which assumes Bell to have been mistaken.
I will leave it to Joy to argue the substance from here as he undoubtedly will when day comes back to Oxford, because it is his argument about Bell, and the retraction of his paper, that we are all trying to assess.
Jay
PS: Administrative point re “This is not an efficient forum in which to answer your question because most readers will not have a copy of your document.” If I (or anyone) provided a working link as I did, then you may assume that the readers all of whom are smart and capable people can download and see that document. And replies to date from multiple people have made clear that they do have a copy. So do not worry too much about that. I assume that most if not all find it easier to look at visually-rendered equations rather than ASCII / latex coding of equations.
Thanks, Jay. As you have asked, let me address to the above claim by Stephen Parrott of my supposed mistake in some detail. To avoid any confusion, let me note that Jay’s equation (1) is my equation (D3) from the Appendix D of this paper:
https://arxiv.org/pdf/1501.03393v6.pdf .
In Gill’s notation, my equations (D3) says that the bounds on the CHSH sum of averages,
E(A1B1) + E(A1B2) + E(A2B1) – E(A2B2) …. (1) ,
is in fact -4 to +4, and not -2 to +2 as claimed by the adherents of Bell’s theorem. This is not a mistake (or confusion, or misunderstanding) on my part at all, but an explicit claim of mine. Here, for simplicity, I am ignoring the fact that the actual local-realistic physical bounds on the above CHSH sum is in fact -2sqrt(2) to +2sqrt(2), provided we do not ignore the geometry and topology of the physical space we live in. Thus, it is my claim that the experimentally observed bounds of -2sqrt(2) to +2sqrt(2) “violating” the Bell-imposed limit of -2 to +2 has nothing to do with quantum entanglement, or non-locality, or non-reality, or lack of determinism, but has to do with the geometry and topology of the 3-sphere. Again, this is not a mistake on my part but an explicit claim, with abundance of evidence presented for it, sometimes with the kind help from others.
Now let me turn the argument presented in the Appendix D of my paper upside down to show, in a different way, where the bounds of -4 to +4 come from, even though physically only -2sqrt(2) to +2sqrt(2) is attainable for the reasons mentioned above. Let me now start with equation (D14), the last equation of my Appendix D. Again in Gill’s notation, it is the single average
E(A1B1 + A1B2 + A2B1 – A2B2) …. (2)
Everyone agrees that the bounds on this single average are from -2 to +2. However, as I have demonstrated in my Appendix D, physically this single average is an absurdity. It is an average of events that cannot possibly exist in any possible physical world, classical or quantum. Therefore anything derived or inferred from the above single average, such as the bounds of -2 to +2, are also an absurdity. Indeed it is not surprising that the bounds of -2 to +2 are “violated” in the actual experiments, and I have claimed that they will also be “violated” in my proposed classical experiment involving exploding toy balls.
Everyone also agrees that the equality between (1) and (2) is a mathematical identity:
E(A1B1 + A1B2 + A2B1 – A2B2) = E(A1B1) + E(A1B2) + E(A2B1) – E(A2B2) …. (3)
Therefore the adherents of Bell’s theorem claim that the bounds on (1) are also from -2 to +2. But how can that be? As I have demonstrated in my Appendix D the bounds of -2 to +2 are derived or inferred from considering an average of physical events that are impossible in any possible world. Even the God of Spinoza cannot make them possible. There for the claims of the bounds form -2 to +2 on the CHSH sum (1) are absurd, even if they are not mathematically wrong.
The question then is, what are the correct bounds on the CHSH sum (1) if not -2 to +2?
Well, I claim that if, for simplicity, we ignore the geometry and topology of the physical space, then the correct bounds are -4 to +4, which are in fact attainable at least in a computer simulation such as this one: http://rpubs.com/jjc/84238 . In fact it is quite easy to verify that the bounds are -4 to +4 by a simple observation, as done in the introduction of my Appendix D. All one needs for this is the assumption that the summation index used for the four separate sums in equation (1) above is not the same, as Jay verified earlier. Each of the four averages ranges from -1 to +1, and therefore CHSH sum can range from -4 to +4.
But what about the mathematical equality (3) that everyone (apart from me, at least) is so keen to exploit. Well, it is just a mathematical equality that leads to a physical absurdity. Once the summation indices on the four averages in the CHSH sum (1) are different from each other, the equality (3) no longer holds, as verified by Jay earlier.
_____________________________________________________________________________
Comment by Stephen Parrott starts here.
I apologize that my previous reply (which is the same as this one) came out with my remarks not distinguished from Dr. Christian’s quote. I think that in trimming Christian’s post, I may have inadvertently deleted a closing “blockquote” HTML tag. I hope this one prints correctly.
I think we are potentially very close to agreement ! If I interpret the above correctly,
Dr. Christian agrees that Bell’s theorem is mathematically correct but somehow doesn’t apply to actual experiments.
His objection seems to be that in actual experiments which approximate mathematical expectations like E(A1, B2) by observed averages, the various expectations (such as E(A1, B2) and E(A2, B2) ) need to be approximated in *different* experiments because the world is
actually governed by quantum mechanics, which does not allow simultaneous measurment of A1, and A2. This is true.
Because of that the *experimental approximation* to a CHSH sum like
E(A1, B1) + E(A1, B2) + E(A2, B1) – E(A2, B2)
indeed could be larger than Bell’s bound of 2, even though the sum as just written (which involves mathematical expectations, not experimental approximations to them) cannot exceed 2.
However, by taking sufficiently large sample sizes, the probability that the experimental approximations for the above sum exceed 2 + epsilon (for given positive epsilon) can be made arbitrarily small. (Richard Gill has already emphasized this.) If Dr. Christian disagrees with this, I hope he will say so clearly, because it would further the discussion.
If that is the only objection to the observed experimental violation of Bell’s inequality, then it boils down to the question of whether the sample sizes were large enough to make the probability of violation small enough. That would require detailed analyses of the experiments. Normally, the experimenters would furnish such analyses (often in the form of statement that the results obtained lie a stated number of standard deviations from the result predicted by a local realistic model).
If there is some other objection to Bell’s conclusion or to claims that experiments violate it (so that a local realistic model is ruled out, up to some small probability that the violation is a statistical fluke), then it has not been expressed clearly enough to
penetrate this thick skull.
If the believers that a local realistic model can explain experimental results want to convince others, they would be well advised to think carefully about how to present their objections so that they cannot be misunderstood. So far as I can see, there are at most six such believers, all headquartered at SciPhysicsFoundations, where they convince each other that they are clear-thinking individuals who could revolutionize physics if only they could get others to listen.
Many *are* willing to listen, but are not willing to invest their time when the probability of a productive outcome seems too small. In particular, one participant often answers specific questions which should have short, simple answers, by referring to such and such of his numerous arXiv postings, which many find so obscurely written that they would require a major investment of time to dissect in detail. If he wants to convince others, he would be well advised to seek more user-friendly ways to present his views.
I completely disagree with your reading of my comments. What I am saying is that the claim by Bell and his followers that the bounds of -2 and +2 on the CHSH sum of expectation values cannot be exceeded by any local-realistic theory is simply wrong. In my comments I have explained precisely why their claim is wrong. Specifically, since the events like A(a1) and A(a2) cannot possibly exist simultaneously in any possible physical world — classical or quantum — Bell’s argument is based on absurdity. Consequently the correct local-realistic bounds on the CHSH sum of expectation values are not -2 to +2 as erroneously claimed by Bell and his followers, but are in fact -4 to +4 (or -2sqrt(2) to +2sqrt(2) if we do not ignore the geometry and topology of the physical space we live in). Is that now clear enough?
The fact that the different measurement arrangements cannot be run simultaneously is irrelevant and does not refute the validity of CH/CHSH. I show that in section 2.3 of https://arxiv.org/abs/1404.4329.
The argument presented in the paper by Graft amounts to using a completely different inequality — one with the bounds of -4 to +4 — when comparing actual experimental results, thus abandoning the bounds of -2 to +2 derived theoretically, and then claim that Bell-CHSH inequality with the bounds of -2 to +2 has been “violated”, when in fact no such thing has happened, or shown to happen.
JC: “The argument presented in the paper by Graft amounts to using a completely different inequality…”
That is false and nonsensical. The paper uses the CH inequality throughout. I clearly showed that incompatibility of experiments is irrelevant.
That changes nothing. Just like the CHSH inequality, the CH inequality too is based on an absurd proposition that A(a1) and A(a2) can exist at the same time. Thus it too is subject to the same criticism I have made in my comment about the CHSH inequality. There is no way to wiggle out of the impossibility of simultaneously observing the events A(a1) and A(a2).
It’s not clear to me. The *hypotheses* of Bell’s theorem assume a mathematical model based on a probability space from which a “hidden variable” lambda is drawn. The response functions like your A(a1), A(a2), etc. are assumed to be random variables defined on that probability space. (They are not “events” as defined in probability theory, but I assume that your use of this incorrect terminology was simply a slip. I assume you meant “random variables instead of “events.) Any collection of random variables on a probability space can be simultaneously measured.
That’s rather long-winded, so let me summarize: You say “since the events like A(a1) and A(a2) cannot possibly exist simultaneously in any possible physical world … , Bell’s argument is based on absurdity”. But Bell’s argument does *not* claim that A(a1) and A(a2) can be simultaneously observed *in the real world* (as opposed to in the hypothetical world of the mathematical model). The question is whether the hypothetical model can reproduce the correlations that we see in the real world. The conclusion of Bell’s theorem is that it cannot.
That said, I certainly can imagine a real world in which A(a1) and A(a2) could be simultaneously observed. That is the “classical” world that I believed in before learning about Bell’s theorem. I can only scratch my head at your statement that “events like A(a1) and A(a2) cannot possibly exist simultaneously in any possible physical world” . I cannot imagine what could lead to such a belief.
I am going to leave this discussion. I think that everything that can be said has already been said.
A(a1) and A(a2) are possible events in spacetime. They are clicks of the detectors located in spacetime. They cannot possibly exist simultaneously in any possible physical world, because a1 and a2 are mutually exclusive directions in the physical 3-space. Since (as you say) you do not understand this, it is unlikely that you will see the flaw in Bell’s logic I have exposed.
Thanks Richard, let me see if I can restate what you are saying and you tell me if I now have this correct. Maybe since it is on everybody’s minds these days, I will work from the example of polling use to predict presidential election outcomes.
We all know that as we increase the “scientifically-selected” sample size of any poll we will get closer to the actual results and reduce the “margin of error.” Same thing when we are “polling” large numbers of coin tosses of Alice and Bob experiments.
So let me go back to my earlier (E is an *average” observed following actual experimental runs, and not an expected value for this runs):
E(A(a)*B(b))_1:10 + E(A(a)*B(b’))_11:20 + E(A(a’)*B(b))_21:30 – E(A(a’)*B(b’))_31:40 (1)
But now, the numbers in the k index such as 1:10 will represent how many *millions* of runs are taken. Once I have recorded E(A(a)*B(b))_1:10 for ten million runs which were actually conducted with a, b, and even if I never conduct experiments E(A(a)*B(b))_11_20 for the a, b orientation combination because I have instead conducted E(A(a)*B(b’))_11:20 in (1) above for the a, b’ orientation, there were so many trials run in E(A(a)*B(b))_1:10 that I can extrapolate that *if I had done* another ten million runs with a and b, the averages observed from the hypothesized E(A(a)*B(b))_11:20 *would have been* very close to the actually-observed E(A(a)*B(b))_1:10 from the experiments that were conducted. So I may then use this statistical extrapolation to include the terms that would have been there had I done the experiments I never did, because I now have enough data to extrapolate to those, and so may make use of the very close approximation (~=):
HE(A(a)*B(b))_11:20 approximately ~= E(A(a)*B(b))_1:10 (2)
Above, I have use the notation “HE” to denote the “hypothesized” average of experiments k=11:20 for a, b that were not conducted, and plain old E to represent the average for the experiments k=1:10 which were actually conducted for a,b. Likewise for all the other experiments that were not done for all a, b, a’, b’ combinations.
So using (2) generalized to all detector orientation combinations, (1) above, which I will separate into two parts to highlight (3a) the experiments which were done and (3b) the experiments which are hypothesized statistical extrapolations but were never done, will be:
E(A(a)*B(b))_1:10 + E(A(a)*B(b’))_11:20 + E(A(a’)*B(b))_21:30 – E(A(a’)*B(b’))_31:40 (3a)
–plus–
HE(A(a)*B(b))_11:40 + HE(A(a)*B(b’))_1:10&21:40 + HE(A(a’)*B(b))_1:20&31:40 (3b)
– HE(A(a’)*B(b’))_1:30
But because
E(A(a)*B(b))_1:10 ~= HE(A(a)*B(b))_11:40 ~= E(A(a)*B(b))_1:40
E(A(a)*B(b’))_11:20 ~= HE(A(a)*B(b’))_1:10&21:40 ~= E(A(a)*B(b’))_1:40 (4)
E(A(a’)*B(b))_21:30 ~= HE(A(a’)*B(b))_1:20&31:40 ~= E(A(a’)*B(b))_1:40
E(A(a’)*B(b’))_31:40 ~= HE(A(a’)*B(b’))_1:30 ~= E(A(a’)*B(b’))_1:40
I can extrapolate (1) very closely to:
(E(A(a)*B(b))_1:10 + E(A(a)*B(b’))_11:20 + E(A(a’)*B(b))_21:30 – E(A(a’)*B(b’))_31:40)
~=
E(A(a)*B(b))_1:40 + E(A(a)*B(b’))_1:40 + E(A(a’)*B(b))_1:40 – E(A(a’)*B(b’))_1:40 (5)
Now, I know from the previous discussion that I think everyone has agreed upon, that the bounds of (1) are -4 and +4, i.e. that:
-4 le E(A(a)*B(b))_1:10 + E(A(a)*B(b’))_11:20 + E(A(a’)*B(b))_21:30 – E(A(a’)*B(b’))_31:40 le +4 (6)
Therefore, I also know that the bounds on the bottom line of (5) are the same:
-4 le E(A(a)*B(b))_1:40 + E(A(a)*B(b’))_1:40 + E(A(a’)*B(b))_1:40 – E(A(a’)*B(b’))_1:40 le +4 (7)
So then I can just go back to using k and simply write (7) above as:
-4 le E(A(a)*B(b))_k + E(A(a)*B(b’))_k + E(A(a’)*B(b))_k – E(A(a’)*B(b’))_k le +4 (8)
This equation is then true for a very large number of runs (which is reminiscent of, and seems to me to actually be one example of, the correspondence principle ). But we need to be careful when we are talking about smaller numbers of runs, because then the laws of averages do not help as much. So (with game 7 of the World Series coming up tonight), although classically we can talk about the position and momentum of a baseball “simultaneously” because “k” is a very large number, when we start to talk the same way about single electrons or even a collection of a modest number of electrons, we need to watch out.
Then, if I use the identity that the sum of the averages equals the average of the sums in the same probability space, I can turn (8) into:
-4 le E[A(a)*B(b) + A(a)*B(b’) + A(a’)*B(b) – A(a’)*B(b’)]_k le +4 (9)
And then, if I do the Bell factorization, this becomes:
-4 le E[A(a)*[B(b) + B(b’)] + A(a’)*[B(b) – B(b’)]]_k le +4 (10)
And then for some odd reason, because B(b) + B(b’) is bounded by +/-2, if B(b) + B(b’)=2 then B(b) – B(b’)=0 and we appear to have a contradictory range for (9)=(10) which is:
-2 le E[A(a)*B(b) + A(a)*B(b’) + A(a’)*B(b) – A(a’)*B(b’)]_k le +2 (11)
and which contradicts (9). And this is where are all running into trouble because there are different views about how to sort this out these tighter and looser bounds .
Do I have it now?
If I do, perhaps where you and Joy and the schools of thought you represent are disconnected is that you are conflating different ends of the spectrum of the correspondence principle. At the very least, from here on, every time somebody posts or talks about one of these Bell equations, they should be clear whether it holds only for huge numbers of runs, or whether it remains valid even for a small number of runs, and even for a single run. In short, perhaps we need to all get in the habit in these Bell discussions, as we do in many other quantum discussions, of being very explicit about what applies to single “quanta” (here, single “runs”) and what only applies in a classical sense to very large numbers of runs.
Specifically, some of these equations may break down for individual runs, but hold up well in the classical limit of large numbers of these runs. And others may be valid in all situations. And if I had to guess right now, I’d say that (8), (9) and (10) are classical equations which break down in the quantum limit of small numbers of runs, and that (11) continues to apply even in the quantum limit of a single run. And I would say that Dr. Parrot’s tight bounds characterize physical reality in the quantum limit of individual EPR experiments and his loose bounds are permitted in the classical limit of huge numbers of EPR experiments. And since Joy is relying on the looser bonds, I’d say that is exploiting a correspondence principle feature of the classical limit for large numbers of runs. Then, we need to process what becomes of hidden variables and local realism in the classical and quantum end of the correspondence. Hidden variables and local realism are excluded in the quantum limit but are permitted in the classical limit? Now I am just thinking out loud again.
But I believe that frames the issues: Niels Bohr. Correspondence Principle. Apply to EPR and Bell.
jay
You asked me if I think you’ve got it right now. Answer: yes.
You’ve essentially given us a novel proof of Bell’s theorem.
There are no contradictions between tighter and looser bounds. People should indeed make clear whether they are talking about finite sample averages or infinite population means.
There is no correspondence principle at work here. We are not assuming quantum theory. Bell investigates what are the logical consequences of the mathematical assumption of local hidden variables. He is talking about theoretical mean values. Not about sample averages.
And the same old mistake in the “novel proof” by Jay lies in this step:
-4 le E[A(a)*[B(b) + B(b’)] + A(a’)*[B(b) – B(b’)]]_k le +4 (10)
On what basis A(a) has been factored out for two different B(b) and B(b’)? Does Bob live in an impossible fantasy world were he can align his detector along b and b’ at the same time while Alice aligns her detector along a? Does Bob live in an impossible fantasy world where he can be in New York and Miami at exactly the same time?
The “novel proof” by Jay is based on exactly the same absurdity as the same old proof of Bell, and therefore the bounds -2 and +2 on CHSH sum derived by Jay are equally meaningless curiosities. It is then no surprise at all that they are frequently “violated” in the real world.
Joy, just to be very clear so we can pinpoint exactly where you and Richard start to have divergent points for view, are you saying, using the numbering in my post on November 2, 2016 at 11:39 am, that
-4 le E[A(a)*B(b) + A(a)*B(b’) + A(a’)*B(b) – A(a’)*B(b’)]_k le +4 (9)
is a correct result, but that:
-4 le E[A(a)*[B(b) + B(b’)] + A(a’)*[B(b) – B(b’)]]_k le +4 (10)
is not?
Thanks,
Jay
No, Jay. (9) and (10) are of course the same. But (10) brings out the absurdity of the Bell-type manipulation very clearly. When you factor out A(a) and A(a’) from the sum and difference of B(b) and B(b’), and write a product like
A(a) * [B(b) + B(b’)] ,
then it becomes more obvious than in your previous equations that such a product is physically absurd. It is not OK to pretend that you (or Bell, or Gill) are just taking some intermediate mathematical steps, precisely because of the identity everyone loves; namely
E(A1B1 + A1B2 + A2B1 – A2B2) = E(A1B1) + E(A1B2) + E(A2B1) – E(A2B2).
This identity says that if what you end up getting on its LHS turns out to be physically absurd, then what appears on it RHS is also physically absurd. It is an absolute equality. Therefore there is no way out of this conclusion.
Now you were able to avoid this trap in your equation (1). But then you found your back into it in your equation (10). How did that happen?
Richard, I want to make sure I understand and can articulate exactly what you are saying here about the infinite population.
Ate you saying that for any *finite* number of samples, the results will deviate from the mathematical probability density functions by some finite “error” owing to randomness, which error grows smaller as the sample size grows larger? And that statistics regards the situation where an *infinite* number of experiments are performed to be equal (perhaps by definition because nobody can ever run an infinite number of experiments?) to a situation in which the empirical results do match the theoretical results exactly? For example, I can roll a pair of dice 36 times, knowing that the 2 through 12 have a theoretical likelihood of 1,2,3,4,5,6,5,4,3,2,1 out of 36 respectively, but find that in a given run of 36 I actually got 0,2,3,4,5,7,4,4,3,2,2, which deviate from the ideal spread at the 2, 7, 8, 12 positions. But were I to do an infinite number of runs, the distribution of results for the infinite sample is *defined* to have no error from the theoretical spread, so as to match 1,2,3,4,5,6,5,4,3,2,1 over 36 exactly, by definition. Is that what you mean by that language about an infinite population? Jay
The law of large numbers is a 300 years old *theorem*, not a definition. https://en.wikipedia.org/wiki/Law_of_large_numbers
It happened between my (6) (which sets (1) between the -4 and +4 bounds):
-4 le E(A(a)*B(b))_1:10 + E(A(a)*B(b’))_11:20 + E(A(a’)*B(b))_21:30 – E(A(a’)*B(b’))_31:40 le +4 (6)
and my and (7):
-4 le E(A(a)*B(b))_1:40 + E(A(a)*B(b’))_1:40 + E(A(a’)*B(b))_1:40 – E(A(a’)*B(b’))_1:40 le +4 (7)
because I included the “hypothetical” experiments which never actually happened via my (5):
(E(A(a)*B(b))_1:10 + E(A(a)*B(b’))_11:20 + E(A(a’)*B(b))_21:30 – E(A(a’)*B(b’))_31:40)
~=
E(A(a)*B(b))_1:40 + E(A(a)*B(b’))_1:40 + E(A(a’)*B(b))_1:40 – E(A(a’)*B(b’))_1:40 (5)
Yes?
Yes, without using the “Law of large numbers terminology,” that is exactly what I was saying. The chart at https://en.wikipedia.org/wiki/File:Largenumbers.svg in the link you posted shows how the one-die line approaches 3.5 almost exactly for large numbers, and extrapolates to be equal to 3.5 as the large number becomes infinite. Since we can never observe an infinite number of rolls empirically, we may say that the average observed value of an infinite number of rolls equals the mathematical mean, and have that be the definition of the average for N->oo.
Put differently: for a finite number of runs, and especially for a small finite number of runs, the observed average will not always be equal to the mean, but it will converge toward the mean for large numbers and become synonymous with the mean for an infinity of runs. We are saying the same thing in different ways, that’s all. I am not a fan of jargon; I always want to have the concepts behind the jargon on the table in plain sight.
I agree totally with your statement agreeing with my statement that “People should indeed make clear whether they are talking about finite sample averages or infinite population means.” And within the former, whether the finite number of samples is small or large. I certainly will do so, and I believe it is critical to do so to get a firm handle on Bell and your dispute with Joy which I think I now have, as I will explain further in some upcoming posts.
Jay
Making clear includes not using the same symbol for both. The symbol E is usually reserved for population means, so another symbol; e.g. Ê should be used for sample averages. or Êᵢ for average of iᵗʰ sample.
Here is a “live” online simulation of my 3-sphere model for the EPR-Bohm correlations:
http://libertesphilosophica.info/eprsim/EPR_3-sphere_simulation5M2.html
All you have to do at the above link is hit the button which says “Begin Simulation.” You will immediately see that it starts out randomly, and then converges to the cosine curve [which is equivalent to converging to the tighter bounds of -2sqrt(2) to +2sqrt(2)] for large numbers.
Give the simulation a few seconds to run, until the number of trials reaches 10 million.
Richard, or anybody else who knows,
I have a question about the experiments which are used to establish Bell theory correlations. This inquiry is simply a point of information:
We often talk about the number of trials as if we have the control to split one singlet into a pair of doublets and then detect that pair of doublets and whether they correlate or anti correlate. But I must assume that in an actual experiment, something gets bombarded, something splits, and we do not know how many actual detection events will be delivered to Alice and Bob. Maybe we get 10 events which are detected, maybe we get a thousand, maybe 1 million, maybe 100 million.
I would appreciate if you could please enlighten me and anybody else about what the on-the-ground in-the-lab deal actually is with these experiments and specifically the numbers of events which are detected and whether than number is controllable, or, as I assume, is whatever you are dealt by the experiment.
To be clear: I am not looking for any lengthy or elaborate discussion about the detector setups. I just want to understand the most sensible language that one can use when talking about the number of trials, in a way that reflects the reality of doing these experiments, and the control or lack thereof that one has over the detected population.
Thanks,
Jay
Nowadays the experiment is typically run for just as long as it takes to get a prescribed total number of measurement pairs (trials), where each pair has been measured with randomly chosen measurement settings. For each trial anew, Alice chooses randomly between settings a and a’, and Bob between b and b’. The four correlations are calculated from the resulting four disjoint sets of measurement pairs. The grand total number of trials is prechosen, but the numbers in each of the four subsets are determined by chance.
For each pair, Bob’s measurement outcome is determined before Alice’s setting could be known at Bob’s location and vice versa. Outcomes are binary (none are missing).
I just realized that the bounds imposed by Bell on the CHSH correlator are in fact -0 and +0.
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=289#p6975
This is becoming increaslingly silly. The expression A(a)*[B(b) + B(b’)] is not a probability.
Who said it is? What I said was A(a)*[B(b) + B(b’)] has probability zero of occurring in any possible world. Is that not clear from what I wrote? This is what I have explicitly written:
“But since b and b’ are two mutually exclusive measurement directions that Bob could have chosen corresponding to two physically incompatible experiments, no event such as A(a)*[ B(b) + B(b’ ) ] can possibly exist in any possible physical world. The probability for the existence of the events such as A(a)*[ B(b) + B(b’ ) ] in spacetime is therefore exactly zero! Consequently, the average on the RHS of the identity (1) is also zero. But if the RHS of the identity (1) is identically zero, then so is its LHS.”
What is unclear about this?
The key sentence in the link Joy provided is the following:
“Now let us try to understand what the RHS of this identity is actually telling us. It is asking us to consider *the averages• of the events such as A(a)*[ B(b) + B(b’ ]”
That seems to be a fair question. He is asking about the average value of that expression over some number of trials. Unfortunately the less than and greater than signs do not pass through to posts on retraction watch, which is why we all have to find other ways to express ourselves when we are talking about averages.
So let me ask the question: please state in direct terms and unambiguously, what is meant by the average of that expression?
Jay
Jay, you will find the precise mathematical definition of the average E at the this link:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=289&p=6978#p6978
What the above definition says is: E(event) = Sum_k (event^k) x (Prob of event^k).
The simplest LHV model you can imagine is the one where Alice and Bob always measure +1, no matter what. In that model the average is 2.
We are not talking about wrong models. We are talking about the proof of Bell’s theorem.
You can put less than and greater than in html. You have to use character codes.
http://rabbit.eng.miami.edu/info/htmlchars.html
< >
A(a)*[B(b) + B(b’)] is not an *event*. It is (according to LHV) some function of lambda, and it takes the values -2, 0 and +2. Nature picks lambda at random, so A(a)*[B(b) + B(b’)] is a random variable taking possible values -2, 0 and 2. It is totally irrelevant whether or not someone could observe it in a real experiment.
From https://en.wikipedia.org/wiki/Expected_value, “the expected value of a discrete random variable is the probability-weighted average of all possible values. In other words, each possible value the random variable can assume is multiplied by its probability of occurring, and the resulting products are summed to produce the expected value”.
I think it would be smart to forget all about averages of repeated measurements, and go straight to the large N limit, and talk about expectation values (like Bell himself did).
None of this has any relevance to the point I am making. The quantity A(a)*[B(b) + B(b’)] — which is a function of possible measurement events which could be observed by Alice and Bob — cannot possibly exist in any possible physical world, just as one cannot be in New York and Miami at exactly the same time. In other words, the probability of A(a)*[B(b) + B(b’)] ever occurring in any possible physical world is exactly zero. Now the proof of Bell’s theorem relies on the average or expectation value of the quantity A(a)*[B(b) + B(b’)]. But an average or expectation value of A(a)*[B(b) + B(b’)] very much depends on the probability of it ever occurring in any possible world, which, as noted, is exactly zero. Therefore the average or expectation value of the quantity is also exactly zero. Thus Bell’s logic necessitates that the CHSH correlator must lie between -0 and +0. There is no way out of this conclusion.
Let me illustrate the point Joy is trying to make as simply as possible, using a very explicit example. An argument in ASCII won’t do it, you need to see real equations with real data, so I created the following two pages for you all to look at: https://jayryablon.files.wordpress.com/2016/11/rw-11-5-16.pdf. Either somebody needs to explain how (7) which says 26/35=2 is a wrong calculation on my part, or if this is a correct calculation, then somebody needs to explain — step by step — how this gets fixed, because this same type of calculation used by Bell. Jay
The point I am making is very simple to understand. What is the probability of a single dice landing on both 3 and 6 at the same time? If you know the answer to this question, then you also know the answer to my question: What is the probability of A(a)*[B(b)+B(b’)] occurring in any possible physical world? The answer is the same for both questions.
I agree. At the referenced link, Christian says (and has said many times in many places)
“But since b and b’ are two mutually exclusive measurement directions that Bob could have chosen corresponding to two physically incompatible experiments, no event such as A(a)*[ B(b) + B(b’) ] can possibly exist in any possible physical world.”
I am going to try to explain why this is wrong. Since this is the *premise* for most of his arguments against Bell’s theorem, those arguments are also wrong.
First of all, I am going to ignore his incorrect terminology which calls A(a)*[B(b) + B(b’)] an “event”. It is a random variable, not an “event” (defined in all probability texts as a subset of a sample space). What he means (I assume) is that in the usual quantum-mechanical application, B(b) and B(b’) cannot be observed simultaneously. No one disputes this. But it is irrelevant to Bell’s theorem.
It is easy to imagine “possible physical worlds” in which anything observable can be observed simultaneously with anything else. Bell’s theoem *assumes* such a “classical” world and derives certain consequences such as the CHSH inequality. If the consequences like CHSH are violated in *our* physical world, that shows that our physical world cannot be the “local realistic” one assumed by Bell’s theorem. Someone asked what is the content of Bell’s theorem, and that is its content.
Here is a physical example. To understand why it is relevant to Bell’s context, it is important to understand that Bell’s theorem has nothing to do with spin, quantum mechanics, or our “real world”.
My car has two odometers (so-called “trip odometers”) to observed two different distances traveled. For example, one distance might be the distance since the last gasoline fillups, and the other the distance traveled on a given day. However, this twin trip odometer is constructed so that these two distances *cannot be observed at the same time*. A switch toggles between them.
Although in the world of my car, those two distances cannot be observed simultaneously, it is perfectly possible to imagine a car with two trip odometers which *could* be observed simultaneously.
In the world of my car, if I am driving along, and I look at at noon to see how far I have gone today, I will get one number. If at the same time I look to see how far I have gone on the current tank of gas, I will get a probably different number. There is no possible way that I can
get these two numbers at the same time. But there is nothing physically impossible about a car which *could* deliver the two numbers at the same time.
You have completely misunterstood Bell’s theorem. It is all about “wrong” models, i.e. local hidden variable models.
And to follow up on this, Bell’s theorem delineates the logical restrictions on what predictions this class of LHV models can produce. Quantum mechanics predicts results outide of these restrictions. So the conclusion is that QM does not belong to the class of LHV models.
Jay,
Someone earlier mentioned we should first go for the low hanging fruit of pointing to the error in Bell’s theorem, rather than discussing the possible merits of Joy’s theory (which we could do later). Given the recent discussions on the forum here, I think we should now lower our ambitions and go for the fallen fruit that is actually on the ground. So, can you and/or anyone please state what they think is the claim of Bell’s theorem in one simple sentence?
Something equally simple and comprehensible to how I would claim the Pythagorean theorem: “The square of the hypotenuse is equal to the sum of the squares of the other two sides.”
“No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.” from Wikipedia
Let me explain the ingredients one uses to get from my post of November 5, 2016 at 5:18 pm to the Wiki statement Fred posted:
I said “any physically-observed correlations which fall *outside* of the outer *mathematical* bounds in (2) cannot be explained by a locally realistic hidden variables theory (LRHV).”
It is widely believed, albeit disputed by Joy, that the CHSH mathematical bounds are L_m=-2 and U_m=+2. It is also observed that L_p=-2 sqrt(2) and U_p=+2 sqrt(2), which is outside these bounds, and which is understood to be a consequence / prediction of quantum mechanics.
If one takes all of this to be true notwithstanding Joy’s objection, then because the physically-observed consequences / prediction of quantum mechanics that the *physical* L_p=-2 sqrt(2) and U_p=+2 sqrt(2), fall *outside* of the CHSH bounds, those QM phenomena cannot be explained by a locally realistic hidden variables theory (LRHV).
That gets us to Wiki: LRHV cannot explain this particular QM prediction.
Jay
HR, excellent idea, let me try to do so. And BTW, it was Dr. Parrot who first suggested we go after low hanging fruit, which was also an excellent idea.
Below is my view of what Bell says, as a three week novice amidst cognoscenti. I first need some simple definitions just for clarity, so we are all using the same language:
Let me start with the “CHSH sum” which I will write with square brackets meaning “average” over some number of trials, large or small, in each disjoint set, as:
CHSH = [A(a)B(b)] + [A(a)B(b’)]+ [A(a’)B(b)] – [A(a’)B(b’)] (1)
This sum falls within certain lower (L) and upper (U) bounds, whereby:
L le CHSH le U (2)
We should also be clear that there are two sets of bounds to keep in mind: a) the mathematical (m) bounds L_m and U_m computed by mathematically using (1) in the most extreme, lowest-probability cases we might wish to hypothesize, and b) the physical (p) bounds L_p and U_p which are obtained when we do actual experiments with a sufficient population sample, harvest data, and plug the data into (1).
Now to the theorem:
“Any physically-observed correlations which fall *outside* of the outer *mathematical* bounds in (2) cannot be explained by a locally realistic hidden variables theory (LRHV).”
The contrapositive logic of this is as follows:
“A locally realistic hidden variables theory can explain any physically-observed correlations which fall *inside* the outer *mathematical* bounds in (2).
Now, from what I have read, Joy and Richard and Dr. Parrott all agree that the physically-observed correlations, when computed by slavishly plugging them into (1), have L_p=-2 sqrt(2) and U_p=+2 sqrt(2), for sufficiently large numbers of trials. The agreement by all three of these fine folks is good enough to make me accept that this is so.
The point of disagreement is that when Bell evaluates CHSH mathematically, he deduces that L_m=-2 and U_m=+2. Richard agrees with the foregoing, and uses some statistical / large numbers arguments to support his view which I will not get into here because you asked for this to be simple. But Joy believes that the Bell derivation of these bounds is in error, mathematically, and that a correct derivation would lead to L_m=-4 and U_m=+4. So where does that place us all at the moment?
If Bell and Richard are correct and L_m=-2 and U_m=+2, then the observed *physical experimental* bounds L_p=-2 sqrt(2) and U_p=+2 sqrt(2) are *outside* the mathematical ones and cannot be explained by any LRHV. So we needn’t spend one moment looking at Joy’s model because it is a “no go” theorem. That is why Dr. Parrott steered us here first, to the low hanging fruit.
However, if Joy is correct, and if L_m=-4 and U_m=+4, then the observed physical bounds L_p=-2 sqrt(2) and U_p=+2 sqrt(2) are *inside* the mathematical ones and *can* be explained by an LRHV theory. If this is so, then the question becomes: find me an LRHV theory which explains why we actually observe L_p=-2 sqrt(2) and U_p=+2 sqrt(2) and not some other numbers. And as soon as you ask that question, Joy will raise his hand and jump up and down and say “I got it!”
So “the fallen fruit that is actually on the ground” is the core question whether L_m=-2 and U_m=+2 (Bell and Gill), or whether L_m=-4 and U_m=+4 (Christian). This is a worthwhile question to resolve one way or the other, and the discussion here ought now stay focused on this single question until it can be resolved one way or the other.
But do keep in mind, the thought that L_m=-4 and U_m=+4 might be true as Joy asserts gives some people a panic attack not unlike the one I am in the middle of with the thought that a particular nameless candidate might become the US President, and what would happen to the world if he does. But science and math are science and math, and I believe in those, and in neither the Bell cool aid nor the Christian cool aid until this question presently presented about L_m and U_m is clearly resolved.
I do call on people here, as scientists, to please set aside the question of what a finding that L_m=-4 and U_m=+4 would mean for Bell (i.e., do not let panic guide your scientific sensibilities), and simply address the question, in a clinical, objective, mathematical fashion, whether L_m=-4 and U_m=+4, or whether L_m=-2 and U_m=+2. And do not wander off to any other fruit until that is done.
I hope that was simple enough, and clarifying.
Jay
Very good, but as a candidate for my own answer I would rewrite your definition to be “Any correlations which fall *outside* of the outer bounds in (2) cannot be explained by a locally realistic hidden variables theory (LRHV).” I would thus drop the qualification “physically-observed”, since the statement would be true even if these correlations (*outside* of the outer bounds) were not physically observed, but rather observed to be within the bounds, i.e, compatible with a local hidden variable theory.
I would like to add one more important point to my post at November 5, 2016 at 5:18 pm excerpted above.
The disagreement between Joy et al. and Bell & Richard et al., boils down to whether L_m=-4 and U_m=+4, or whether L_m=-2 and U_m=+2. I have suggested we take a close look at how the L_m=-2 and U_m=+2 bounds of the prevailing view are derived right down to the raw binary data, and am doing so and will continue to do so in other posts here.
But it occurred to me that my statement in the final paragraph above about “what a finding that L_m=-4 and U_m=+4 would mean for Bell” is an overstatement. So because it is always good to calm the waters and not roil them, it is important to be very clear what the consequences would be should the L_m=-2 and U_m=+2 bounds be retracted as a consequence of the discussions here.
Such a change in the prevailing view from L_m=-2 and U_m=+2 to L_m=-4 and U_m=+4 would actually not change Bell’s theorem itself one iota. As Dr. Parrot has noted earlier, Bell is just a theorem which says “if not A then not B,” and contrapositively, “if B then A.” On its own, Bell’s theorem it does not settle whether we have “A” or “not A.” Correctly determining L_m and U_m for the CHSH linear term combination is what does that.
Bell’s Theorem, which I need not remind this crowd, started with EPR, and explained the precise conditions under which local realism can, and cannot, be used to explain quantum mechanics, especially entanglement correlations. In the event that we were to validate here that L_m=-4 and U_m=+4 as maintained by Christian (and I will not quit this discussion until there is an agreement one way or the other on this question between Christian and Gill also blessed by someone of Parrott’s stature), Bell’s theorem will still remain one of the preeminent theorems of modern physics. Nobody will have “disproved” Bell’s theorem. Nothing would change on this score, because Bell is still the “input / output” machine that answers the EPR paradox. All that would change is that the input to Bell would have been found to be different.
So at the moment, the statement of Bell which HR and I agreed upon is:
“Any correlations which fall *outside* of the outer *mathematical* bounds in (2) cannot be explained by a locally realistic hidden variables theory (LRHV).”
This is the prevailing *application* of Bell today, because the observed *physical quantum mechanical* bounds L_p=-2 sqrt(2) and U_p=+2 sqrt(2) fall outside of what are presently thought to be the mathematical bounds L_m=-2 and U_m=+2. But were this to reversed in favor of Christian’s proposed L_m=-4 and U_m=+4, then it is the contrapositive of what HR and I agreed upon as a Bell statement that would go into prevailing application, namely:
“A locally realistic hidden variables theory can explain any correlations which fall *inside* the outer *mathematical* bounds in (2).”
Local realism would be restored to physics, and the question would open up as to what is the right theory to use for this (enter Christian with hand raised). But it would be Bell that remains the midwife of that headline news.
Moral of the story: Bell’s Theorem will always be here and is not going away. Only the answer that Bell provides to EPR regarding local realism and hidden variables would change.
Jay
I would actually go first in the opposite direction:
“Any physically-observed or theoretically-deduced correlations which fall *outside* of the outer *mathematical* bounds in (2) cannot be explained by a locally realistic hidden variables theory (LRHV).”
Here I have added “or theoretically-deduced” to my earlier definition.
A) So if you agree that the only two types of correlations are those which are “physically-observed or theoretically-predicted,” then we can remove those specifics and just say:
“Any correlations which fall *outside* of the outer *mathematical* bounds in (2) cannot be explained by a locally realistic hidden variables theory (LRHV).”
This is your definition, and if you agree with paragraph A above, then it will become mine as well.
J
Yes, I agree with A).
I fully agree with that statement. But the question to which we have narrowed everything down, in significant part at your good suggestion from some days back, is what exactly are “the consequences of CHSH?”
Denoting:
CHSH == [A(a)B(b)] + [A(a)B(b’)]+ [A(a’)B(b)] – [A(a’)B(b’)],
Bell says that -2 le CHSH le +2. Because the physical world based on quantum mechanics exhibits a larger range from -2 sqrt(2) to +2 sqrt(2) which you pointed out in a post on November 1 and nobody here disagrees with, then these limits would assuredly rule out local realism. And having picked this low hanging fruit we could all go home.
But Christian claims that -4 le CHSH le +4. *IF* this were to be the case (and I am not saying it is or it isn’t) then the quantum results from -2 sqrt(2) to +2 sqrt(2) would become eligible for explanation using local realism. So the question at play has now been narrowed to whether Christian is right or wrong about those wider bounds.
While I am unconvinced of the reasons Christian has given for claiming the wider bounds, and feel his argument about two measurements being taken “simultaneously” and being “incompatible” have a ring of an uncertainty relation between canonical variables which I do not believe applies here, for me the jury is still out on whether he is reaching a correct conclusion about the wider bounds, even if he is articulating the wrong reasons. He may well be a good bloodhound who has picked up a scent of something amiss but not put his finger completely on the right articulation of what he is sniffing out and has perhaps articulated his sensibility in a way that has driven other some people in an opposite direction. Just let’s be patient and bring this to ground one way or the other.
Earlier, you narrowed the discussion down from talking about Joy’s S^3 theory to talking about Bell alone as a “no go” theorem. Now, the discussion is further narrowed to whether the mathematical bounds are -2 le CHSH le +2 or -4 le CHSH le +4. In a former “life” I spend many a long day narrowing, isolating, pinpointing, and fixing bugs in large bodies of computer code. My debugging sensibility says that the disagreement between “Christian world” and “Bell world” boils down to this line of code.
The two-page study & discussion piece I have developed and posted earlier today, for the purpose of evaluating this line of code, which is my first not last step of digging into and disassembling this code line, is at https://jayryablon.files.wordpress.com/2016/11/rw-11-5-16.pdf. As you will see it would be impossible to lay this out in anything other than an attachment with visual equation rendering, which is why I have done it that way. So please do assume that everyone can download and see this, or if they cannot, will say so.
What now we need to study are the “bits and bytes” in this line of code, at “machine language” level. We are not at the 20,000 foot blue sky level about Christian or Bell or local realism etc. We are at ground level, and digging below ground to systematically study the tree roots, and using magnifiers and microscopes to do so.
My apologies, I noticed a couple of copy and paste errors in the equations, in the indexes and in the use of primes. A corrected document is here: https://jayryablon.files.wordpress.com/2016/11/rw-11-5-16-2.pdf. Please use this.
Oops, please also ignore the 1/5 appearing in equation 6. That also needs to come out, but there is no reason to post a new document just for that.
Different views are being put forward by different people. Jay is very good at sorting those out. Therefore I will stick to my point of view and try to summarize here, as clearly as possible, one of my several refutations of Bell’s theorem that is being currently discussed:
I think no one is (or should be) disputing the fact that A(a)*B(b) and A(a)*B(b’) are two only counterfactually possible events in space time. Each represents a simultaneous click of the space-like separated detectors of Alice and Bob. But they cannot both exist simultaneously, because they involve clicks of the detectors along two mutually exclusive directions b and b’ that Bob could have chosen to align his detector. The resulting experiments are then physically incompatible experiments. The question then is: Is it legitimate to even consider a hybrid quantity like A(a)*B(b)+A(a)*B(b’) = A(a)*[B(b)+B(b’)] as a physically meaningful quantity? Note that I am not stressing the fact that it is not simultaneously observable. Of course it is not simultaneously observable. But that is the least of the problems with writing a sum like A(a)*B(b)+A(a)*B(b’) = A(a)*[B(b)+B(b’)] of two only counterfactually possible events A(a)*B(b) and A(a)*B(b’) in space and time. Since such a quantity cannot possibly exist in any possible physical world, what on earth does such a quantity even mean? It is a Unicorn.
But I am willing to go along with Gill and Parrott for the sake of argument and think of it simply as a random variable in a probability space under consideration. The question then is: What is the probability associated with the quantity A(a)*[B(b)+B(b’)] ? Clearly, to anyone who is remotely respectful of the fact that we are concerned about actually performable physical experiments in a possible physical world, the probability associated with the quantity A(a)*[B(b)+B(b’)] is the same as that associated with a single dice landing on both 3 and 6 at the same time. I think any schoolchild would agree that that probability is identically zero in any possible world. If anyone claims that the probability associated with A(a)*[B(b)+B(b’)] is not zero, then please enlighten me of what value that probability has, and why it is not zero.
Now Bell’s theorem is proved by considering the average of the quantities such as A(a)*[B(b)+B(b’)]. So let me write down a simple expression for this average (as far as I can see, replacement of average with expectation value will not change the essence of my argument):
E( A(a)*[B(b)+B(b’)] ) = Sum_k { ( A(a)*[B(b)+B(b’)]^k ) x ( Probability of A(a)*[B(b)+B(b’)]^k ) } ,
where, although not explicated, the application of the law of lager numbers is understood.
But since the Probability of A(a)*[B(b)+B(b’)]^k is identically zero, it is evident that the average (or expectation value) E( A(a)*[B(b)+B(b’)] ) is also identically zero. It therefore follows from the much discussed mathematical identity that the upper and lower bounds imposed on CHSH by Bell’s logic and mathematics are not -2 and +2, but -0 and +0. QED.
I hope I do not have to spell out the funny implications of this conclusion. 🙂
There are just three possible events and just three probabilities associated with the random variable X = A(a)*[B(b)+B(b’)].
According to local hidden variables, A(a)*[B(b)+B(b’)] is just some function of the hidden variable lambda. We don’t get to observe lambda and we don’t get to observe X. But if local hidden variables are true, then lambda and X(lambda) do exist. And X can take the values -2, 0 and 2. The set of all possible values of lambda can therefore be split into three disjoint subsets: the subset of all lambda where X(lambda) = -2, the subset where X(lambda) = 0, and the subset where X(lambda) = +2.
Those three subsets can be called “events”. The event where X = 2, etc. Which of the three events actually happens in any single trial is not observed by Alice and Bob. But when nature does pick a value of lambda, it will fall in just one of these three subsets, and it does that with probabilities which I’ll denote p(+2), p(0) and p(-2) respectively.
Those three probabilities are equal to the integrals of rho(lambda) d lambda over each of the three subsets. The three probabilities are nonnegative and add up to +1.
Finally, E(X) = 2 p(2) – 2 p(-2).
Christian seems to confuse “event” and “random variable” and still does not have the good formula for “expectation value”.
I disagree with Gill on several counts. In a local hidden variable theory lambdas of course do exist by definition, but the function X(lambda) = A(a, lambda)*[B(b, lambda)+B(b’, lambda)] does not have any physical meaning. Nor can X take the values -2, 0 and +2 in a physically meaningful sense.
There are only two particles at disposal to Alice and Bob for each run of the experiment, not three. Such pairs of particles can be identify with the hidden variable lambda^k, or just with the index k as I have done in my Appendix D. Now X(lambda) defined by Gill is a function of three possible detection events in spacetime, or of three clicks recorded by the two detectors detecting the two particles at disposal to Alice and Bob, for each lambda (or k). These clicks of the two detectors are recorded as the numbers A(a), B(b) and B(b’), where B(b) and B(b’) are only counterfactually possible numbers. Therefore there is no physical sense in which X can take the values -2, 0 and +2. Another way of saying the same thing is that the probabilities of the function X(lambda) taking the values -2, 0, and +2 are identically zero: p(+2) = 0, p(0) = 0 and p(-2) = 0, unless of course Alice and Bob belong to a world in which the probability of finding a single dice landing on both 3 and 6 at the same time is non-vanishing. But since all three of these probabilities must be identically zero, the average or expectation value of X is also identically zero. Consequently the CHSH correlator must lie between -0 and +0.
I should stress that I have no problem with what Gill has written if X(lambda) is defined by involving only two detection events, such as x(lambda) = A(a, lambda)*B(b, lambda) or x(lambda) = A(a, lambda)*B(b’, lambda). These two definitions have perfectly reasonable physical meanings. But an expression like X(lambda) = A(a, lambda)*[B(b, lambda)+B(b’, lambda)] is physically self-contradictory even for any local hidden variable theory. It is not demanded by either Einstein’s or Bell’s conception of local realism.
Nobody is saying that X has some physical meaning. But if we believe in local hidden variables, then lambda exists, the function X exists, and X(lambda) exists. Once nature has chosen lambda, X(lambda) is determined. Even if Alice and Bob have gone home and switched off their detectors.
I am sure X(lambda) can be assumed to exist as a function. But the probabilities of it taking the values -2, 0, and +2 are identically zero: p(+2) = 0, p(0) = 0 and p(-2) = 0. Unless of course Alice and Bob belong to a world in which the probability of finding a tossed coin on heads and tails at the same time is non-vanishing.
A(a, lambda), B(b, lambda) and B(b’, lambda) all exist, and are all equal to +/-1, whatever Alice and Bob actually measure. Clearly X(lambda) can only be -2, 0, or +2.
I agree entirely: “A(a, lambda), B(b, lambda) and B(b’, lambda) all exist [at least counterfactually], and are all equal to +/-1 [by construction], whatever Alice and Bob actually measure. Clearly X(lambda) can only be -2, 0, or +2.”
But that is not going to rescue Bell’s theorem. Because the probabilities p(-2), p(0) and p(+2) of X(lambda) taking the possible values -2, 0, and +2 are all identically zero: p(+2) = 0, p(0) = 0 and p(-2) = 0. Unless of course the probability of my being in New York and Miami at exactly the same time can be non-vanishing in the local-realistic world we live in.
The meaning of that depends on the meaning of “theory of local hidden variables”. Usually that is understood to imply that ordinary logic and mathematics can be soundly applied to the hidden variables in the same way they are applicable to obvious variables.
On November 5, 2016 at 10:30 pm I posted a discussion note using two unequal sample sizes as between the (a, b) and (a, b’) measurement pairs. I pointed out how the average of sums = sum of averages rule does not work for such samples. I have updated that note and placed it at https://jayryablon.files.wordpress.com/2016/11/discussion-note-1.pdf in order to explain why this is so and how to use statistical reweighing and normalization to restore this rule. You may discard the earlier note and use this instead, as it is more complete.
I will have more to follow. Referring to my posts on November 5, 2016 at 5:18 pm and November 5, 2016 at 10:17 pm, we have narrowed all of the issues down to whether:
L_m = -2 and U_m = +2 (1)
are a correct set of mathematical bounds for
CHSH = [A(a)B(b)] + [A(a)B(b’)]+ [A(a’)B(b)] – [A(a’)B(b’)] (2)
in
L_m le CHSH le U_m, (3)
It is my intention, in gory binary detail, to walk through the usual calculation that leads to (1) above, to ascertain — and hopefully reach complete consensus among participants here — whether or not (1) are correct mathematical bounds. It is (1) that is the key line of “code” that we are “debugging.” I will take it a step at a time, and this note at https://jayryablon.files.wordpress.com/2016/11/discussion-note-1.pdf is the first step.
Jay
What you call “the usual calculation” is not the usual calculation. Don’t use unequal sample sizes.
In (2), the square brackets stand for expectation values, not for sample averages. Write them out as integrals and derive (1) by a couple of lines of algebraic manipulation (sum of integrals equals integral of sums …).
It seems to me that the precise point of disagreement over the validity of Bell’s theorem has now been pinpointed. I would therefore like to prove my claim above more explicitly:
In the notation introduced by Gill in response to my previous posts, let X = A(a)*[B(b)+B(b’)] be the random variable of interest and let p(X) be the probability of X taking the values +2, 0, or -2. The average or expectation value of X is then given by
E(X) = Sum_i [ X_i p(X_i) ] = +2 p(X = +2) – 2 p(X = -2) . ……………… (1)
Now Gill claims that the probabilities, p(X = +2), p(X = 0) and p(X = -2), are all non-negative and add up to +1. I, on the other hand, claim that the probabilities, p(X = +2), p(X = 0) and p(X = -2), are all identically zero:
p(X = +2) = 0, p(X = 0) = 0 and p(X = -2) = 0 . ………………………….. (2)
If my claim is true, then at least one of my objections to the validity of Bell’s theorem stands.
To prove my claim (2) above, let me now rewrite X more explicitly as
X(a, b, b’; k) := u(a, b; k) + v(a, b’; k) := A(a; k)*B(b; k) + A(a; k)*B(b’; k) , ….. (3)
where k is the hidden variable, or an initial state, or a run, or a given particle pair. Written this way, it is now very clear that the random variables u and v are only counterfactually realizable. In other words, they define two incompatible experiments for any given k.
The question now is: What is the probability of occurrence for the “event” X defined in (3)?
Now at least in my view, this probability is given by p(u and v) — i.e., the probability of X occurring is the same as the probability of u and v occurring simultaneously. But as already noted, u and v represent two mutually incompatible experiments that cannot be realized simultaneously in any possible physical world, classical or quantum. Therefore it is clear that
p(u and v) = 0 , …………………………………………………………………….. (4)
regardless of the values u and v may take. Note that I have interpreted “+” in (3) as “and” in (4). I will maintain this equivalence between “+” and “and” unless persuaded otherwise.
Joy, you interpret “+” as “and”. Moreover you interpret “A(a, k)” as “For the k’th trial, Alice chooses to use setting a”.
I think both interpretations are wrong. My interpretation of A(a, k) is “the measurement outcome Alice would have seen, if she had used setting a (rather than whatever setting, if any, she did actually use)”. And for me, + means +.
I did not invent the notion of local hidden variables. But that is what Bell is assuming, in deriving CHSH. And that assumption implies, in my opinion, the existence of A(a, k) according to my interpretation.
The physical realisability of *actually* measuring A with several different settings at the same time is, in my opinion, completely irrelevant.
By A(a, k) I do not mean “for the k’th trial, Alice chooses to use setting a”. What I mean by A(a, k) is “the measurement outcome Alice would have seen, if she had used setting a (rather than whatever setting, if any, she did actually use)” for the k’th trial. If there was any ambiguity about what I meant by A(a, k), then now it has been removed.
Let me now be more explicit also about what I mean by B(b, k) and B(b’, k).
By B(b, k) I mean “the measurement outcome Bob would have seen, if he had used setting b (rather than whatever setting, if any, he did actually use) for the k’th trial.”
And by B(b’, k) I mean “the measurement outcome Bob would have seen, if he had used setting b’ (rather than whatever setting, if any, he did actually use) for the k’th trial.”
Thus, in an abstract mathematical space of all possible outcomes, the outcomes A(a, k), B(b, k) and B(b’, k) do indeed exist as counterfactually possible events in spacetime, whether they are actually observed by Alice and Bob or not. Therefore, there is no difficulty in defining the function X in this abstract space as
X(a, b, b’, k) := u(a, b, k) + v(a, b’, k) := A(a, k)*B(b, k) + A(a, k)*B(b’, k) . …….. (1)
And this function X, depending on the values of its arguments, can only take values +2, 0, or -2, because u and v can only take values +1 or -1, and that in turn because A and B can only take values +1 or -1. So, to stress the obvious, I have absolutely no problem at all with the existence of the function X and the fact that it can only take values +2, 0, or -2.
But the question now is: What is the probability p(X) of X, in the standard sense of what these terms mean in the context of computing the expectation value E(X) = Sum_i [ X_i p(X_i) ] ?
In other words, in the light of (1), what are the explicit values of the probabilities p(u_i + v_i) ?
Now I have claimed, and proved, that p(u + v) is identically zero: p(u + v) = 0. Moreover, I have provided explicit physical reasons for my claim, with real-life examples.
Gill, on the other hand, has claimed that p(u + v) is non-negative and adds up to +1. In other words, according to Gill p(u_i + v_i) can be anything between 0 and +1. But that can be said about the probability of almost anything. I have provided a very specific value for p(u + v) and justified this value in several different ways. Without equally specific values for p(u_i + v_i) from Gill it is impossible, in his case, to evaluate the expectation value
E(u + v) = Sum_i [ (u_i + v_i) p(u_i + v_i).
In other words, without specific values for p(u_i + v_i) Bell’s theorem remains unproven.
The probabilities in question are determined by the probability distribution rho of the hidden variable, and the functions A and B. One can write down formulas for them, if one is so inclined. Waste of time. All we need to know is the linearity of expectation value.
By definition, the sum of all p’s is 1 or they are not probabilities.
Sometimes abstract ideas and complex notations obscure simple facts. So let me explain what I find problematic in Gill’s claim using a homely example. Consider an ordinary dice. Its six faces are marked with 1, 2, 3, 4, 5 and 6 dots. The “hidden variable space” in this homely case is the space of all possible faces on which the dice can land. We can write down the six numbers representing the number of dots on a piece of paper, and that is then our abstract “hidden variable space” that Gill is talking about. Now one can ask: What is the probability of the dice landing on 3? Or: What is the probability of the dice landing on 5? Etc. But following Bell, Gill wants to ask a different question: What is the probability of the dice landing on (3+5)? To me, at least, it is quite obvious that the probability of the dice landing on (3+5) is identically zero — not non-negative adding up to +1 as Gill claims.
The hypothesis of local hidden variable theory is supposed to be about the world we live in, not about some impossible world that assigns non-negative probabilities to a dice landing on (3+5).
Sometimes abstract ideas and complex notations obscure simple facts. So let me explain what I find problematic in Gill’s claim using a homely example. Consider an ordinary dice. Its six faces are marked with 1, 2, 3, 4, 5 and 6 dots. The “hidden variable space” in this homely example is the space of all possible faces on which the dice can land — or equivalently, the set of all possible outcomes, {1, 2, 3, 4, 5, 6}, of a throw of the dice . We can write down these six numbers representing the number of dots on the dice on a piece of paper, and that is then our abstract “hidden variable space” analogous to what Gill is talking about. Now we can ask: What is the probability of the dice landing on 3? Or: What is the probability of the dice landing on 5? Etc. But following Bell, Gill wants to ask a different question: What is the probability of the dice landing on (3+5)? To me, at least, it is quite obvious that the probability of the dice landing on (3+5) is identically zero — not non-negative adding up to +1 as Gill claims.
The hypothesis of local hidden variable theory is supposed to be about the world we live in, not about some impossible world that assigns non-negative probabilities to a dice landing on a non-existent face like (3+5).
I didn’t notice any “abstract ideas and complex notations” in Bell’s theorem. There’s only some *very* elementary probability theory.
The sum of the probabilities must be one. Either at least one of the mentioned must have a non-zero probability or there must be at least one alternative not yet mentioned.
I disagree. p(+2), p(0) and p(-2) are all identically zero. The alternative not mentioned explicitly is that the assumptions underlying Bell’s theorem are inherently inconsistent.
If the sum of the probabilities of all possibilities is zero then the only possible interpretation of that is that the probability that the world exists is zero.
I like that. That is exactly what it means: “…the probability that the world exists is zero.”
Richard, I agree as we discussed offline that we need to use equal same sizes. I started with unequal sizes to illustrate this point, i.e., to show that we need to, and why we are allowed to, use equal sample sizes.
Thank you for the clarification that this is an expectation value not an average. I assume Joy agrees on that point?
If so, then because the average harvested from individual trails converges to the expectation value in the limit of a very large number of runs (which as we have discussed should be in the zone of millions or more), I believe I can still stick to my plan to a) illustrate how the math works for smaller numbers of equally-sized samples among the four CHSH disjoint pair sets, and thereafter b) show how this gets extrapolated to very large sample sizes. You have made very clear to me that in the end we have to get to large sample sizes, and I will not neglect this. Richard, do you agree with this being a valid strategy?
Jay
Query, I am not sure I agree re integrals: An integral is taken for a *continuous* mathematical function, and it is a limit as delta x approaches zero for the dummy integration variable x which is also continuous. The random variables inside the CHSH expectation brackets are *discrete*, so we need to use sums instead. For example, for five trials, you have the allowable values -5, -3, -1, +1, +3, +5. And even for 10 million trials, the possible values are from -10 million, -10 million +2 … 10 million -2, 10 million. So I believe it is mathematically correct to set CHSH up as a sum, and then look at the limiting case of large sample sets.
Both approaches are correct, and will eventually lead to the same conclusion. Using sums is more cumbersome though, since using integrals already has the limiting case “built in”.
Good. Then on the philosophy of “measure twice, cut once” I am willing to take the more cumbersome route using discrete random variables, and then using that to confirm (or not) that they reach the same conclusion, i.e., show (or not) that you get to the same result by two different paths.
Then it’s a good idea to read Richard’s paper https://arxiv.org/abs/1207.5103 (if you have not already done so), since it might save you some work in the process.
Yes, HR, you and I are on the same page. I am aware of Richard’s proof in section 2, and that is the exact same proof I want to study in detail using discrete random variables in the large sample limit.
I hope you do not rely on that paper too much, Jay, because it has been heavily criticized on PubPeer by various authors, and several mistakes in it have been revealed:
https://pubpeer.com/publications/D985B475C637F666CC1D3E3A314522#fb27706
I now have finished preparing a second discussion note dealing with equal sample sizes, posted at https://jayryablon.files.wordpress.com/2016/11/discussion-note-2-1.pdf. I always stand to be corrected, and if I have gotten something wrong I would like to be corrected.
But it looks to me as discussed starting around (25) of this note, that the mathematical bounds:
L_m = -2 and U_m = +2 (1)
on
CHSH = [A(a)B(b)] + [A(a)B(b’)]+ [A(a’)B(b)] – [A(a’)B(b’)] (2)
are erroneously derived, and that in fact all we an say is that
L_m = -4 and U_m = +4 (3)
This is what Joy Christian has maintained all along. If this is so, then by the Bell’s Theorem contra-positive logic previously discussed here, local realistic hidden variable theories are now back on the table for understanding quantum mechanical entanglement correlations, and Joy gets to raise his hand and say “lookee here” with his S^3 model.
To be sure, while I now have to agree with Joy’s assertion (1) is wrong and that (3) is correct, I believe that his reasons are incorrect. He arrived at the right conclusions but for the wrong reasons. To use one of his analogies, while I cannot roll one die and get a “2” and a “5” at the same time, I can roll the die and get a “2,” then roll it again and get a “5,” and then write the sum “7=2+5.” Simultaneity and incompatibility having nothing to do with this. And when we perform the Bell factorization that leads to (1), all we are doing in essence is adding two successive die rolls (and here, since the results are binary, coin flips).
The reason that (1) appears to fail as the bounds for CHSH, is rather that a) the usual calculations treat A and B as if they are ordinary binary numbers rather than I-dimensional binary-valued vectors with I being the number of trials included in each measurement set, and b) the usual calculations treat the B in the first and second CHSH sets as if they are interdependent upon (and in fact equal to) the B at like-orientation in the third and fourth sets, when in fact they are entirely disjoint from one another and therefore cannot be used together at all to obtain the limits (1), no mater how large or small the sample set.
I do not believe one needs any esoteric arguments to reach this conclusion. It is simple math. The limits (1) are effectively derived by something akin to saying A=B, when it can be proven when one exercises great care and dives into the details of what these symbols represent as mathematical vectors operated upon by linear algebra, that A not equal B.
Again, I always stand to be corrected, and as a Bell novice advancing but still learning, would like to know if I am missing something here.
Jay
Jay, as far as I can see from a first reading, nowhere do you use the fact that the model should be local. For non-local models, of course the bounds are -4, +4.
The variables defined by Jay are all manifestly local and strictly adhere to the definition of locality specified by Bell. Jay is simply following the notations I used in my Appendix D.
This is not a matter of notation.
Nor is Jay assuming non-locality anywhere in his analysis. As I stressed, Jay’s variables are all manifestly local.
You were not able to derive Bell’s bounds because you did not use the assumption of local hidden variables and you did not use probability theory (taking the limit of larger and larger numbers of observations).
Thank you very much, Jay, for all your efforts, and for your detailed analysis. I agree with most of your analysis, apart from your above quoted critique of my argument. We are not allowed to roll the die a second time. The outcomes “2” and “5” are supposed to be two counterfactually possible outcomes of a single role of a single die. To quote Bell himself from his response to one of his early critics: “But by no means. We are not at all concerned with sequences of measurements on a given pair of particles [or on a “die” in our case]. We are concerned with experiments in which for each pair the spin of each particle is measured once only.” In other words, the quantity B(b) + B(b’) appearing in the CHSH correlator — your eq. (2) — is representing two counterfactually possible outcomes of a spin measurement occurring at the same time. It is like asking: What if I had visited New York at noon on the 4th of July instead of Miami at noon on the 4th of July? If we denote these two possibilities as B(b) and B(b’), then it makes no sense to write a quantity like B(b) + B(b’), because that would represent a meaningless of impossible “event” in spacetime.
Richard, referring to your paper at the link HR provided above, please point out to me what aspect of your proof of Bell’s bounds +/-2 in section 2 uses the assumption of local hidden variables. By my reading, you are using “two fair coins” just as I am doing. Also, please point out where you use probability theory based on large numbers. I ask all this because I want to see in very specific mathematical terms what I may have omitted from my analysis. Thanks, Jay
I second Jay’s request.
If Gill and/or HR can show us an explicit calculation in the manner of Jay’s detailed analysis, then we would either understand what they are saying, or may be able expose hidden assumptions in their claim. Until then it seems to me that their’s is an empty claim.
Let’s take this step by step. Say you are handed a local hidden variable model by someone, that you now can use to predict the outcomes of the first run of an Alice-Bob experiment. Alice and Bob both have two choices for their detector settings. You have no idea what settings they will choose, so you better use your model to predict for all eventualities. But because the model is supposed to be local, Bob’s choice of setting should not influence Alice’s result (no matter her setting), and vice versa. So the predictions you have to make is only the four predictions A(a), A(a’), B(b), B(b’) for the first run. Are you with me so far?
You asked me where I used the assumption of local hidden variables, and where I used probability theory (law of large numbers) in my paper https://arxiv.org/pdf/1207.5103v6.pdf
The local hidden variables assumption comes in the assumption that for each trial there exists a quadruple A, A’, B, B’ of numbers +/-1. These are the numbers written on Stephen Parrott’s slips of paper. For instance, A’ is what “what Alice’s measurement outcome would be, if she chooses a’ as her setting”. B is “what Bob’s measurement outcome would be, if he chooses b as his setting”.
The two coin tosses per trial encode Alice and Bob’s choices of which setting to use.
My Theorem 1, see equation (3), is a finite N probability statement: if we do N trials, and if N is very large, then the chance is tiny that CHSH will deviate above +2 more than by some specified amount. In equation (4) I draw the obvious conclusion about the limiting situation when N converges to infinity.
It is worth recalling, Jay, that — while “large N limit” is being insisted upon by the followers of Bell — in the recent so-called “loophole-free” experiments (which have been referred to by Annals of Physics as “violating local realism”) used only 256 events in their experiments. That is a far cry from the 12 million plus events you are being asked to consider. 🙂
Let me see, and thank you for indulging we with a step by step:
I infer that there is an assumption that Alice has a 50% chance of choosing a and 50% of choosing a’, and that Bob has a 50% chance of choosing b and 50% for b’, for how they orient their detectors, so that each of the ab, ab’, a’b and a’b’ combinations has a 1/4 chance of being selected on the first run. And likewise for successive runs. I infer that this is the origin of the “equal probability of 1/4, for each row of the table” which Richard refers to in the fourth line, right hand column, on page 3 of his paper, and that my “predictions” are based on this probability. I also infer that although I am not permitted to know what choice Alice and Bob each make for any given run, that I am allowed to infer that after large numbers of runs, there will have been roughly equal numbers of ab, ab’, a’b and a’b’ combinations employed at 25% each. And if I have correctly inferred all of these assumptions about the experimental design and “rules,” then I deduce that I can treat each run (with 25% assigned to each detector combination) *as if* it was four separate runs where each of ab, ab’, a’b and a’b’ was in fact chosen (still 25% = 1/4), because I am post-facto applying the statistics of large numbers to each run. Anything amiss here?
Well, we’re jumiping ahead here, but ok. Point being that you now have a 4xN matrix, where each row in the matrix encodes the model predictions for any Alice/Bob settings for each run of the hypothetical experiment. If you simply apply the CHSH equation to the entire matrix, the bounds for the average [-2,+2] are trivial, and absolute. (You can check this yourself).
Now it is a rather simple exercise to see that randomly sampling from this matrix (as you would do in a simulated experiment) would give results very close to using the whole matrix (up to statistical “polling” errors, so to speak in these Trump days).
HR, I am very keen on looking at the exact data structures. So I did 48 successive coin flips and used them to create the 4×12 matrix at https://jayryablon.files.wordpress.com/2016/11/matrix.pdf. Let’s talk this through using that matrix as our data sample (and wow did the pollsters manage to make some huge errors). J
To bad we are in different time zones, so I am writing this in bed, but how about you generate a 4×12 million matrix while I sleep? (Tongue in cheek :-)) But seriously, would be instructive though.
Yes, that is valid prediction matrix. But 12 runs is to few to enable us to draw any meaningful statistical conclusions.
And note also that it is the crucial assumption of locality that makes four entries for each run suffice. For a non-local model, we would have to predict eight numbers to cover all eventualities; A(a,b), A(a,b’), A(a’,b), A(a’,b’), B(a,b), B(a,b’), B(a’,b), B(a’,b’).
Bell’s argument won’t work with an 8xN matrix. But because of the locality assumption A(a,b) = A(a,b’), etc., we only need four values to make an exhaustive prediction.
Jay,
It is good that you are thinking carefully and trying to get to the root of the proof of Bell’s theorem. You are right that in the context of your setup, a factorization used in the usual proof your (13) would not be valid. But in your admirable attempt to understand the problem by microanalyzing it, you are losing sight of the forest while examining the trees.
The usual proof of Bell’s theorem assumes a *hypothesis* which never appears in your analysis—the assumption that the your A1(a), etc., are obtained from a so-called “hidden variable” lambda. That is physicists’ language. In mathematical language, the A1(a), etc.,
are random variables on a probability space. I will explain this more fully below, but first I state the problem with your analysis.
Your analysis never uses the crucial *hypothesis* of hidden variables. That is why you will never obtain a valid proof of Bell’s theorem from it—because Bell’s theorem *is not true* with the hypothesis of hidden variables omitted. Your analysis is like trying to prove
the Pythagorean theorem for general triangles, omitting the hypothesis that one angle must be a right angle.
Now I will attempt to explain the hypothesis that A1(a), etc., are random variables on a probability space. Any mathematician will immediately recognize exactly what this means,
but for non-mathematicians it may seem like vague jargon.
Think of the probability space as an urn containing slips of paper. The slips are the “hidden variable”. On each slip is written something like “If Alice chooses *a*, return +1, if a’, return -1; if Bob chooses *b*, return -1, if b’, return +1. In all there are 2^4 = 16 possible slips.
The proportion of each is its probability.
An experiment draws one slip. If we read off the slip *only* the part that says “If Alice chooses *a*, return … “, we get the random variable A(a). It is a function on the set of slips with possible values +1 or -1. There are four such random variables, A(a), A(a’ ), B(b), B(b’ ).
According to quantum mechanics, Alice has access only to *one* of the random variables A(a) and A(a’); Bob only to one of B(b) and B(b’). But in principle, somebody could have access to all four. That is the main *hypothesis* of Bell’s theorem.
If you assume from the start that it is self-contradictory for Alice to access both A(a) and A(a’) simultaneously, as Christian does and you seem to, then you will not be able to prove Bell’s theorem. That it is not possible according to quantum mechanics does not make it self-contradictory.
[Incidentally, I hope that Christian will stop endlessly repeating the analogy of the impossibility of being simultaneously in New York and Miami, as if it proves something. It insults the intelligence of the readers. Yes, we do understand and agree that one cannot be simultaneously in N.Y. and Miami ! )
Stephen, let me stop right here, to make sure I am understanding this, and please forgive me insofar as I do like to start with concrete data and then work back to the abstract conclusions. I see eight slips which are:
a -> +1 (if Alice Chooses a then return +1)
a -> -1 (etc. for the rest)
a’ -> +1
a’ -> -1
b -> +1
b -> -1
b’ -> +1
b’ -> -1
But you say there are 2^4 =16. What eight slips am I overlooking? Thanks, Jay
One slip has four “binary” numbers on it. An example is:
If a, then 1, if a’ then -1, if b then -1, if b’ then -1.
If we agree that the order is always as just given, we can simplify the notation by writing
(1, -1, -1, -1)
instead of the above. So there are 2x2x2x2 = 16 slips, namely (+1, +1, +1, +1),
(+1, +1, +1, -1), etc. Your first slip (if a then 1) is not a “slip” according the the definition just given.
_______________________________________________________________________
The historical “hidden variable” terminology seems to me unfortunate. A much cleaner approach is to speak of a typical “lambda” (value of hidden variable or outcome in a probability space) as a four-tuple
lambda := (i, j k, m) with each entry either +1 or -1 .
This describes the set of outcomes very concretely as a 16-element set. The seemingly more complicated integral formulation is easily seen to be essentially equivalent. The probability
of the above typical outcome can be denoted
p( i, j, k, m)
instead of the more complicated integral notation of the traditional “hidden variable” approach.
The probability that Alice chooses to measure *a* and gets outcome -1 while Bob measures
*b’ * and gets +1 is then
sum_{j, k} p(-1, j , k, +1) (equation (1) )
which is a generalization of a so-called “marginal” probability.
If we assume that Alice cannot simultaneously observe *a* and *a’ * and Bob cannot observe
*b* and *b’ * , then physically we are given only four such “marginal” probabilities instead of the full p(i, j, k, m). But (and this is the important point), we *cannot assume* that the given “marginals” are true marginals obtained as in equation (1). If they always *can* be so obtained, we say that there is a “(local) realistic” model. This is equivalent to the historical “hidden variable” terminology.
Because there is no reason to assume that such given “marginals” are obtained as the true
marginals of equation (1), it is better to use a different name for them such as “pseudo marginals”. The rules of quantum mechanics furnish four such pseudo marginals. Can they always be obtained as the true marginals of equation (1), for some p( i , j , k , m) ?
It turns out that they *cannot* always be so obtained. It turns out that a *necessary* condition that they be so obtained is that they satisfy the CHSH inequalities. This is one way to state Bell’s theorem, in my view a much cleaner way than the language of “hidden variables”.
One reason that it is cleaner is that one can present a Bell denier with four quantum pseudo marginals (presented as 2×2 matrices) which do not satisfy CHSH and challenge him to produce a p(i , j, k, m) which reproduces them as true marginals. This is a a very concrete problem which could probably be solved by computer, and if he can’t do it one has to question his claims.
A computer solution requires solving a linear system for the 16 probabilities p( i , j , k, m) , which of course any computer algebra system can do. What makes it slightly tricky is that one needs a solution with all p( i , j , k , m) non-negative. I don’t know if there are packages that can always do this. One way to definitively solve the present Bell “controversy” (which of course is not a real controversy among the vast majority of physicists) would be to explicitly produce p( i , j , k , m) for four quantum marginals which do not satisfy CHSH. (See p. 164 of Asher Peres’ book on quantum mechanics for such marginals expressed in different language.) That would also make history. No journal would dare reject it.
If a computer expert like you could do that, there could be no argument that Bell’s theorem is incorrect. Moreover, although obtaining p( i , j , k, m ) would require programming, readers would not have to rely on its correctness because the solution could easily be checked directly with standard computer algebra systems such as Maple, or even by hand.
Ha! Glad to know I am not the only crazy person who checks my incoming and writes messages in bed. 🙂 Well, I do not think that overnight provides enough seconds to do 12 million anything, we let’s work from 12 and then use all of our overactive imaginations to extrapolate what 12 million might look like. J
I am pleased to note that at least one proponent of Bell’s theorem has accepted the fact that if it is self-contradictory for Alice to access both A(a) and A(a’) simultaneously, then it is not possible to prove Bell’s theorem.
But contrary to the claim by Stephen Parrott, quantum mechanics has nothing to do with the fact that Alice cannot access A(a) and A(a’) simultaneously. A(a) and A(a’) are only counterfactually possible measurement results, along two mutually exclusive detector directions, a and a’. Since a and a’ are classical, macroscopic directions, no experimenter has the ability to align his or her detector along both a and a’ simultaneously. Therefore, regardless of any physical theory — classical or quantum — it is self-contradictory for Alice to access both A(a) and A(a’) simultaneously. Consequently, as acknowledged by Stephen Parrott, it is not possible to prove Bell’s theorem.
The fact that we are “unable” to simultaneously determine the values of a particle’s spin components in direction a and a’ does not a priori exclude that both particle’s spin components have certain “pre-existing” values before any measurement is performed. That’s the reasoning Bell’s inequalities are based upon.
I am well aware of that. Please see my example below:
No idea what the dice and the inscriptions on its faces really stand for when thinking about the outcomes of measurements on an ensemble of identically prepared, interacting two-particle systems. Sometimes there is need for straightforward definitions.
I have reviewed recent comments by Richard and HR and Stephen and Joy and others. What comes to mind is the old adage that “everybody is entitled to their own opinions but not to their own facts.” In the case of what people are saying about Bell, I will re-frame this adage to say that “everyone is entitled to their own interpretations but not to their own data.” So, I would like to see if I can obtain universal agreement from this crew about the data that we are all discussing here, without yet crossing into how we carry out calculations based on that data and how we interpret that data and the consequences of our calculations.
With that in mind, I uploaded a 1-sheet file to https://jayryablon.files.wordpress.com/2016/11/bell-dataset.pdf, which contains two matrices. The first, which I call “slip dataset,” is 4 x 16 and represents each of the 16 “slips” in the “urn” that Stephen has been talking about, for all of the possible A, A’, B, B’ combinations. The second, which I call “slip and selection dataset,” is 4 x 64 and represents these same 16 “slips” replicated by 4 to represent the four “choices” AB, AB’, A’B, A’B’ that can be made to observe these slips. I have used a “0/” (0 with strikeout) to represent “null,” meaning that based on the values Alice and Bob actually “get to see” based on how they align their detectors, these are the values that they never get to see because these were not chosen. Again, no interpretation, just descriptive statements about the data.
Might we all agree on the following:
1) As to the slip dataset, if we were to do 16 million trials rather than 16 trials, we can *expect* on average to draw 1 million slips containing each of these 16 value combinations?
2) The harvest order of the 16 million trial results does not matter, so we can reorder everything into batches of 1 million and then use each slip dataset column to represent the expected 1 million trial results for each of the 2^4=16 slip types?
3) As to the slip and selection dataset, we may have Alice and Bob each flip a fair coin to instruct them how to align their detector for each trail, so we may also *expect* that on average 25% of their trials will be for each of the paired AB (heads heads), AB’ (heads tails), A’B (tails heads), A’B’ (tails tails) outcomes?
4) Therefore, were we to now do 64 million trials rather than 64 trials, we can expect on average to have 1 million occurrences of each of the 64 columns of the the slip and selection dataset?
5) Once again, the order does not matter, so we can reorder everything into batches of 1 million and then use each slip dataset column to represent the expected 1 million trial results for each of the 2^6 = 16 slip types x 4 coin toss results?
Again, no interpretation yet, I just want to see if all agree about the data that we are working with here.
Jay
PS: You will note that in the slip dataset, the pattern amounts to hexadecimal counting from 0 to 15.
Yes, but 1) can be made even more general. For a local hidden variable model it is not necessary that all 16 value combinations have the same (1/16) probability to be drawn, the model could make some much more frequent than others. You will eventually see that this does not change the conclusion of Bell’s theorem.
Jay,
So far as I can tell, You have the right mathematical model ! This is very important because once one sees a clear way o think about a problem, the problem is often half solved.
For future reference, I will just note that your statements (2) and (5) are not clear to me. I’m not saying that they are wrong, but the algorithm which you intend to produce the reordering is not clear to me. I just want to make sure that initial agreement does not permit later illegitimate manipulations, something like the unequal sample sizes previously used.
I also want to emphasize what HR said. If you start with an urn with probability 1/16 of each possible slip, you will obtain a CHSH sum of approximately 0 for a large sample sizes. Physically, this is not so interesting.
What the Bell deniers want is to get a CHSH which is significantly larger than 2 (or less than -2). You will be very unlikely to do that with *any possible proportions of slips in the urn* and for sufficiently large sample sizes. This is what Theorem 1 of Gill’s paper http://www.arXiv.org/abs/1207.5103 makes precise. It goes beyond merely saying that the sample size has to be large enough. It actually specifies how large a sample size will guarantee a preassigned (small) probability of violating CHSH.
I’m not sure what sort of experiment you have in mind to do with your urn of slips, but I do have a suggestion for one. Write down four quantum pseudo-marginals which violate CHSH, such as those given (in different notation) on p. 164 of Peres’ book on quantum mechanics. Ask Christian and his crew what should be the proportions of slips in the urn
to obtain these pseudo-marginals. (If his theory doesn’t permit specifying these, that would seem suspicious.) Then write a simple program to draw large numbers of slips from that urn and calculate CHSH.
Jay,
So far as I can tell, You have the right mathematical model! This is very important because once one sees a clear way to think about a problem, the problem is often half solved.
For future reference, I will just note that your statements (2) and (5) are not clear to me. I’m not saying that they are wrong, but the algorithm which you intend to produce the reordering is not clearly specified. I just want to make sure that initial agreement does not permit later illegitimate manipulations, something like the unequal sample sizes previously
used.
I also want to emphasize what HR said. If you start with an urn with probability 1/16 of each possible slip, you will obtain a CHSH sum of approximately 0 for a large sample sizes. Physically, this is not so interesting.
What the Bell deniers want is to get a CHSH which is significantly larger than 2 (or less than -2). You will be very unlikely to do that with *any possible proportions of slips in the urn* and for sufficiently large sample sizes. This is what Theorem 1 of Gill’s paper http://www.arXiv.org/abs/1207.5103 makes precise. It goes beyond merely saying that the sample size has to be large enough. It actuallyspecifies how large a sample size will guarantee a preassigned (small) probability of violating CHSH.
I’m not sure what sort of experiment you have in mind to do with your urn of slips, but I do have a suggestion for one. Write down four quantum pseudo-marginals which violate CHSH, such as those given (in different notation) on p. 164 of Peres’ book on quantum mechanics. Ask Christian and his crew what should be the proportions of slips in the urn
to obtain these pseudo-marginals. (If his theory doesn’t permit specifying these, that would seem suspicious.) Then write a simple program to draw large numbers of slips from that urn and calculate CHSH.
Jay, I am not sure what your data set is saying. AB, AB’, A’B and A’B’ should be simultaneous clicks of the detectors of Alice and Bob. That does not seem to be respected in what you have presented. There are no null events in the actual experiments. If A clicks, then either B or B’ must also click — for example. I don’t see that respected in your data structure either.
The 16 columns of the first matrix should be allowed to have arbitrary frequencies. Pick any 16 nonnegative whole numbers. Call them n1, n2, …, n16. Duplicate slip 1 n1 times. Duplicate slip 2 n2 times. … Duplicate slip 16 n16 times. You now have N = n1 + n2 + … + n16 columns/slips.
The 64 “slip and detection” columns also needs expansion in the same way, using the same set of 16 numbers n1, n2, …, n16 four times; once for each of the four groups. You now have 4N “slip and detection” columns.
This way, when we pick a column at random from the expanded “slip and detection” data set, we are choosing whether to measure ab, ab’, a’b, or a’b’ with equal probabilities 1/4. Our choice of *which* settings to use is independent of the “hidden” values of (A, A’, B, B’). The 16 possible hidden values have probabilities n1/N, …, n16/N.
For instance, the first column:
Entry for A is “-“: A was measured and the outcome was “-”
Entry for A’ is “0”: A’ was not measured
Entry for B is “-“: B was measured and the outcome was “-”
Entry for B’ is “0”: B’ was not measured
Under what circumstances, physical or otherwise, might some of the 16 value combinations occur as as to be “much more frequent than others”?
For example, suppose you are given a quantum state. From that, you can calculate the quantum pseudo-marginals. From those, you can calculate the correlations E(A(a)B(b)), etc., and hence the CHSH sums.
Bell’s theorem says that *if* there exists an urn model (i.e., local realistic model), then the CHSH sums must be no more than 2 and no less than -2. The converse is a much more difficult theorem of Arthur Fine, proved around 1982. That converse says that if the CHSH sums are no more than 2 and no less than -2, then there does exist an urn model which reproduces the given quantum pseudo-marginals. (Actually, both Bell’s and Fine’s theorems apply to all pseudo-marginals, not just those predicted by quantum theory.)
So, for *some* (but not all) quantum pseudo-marginals, there *does* exist an urn model. Of course the proportions of the various slips in the urn will not usually be 1/16 (because if they are all 1/16, then the CHSH sum will be zero).
Remember that Bell’s theorem applies to any local hidden variable model. For instance, the model that predicts that Alica and Bob always measures spin up will only use the slip (+, +, +, +).
Very well, although I have explained my point in detail in the Appendix D of my paper which triggered this discussion ( i.e., this one: https://arxiv.org/abs/1501.03393 ), let me explain my point again using the language traditionally used to prove Bell’s theorem ( see Bell’s famous paper of 1964 ), using the “large N limit” ( i.e., integration ) as well as the usual assumption of “local hidden variables”, as insisted upon by the proponents of Bell’s theorem.
Let me reproduce the key part of the derivation of the bounds -2 and +2 on the CHSH correlator from Bell’s paper of 1964. It involves the mathematical identity we discussed previously, namely E(X) + E(Y) = E(X + Y), and an average of the random variable X(a, a’, b, b’, k) with k as a hidden variable (usually denoted by lambda), which is defined as
X(a, a’, b, b’, k) := A(a, k)*B(b, k) + A(a, k)*B(b’, k) + A(a’, k)*B(b, k) – A(a’, k)*B(b’, k),
where A and B are +1 or -1 and a, a’, b and b’ are possible measurement directions that may be freely chosen by Alice and Bob. The mathematical identity of interest is then
Int_K A(a, k)*B(b, k) rho(k) dk + Int_K A(a, k)*B(b’, k) rho(k) dk + Int_K A(a’, k)*B(b, k) rho(k) dk – Int_K A(a’, k)*B(b’, k) rho(k) dk = Int_K X(a, a’, b, b’ ) rho(k) dk ,
where Int_K denotes integration ( i.e., “large N limit” ) over the space K of hidden variables k.
Let me stress that the above is the standard mathematical identity, frequently used in the derivation of Bell’s theorem, and can be found even in Wikipedia. The RHS of the above identity is simply an “average” of the random variable X(a, a’, b, b’, k) defined above.
Now it is claimed by the followers of Bell that X(a, a’, b, b’, k) is just a random variable on the space of all counterfactually possible outcomes that may be observed by Alice and Bob ( as in my dice example ). And — they claim — the bounds of -2 and +2 on the CHSH correlator follows immediately from its “average” on the RHS. Well, they indeed seem to be, because, as noted, the equality between the LHS and the RHS in the above equation is a strict mathematical identity (as frequently stressed by the followers of Bell), and it is easy to see that the RHS of the equation is bounded by -2 and +2. But that is just the problem.
Since the above equality is a strict mathematical identity, we can just take its RHS as our starting point and ask: What is exactly being averaged in it?
Well, what is being averaged is a quantity that cannot possibly exist in any possible physical world, as demonstrated in my paper linked above. Spacetime events such as B(b) and B(b’ ) are only counterfactually possible measurement results, along two mutually exclusive detector directions, b and b’. Since b and b’ are classical, macroscopic directions, no experimenter has the ability to align his or her detector along both b and b’ simultaneously. Therefore, regardless of any physical theory — classical or quantum — it is impossible for Bob to measure both B(b) and B(b’ ) simultaneously, just as it is impossible for a die to land on 3 and 6 simultaneously. Therefore X(a, a’, b, b’, k), which involves sums like B(b)+B(b’ ), is a totally fictitious quantity that cannot possibly exist except in some fantasy world, and therefore the bounds of -2 and +2 derived from X are equally fictitious. They have nothing whatsoever to do with any possible physical world, classical or quantum. They are merely mathematical curiosities. They by no means rule out any local hidden variable theories. They by no means forbid a strictly local, realistic, and deterministic derivation of the correlation E(a, b) = -a.b. Bell’s so-called “theorem” has no relevance for physics whatsoever.
Stephen, please review (14) to (19) and related discussion in https://jayryablon.files.wordpress.com/2016/11/discussion-note-2-1.pdf to see what sort of correlated reordering I am talking about. All I am saying is that if I draw some population of slips from the urn and then tally up the results, the order in which I have drawn them does not matter, and I can rearrange my data sets as if the slip which was drawn second is treated as if it was drawn third, and the third drawn slip treated as if it was drawn second, etc., etc. as much as is needed to segregate / bundle all like results, without impacting the overall numbers (random variables) that emerge from taking any dot products. Thanks, Jay
The patience of you chaps is admirable, if nothing else. I keep coming back and seeing where it’s at; given the history, I certainly don’t expect Joy to change his mind, but if Jay gets to understanding Bell’s theorem then at least time is not completely wasted 🙂
Anyway, I do have a question for you all now because half the time it’s not clear to an outsider, if you are all even talking about the same thing.
Bell’s theorem is a theorem, in the clear mathematical sense; does everyone agree on that? The theorem, like all theorems, says “If [hypotheses are true] then [a conclusion follows]”. It doesn’t necessarily say anything about the real world whatsoever; except if you measure the hypotheses to be true or the conclusion to be false, then you could infer something about the other. So that said, the truth of the theorem can be discussed without any mention of real world data; this is the nature and perhaps the beauty, of mathematics. It is then a different question to ask how the theorem applies to the real world. So may I ask all those arguing, in a one word answer (no more), do you believe that the mathematical theorem (again, ignoring anything in the real world) holds true?
Can you state the “theorem” in mathematical terms only?
You’re the ones arguing about it. I’m not claiming to be an expert, and I don’t want to get a detail wrong. I’m just trying to determine if you are actually arguing about Bell’s theorem, or about the physical consequences of it. As I understand it, the theorem is roughly “If you assume you have a theory with hidden variables (which can be made precise using the language of probability theory) then an inequality holds between certain quantities”
Of course this is completely mathematical, and should not require any notion of physical data to discuss it.
But what I think is happening, is that Joy is arguing that we can’t perform mathematical operations in the proof because we want to later interpret these things as physically measurable things. This is still wrong, but I think for a lot of this arguement to move on, the participants need to recognise that Joy isn’t thinking about the theorem as an actual theorem.
Perhaps an analogy to consider is someone criticising a geometric proof equating areas of certain shapes, because along the way we had to use trigonometry and in particular the number ‘pi’. However, since in the real world we cannot ever measure a length ‘pi’, it is illegal mathematics to use this number in our working, since these numbers are to correspond to physical lengths. Of course this is preposterous, but I don’t see how it it any more preposterous from saying that we can’t use the rules of mathematics in a mathematical proof if we later plan on interpreting intermediate steps as physical things.
If a theorem is based on a proposition P, but in the proof of that theorem one is forced to use the proposition not-P, then one should immediately recognize that something is seriously wrong. But that is exactly what the followers of Bell are refusing to recognize.
In my comment of “November 12, 2016 at 12:47 am” I have shown that Bell’s theorem starts out assuming Realism (i.e., proposition P) on the LHS of a mathematical identity, but are then forced to use Anti-Realism (i.e., proposition not-P) on the RHS of that identity. Anti-Realism gets surreptitiously smuggled into the proof of their so-called “theorem” when they are forced to use impossible measurement results such as B(b)+B(b’ ) on the RHS.
Bell’s so-called “theorem” is not just a mathematical theorem. It is a claim about whether or not certain correlations observed in Nature are explicable purely locally, and realistically. As such, one is not allowed to smuggle-in Anti-Realism in the proof of Bell’s “theorem.”
It is a mathematical theorem, that in turn has consequences for physical theories. But really you should be discussing it purely as a mathematical theorem first, because the theorem itself exists and can be verified independent of it’s physical implications.
So, just to clarify what your qualm with the theorem is, it is the following?
The hypothesis of the theorem is “assume local hidden variables”, which are defined mathematically, so this is your proposition P. The conclusion of the theorem is Bell’s inequality. You need to look at this as a separate issue entirely from anything physics; assuming the mathematical formulation of locally hidden variables, you can show the inequality holds.
Then you claim that somewhere in the proof of this inequality, Bell uses a result that relies on the non-existence of local hidden variables? That is, he uses “not P”. In which case, you would be correct to say this is a problem with the proof.
Yes.
It depends a bit what you mean by Bell’s theorem. Bell himself meant by it the statement “If [local hidden variables] then [CHSH inequality] follows”. It is a true theorem from probability theory originally proved by George Boole in 18-something (actually, Boole presents it as an exercise for the reader). Of course one has to give precise mathematical definitions of the technical terms mentioned in the statement of the theorem.
I like to think of it as a theorem in computer science, more specifically, in the theory of distributed computing. Think of a network of three computers: a “source computer”, repeatedly sending information to two “measurement station computers”. An outside observer repeatedly and completely at random sends inputs a or a’, and b or b’, to the two measurement station computers. Those two computers repeatedly output +/-1. The whole thing is neatly synchronised in a sequence of “trials”. The two measurement stations are not allowed to talk to one another during each trial (but they can talk to one another between trials, so as to keep one another up to date concerning results so far).
We say that there has been a SUCCESS if the two outputs are the same when the inputs are (a, b), (a, b’) or (a’, b) and different when the inputs are (a’, b’).
We say that there has been a FAILURE otherwise.
Bell’s theorem says that it is impossible to write computer programs such that the long run probability of SUCCESS is larger than 0.75.
This first paragraph:
“It depends a bit what you mean by Bell’s theorem. Bell himself meant by it the statement “If [local hidden variables] then [CHSH inequality] follows”. It is a true theorem from probability theory originally proved by George Boole in 18-something (actually, Boole presents it as an exercise for the reader). Of course one has to give precise mathematical definitions of the technical terms mentioned in the statement of the theorem.”
is what I’m getting at. Assuming “locally hidden variables” is well-defined, and everyone agrees on the definition, then do people agree that the inequality holds? Because this relies only on mathematics, and doesn’t have anything to do with who can measure what in Miami. I think you’re right in trying to explain it as a theorem outside of the context that it is used, such as a computer science result, and go from there. Anyway, good luck 😛
Stephen and HR have agreed with the datasets I posted, and while Joy and Richard have not yet weighed in, I am assuming that they will also agree with those datasets, and that their disagreements will reveal themselves to be rooted in how those datasets are interpreted and used to draw conclusions about nature.
On the basis of this assumption, and before calculating anything from those two datasets, I would like to discuss a) locality, b) realism and c) hidden variables, but in a very specific way:
For a person unfamiliar with the details of Bell (such as me about 3-4 weeks ago before I took a deep breath and plunged into this discussion) but knowledgeable about other areas of physics, a) locality is suggestive that effects can be traced to causes and that macroscopic regions of spacetime can be studied with integrals of the form $ F(t,x) dx dt where F(t,x) are functions of space and time locally defined and integrable over space and time, and that there is no “spooky action at a distance” per AE; b) realism says that if a tree falls in the woods and nobody sees it, we can still make statements about the fall of the tree as having happened in reality, as opposed to being unable even in principle to say what an electron’s simultaneous position and momentum “really” are; and c) hidden variables may bring to mind such things as DeBroglie / Bohm’s pilot waves.
But to the extent that Bell’s theorem teaches us anything about a) locality, b) realism and c) hidden variables, and to the extent that the datasets in https://jayryablon.files.wordpress.com/2016/11/bell-dataset.pdf are correct and all of the Bell conclusions are derived by mathematical deductions from these datasets, it should be possible to discuss these three notions of a) locality, b) realism and c) hidden variables *strictly in terms of features of these datasets*.
I have my own views as a Bell novice regarding what these datasets might encode about a) locality, b) realism and c) hidden variables, such as that the “null” data items are “hiding” the unseen outcomes which unseen outcomes may be “hidden variables” but are still “real” just like the fall of the tree in the woods and therefore must be entered into any calculations that one does with the datasets. And I know that EPR also gave some definitions to these three ideas. But I prefer not to elaborate yet. Rather, I would like to ask Joy and Richard and HR and Stephen and anyone else who chooses, to please answer the following:
When you look at those datasets — and never mind the calculations you think should be done with those datasets or the conclusions that should be reached — how do a) locality, b) realism and c) hidden variables as distinct ideas, get represented in, i.e., encoded into, those very datasets?
Thanks,
Jay
Jay asks how locality, realism and hidden variables should get represented into his datasets.
In fact he has already encoded all three concepts in the datasets, bravo!
Hidden variables: the 16 columns of the “slip dataset” are the 16 possible values of a hidden variable (as Jay remarked it can be thought of as a number from 0 to 15 encoded in binary).
Realism: each slip contains the information what each measurement outcome would be, for each of the two possible settings of Alice and Bob. It is predetermined, it exists independently of what (if anything) Alice actually chooses to measure. Her act of measurement merely reveals a value which is already in physical existence.
Locality: Alice’s outcome depends on Alice’s setting, but not on Bob’s setting.
Because we are restricting Alice and Bob each to two possible measurement settings, our “hidden variable” can without loss of generality be taken to be the outcome of a 16 sided dice. We are not interested in fine detail of the hidden variable. We just need to know how often each of the 16 values of the quadruple
(A(a, lambda), A(a’, lambda), B(b, lambda), B(b’, lambda))
occurs.
Presently, Jay is assuming a fair 16-sided dice. This restriction to a fair dice needs to be removed.
In answer to Jay’s question, this is precisely my view of the meaning of “realism” and “locality”, stated more concisely and eloquently than I probably could.
While writing, I’d like to mention how much I appreciate HR’s pithy comments, which generally get immediately to the heart of a question in very few words.
As far as locality goes, I’ll just copy my reply in an earlier post above:
“And note also that it is the crucial assumption of locality that makes four entries for each run suffice. For a non-local model, we would have to predict eight numbers to cover all eventualities; A(a,b), A(a,b’), A(a’,b), A(a’,b’), B(a,b), B(a,b’), B(a’,b), B(a’,b’).
Bell’s argument won’t work with an 8xN matrix. But because of the locality assumption A(a,b) = A(a,b’), etc., we only need four values to make an exhaustive prediction.”
Realism, or hidden variables, or counterfactual definiteness (they all have the same operational meaning), simply means that it is possible for the model to make complete predictions, i.e. the model can make predictions for all possible settings Alice and Bob may choose, even if their choices are not known in advance.
Joy,
In the second dataset at https://jayryablon.files.wordpress.com/2016/11/bell-dataset.pdf, while I used the “null” symbol “0/”, and also used the word “null,” on reflection I think the better way to state this is that “0/” represents the outcomes which were not observed because that detector direction was not chosen. So for example, in the AB section, I have a “0/” placed over all of the A’ and B’ outcomes because those were not seen. And while I need to be careful with this next quoted word, those 0/ outcomes are the ones which are “hidden” from Alice and Bob. Does that clarify, and does that makes sense to you?
Jay
PS: What I am really trying to do in the second dataset is enumerate all 64 possible combinations of outcomes including the 16 possible +/- detections for all of A, A’, B, B’, times the four possible choices AB (not A’ or B’), AB’ (not A’ or B), A’B (not A or B’) and A’B’ (not A or B) which may be selected by Alice and Bob and therefore “visible” or “hidden.”
OK, I get it now. By AB etc. in the column headings in the second data-set you don’t mean the actual product A*B, as I incorrectly thought. Fine. I am happy with the data set now.
Correct. Those column headings are not products. To avoid what originally confused you, I should have written them more expansively as:
select A, B (not A’ or B’)
select A, B’ (not A’ or B)
select A’, B (not A or B’)
select A’, B’ (not A or B)
Very good! Joy and Richard and Stephen and HR now all agree that the underlying data sets at https://jayryablon.files.wordpress.com/2016/11/bell-dataset.pdf correctly enumerate all of the individual outcomes we can find in an EPR experiment.
Next, I would like to poll for views on the question I posted earlier on November 12, 2016 at 10:11 am, which I excerpt below:
So while I asked for no interpretation earlier, now I am asking explicitly for interpretation. I do this keeping in mind that Stephen articulated what is for me a core operating philosophy with which I could not agree more:
And I am deliberately asking for this *before* trying to calculate anything, because we need to trace how these interpretations wind through the calculations where it is easier for them to get tangled up and disguised.
So I now ask for everybody’s views about locality and realism and hidden variables in reference to these datasets, because my “debugging” sensibilities tell me that the disagreements between Joy and Bell et al. are interpretive, and are already rooted in the way in which everyone would talk about the encoding of locality and realism and hidden variables in reference to these datasets upon which they have all agreed. And I have a feeling that once everybody talks about locality and realism and hidden variables in reference to these datasets, we will be able to identify with some precision where the path splits between Joy and everyone else.
Jay
Richard, that is exactly how I look at those same datasets.
Richard, that is exactly how I look at those datasets, and why I set them up that way. Since you and Joy are on the same page here, now, before doing any calculations, we can get to the interpretive questions about locality, realism, and hidden variables which I posted earlier today. Jay
Now we have come so far with the mathematical setup in this discussion that we can pinpoint the source of the misunderstanding the regulars over at sciphysicsforums.com.
They mistake the 4xN matrix for a matrix of *results of actually performed experiments*, and thus deem it “unphysical”, since obviously Alice cant choose settings a and a’ in the same run. We now know better; the 4xN matrix is a matrix of model predictions, quite independent of any experiment that has yet not been performed.
It is as if I say “If Alice goes to New York tomorrow, I forecast she will experience a temperature of 55F, but if she goes to Los Angeles instead, I forecast a temperature there of 80F.”
And then someone says, “Nonsense, she can’t be in both cities at the same time.”
No, she can’t. And it’s also completely irrelevant to my forecast.
I disagree. If it is irrelevant, then why don’t you prove Bell’s “theorem” without using impossible events like B(b)+B(b’ )? Such a proof would immediately settle this dispute.
But the proof does not use impossible events like B(b) and B(b’). It only uses the very possible *predictions* for both B(b) and B(b’).
No one has said that B(b) and B(b’ ) are impossible events. Their conjunction B(b)+B(b’), however, is evidently an impossible event in any possible world. Are you saying that the proof of Bell’s “theorem” does not use impossible events like B(b)+B(b’ )? If so, then please provide evidence.
B(b) and B(b’) are not events. They are predictions. What Bob would see if he used setting b; and what Bob would see if he used setting b’. B(b) + B(b’) is not an event either. It is just the sum of two predictions. It doesn’t “mean” anything. It is just some function of the local hidden variable lambda. According to local hidden variables: lambda exists; the functions B(b) and B(b’) exist; hence the function B(b) + B(b’) exists.
I disagree. While local realism, as defined by EPR, demands the counterfactual existence of possible events like B(b) and B(b’ ), it does not demand the existence of their conjunction B(b)+B(b’ ) even counterfactually. In fact the latter “event” cannot possibly exist in any possible physical world, even counterfactually. It is like the meaningless sum 3+6 of the possible landing of a die on its face 3 or 6. Even though both 3 and 6 exist within the set {1, 2, 3, 4, 5, 6} of all possible landings of the die, the “event” 3+6 is not a part of this set.
By saying that the sum B(b)+B(b’ ) does not mean anything does not make it mean nothing. It is an additional assumption smuggled-in in the middle of the proof of Bell’s theorem in order to get the desired bounds of -2 and +2. Without this additional assumption — which is not demanded by the local realism defined by either EPR or Bell — the bounds on the CHSH correlator would be -4 and +4, not -2 and +2.
To recognize this more clearly, we can look back to the much discussed mathematical identity, and — since it is a strict mathematical identity — we can simply take the integral
Int_K {A(a, k)*B(b, k) + A(a, k)*B(b’, k) + A(a’, k)*B(b, k) – A(a’, k)*B(b’, k)} rho(k) dk
as our starting point, where Int_K denotes integration over the space K of the hidden variables k. We can now ask what it is that is being averaged over in this integral. It is easy to see that it is an impossible, entirely fictitious event that is being averaged over, and therefore the bounds of -2 and +2 are equally fictitious.
I have to repeat: B(b) is not an *event*, it is the *prediction* of a number. B(b’) is not an *event*, it is the prediction of a *number*. B(b) + B(b’) is the sum of two predicted numbers.
I disagree. While B(b) and B(b’ ) are counterfactually possible events in spacetime, their sum B(b)+B(b’ ) is not possible even counterfactually, as I explained above.
Seems to me that sign(s_1^k . a), sign(s_1^k . a’), sign(s_2^k . b) and sign(s_2^k . b’) are four *numbers*, not events. See Christian’s https://arxiv.org/pdf/1405.2355v6.pdf, equations (60) and (64).
Of course it would be an *event* to measure particle 1 in direction a’ and see the value -1.
It doesn’t make sense to add events, whether counterfactual or real. But you can add numbers.
sign(s_1^k . a), sign(s_1^k . a’), sign(s_2^k . b) and sign(s_2^k . b’) are not a part of my 3-sphere model. They are Bell’s definitions of the measurement functions.
A click of a detector is an event in spacetime. This event can of course be represented by a number, like +1 or -1. But a click of a detector does not cease to be an event in spacetime just because it has been represented by a number. If we treat B(b) as a pure number, then it is easy to forget what it actually represents.
The bottom line is that B(b)+B(b’ ) is not a part of the set of all counterfactually possible events in spacetime like B(b) and B(b’ ), just as 3+6 is not a part of the set {1, 2, 3, 4, 5, 6} of all possible outcomes of a die throw.
To put an even fine point on my query of November 12, 2016 at 1:33 pm, still using https://jayryablon.files.wordpress.com/2016/11/bell-dataset.pdf which all have agreed to as the underlying dataset, let me take a close look at the first column in both datasets using Richard’s above “for instance.” I do this adopting the “urn with slips” metaphor provided by Stephen on the assumption that all agree this is a good proxy for the actual EPR experimental setup, and that we would hear if someone thought not.
So let us suppose from a single harvested datapoint, Alice and Bob draw the slip in the first column of the slip dataset for which:
(A,A’,B,B’)=(-1, -1, -1, -1) (1)
And let us further suppose that they have used A and B for their measurement direction, and not A’ and B’, so that the slip from amidst the 64 slips in the slip and selection dataset is:
(A,A’,B,B’)=(-1, not observed, -1, not observed) (2)
This slip, along with all the others, will then be pumped into a mathematical calculation. And that calculation — whatever its specifics — will be influenced by how that slip (and others) are input into the calculation.
Objectively speaking, I see one of three ways to input the slip to our calculations. Maybe there are more, but these are the options I see right now:
1) Use all values of the slip (1) in the calculations, no matter what the measurement selections are, so that the input to our calculations is:
(A, A’, B, B’) = (-1, -1, -1, -1) (1 again)
2) Treat the two unobserved values as mathematical “zeros” so they do not have any effect on the calculation, and the slip is input as:
(A, A’,B , B’) = (-1, not observed, -1, not observed)=(-1, 0, -1, 0) (3)
Then, when we take averages (division by number of trials) do we include or exclude these zeros in the number of trials, which depending on that choice, will cut the result or not by a factor of 2?
3) Treat the two unobserved values as *indeterminate* values, so that the slip is input as:
(A, A’, B, B’) = (-1, not observed, -1, not observed)=(-1, indeterminate, -1, indeterminate) (3)
Same question about averages as in 2, but now there may be some indeterminate results passing through and then output by our mathematics. And if that is so, can we live with the indeterminacy, or not? And if we can live with it, how?
Clearly, no matter what specific calculation we perform using these slips, the result will vary depending upon whether we use option 1, 2 or 3, and upon what we include and exclude from the sample size when we calculate averages.
Conclusion: I have a feeling that Joy, and Richard et al., will advocate for different ways to input the slips to our calculations, which will produce different results. And I have a feeling that locality and realism and hidden variables will enter mightily into how they justify their view of how to input the slips.
Jay
In the actual experiment, slips are drawn from urn 2, the “slip and selection dataset”.
For instance, on the first trial, the following slip might be drawn:
(A,A’,B,B’)=(-1, not observed, -1, not observed)
meaning: Alice chose setting a; Bob chose setting b; they both observed outcomes -1.
At the end of the experiment, this is what the experimenter does: for each of the four setting-pairs, count how many times each of the four possible outcome-pairs occurred.
This results in 16 numbers like N(+, +; a, b): the number of times Alice chose setting a, Bob chose setting b; Alice observed +, Bob observed +.
Define N(=; a, b) = N(+, +; a, b) + N(-, -; a, b),
define N(!=; a, b) = N(+, -; a, b) + N(-, +; a, b)
Define N(a, b) = N(=; a, b) + N(!=; a, b)
and similarly for the other setting pairs (a, b’); (a’, b); (a’, b’).
The experimenter then computes 4 observed correlations
E(a, b) = (N(=; a, b) – N(!=, -; a, b)) / N(a, b)
and similarly for the other setting pairs (a, b’); (a’, b); (a’, b’).
Finally take a look at the usual CHSH quantity.
But before you get excited at what you find, consult a statistician, and compute the standard error of your observed CHSH.
If “observed CHSH” is outside of the interval from -2 to +2 by a decent number of standard errors, and if your experiment was impeccably performed according to a strict protocol which makes it impossible to cheat, write to the newspapers that local realism has been disproved (you have observed a violation of the Bell-CHSH inequality).
If it is outside of the interval from -2 sqrt 2 to + sqrt 2 by a decent number of standard errors, and if your experiment was impeccably performed according to a strict protocol which makes it impossible to cheat, write to the newspapers that quantum mechanics has been disproved (you have observed a violation of the Tsirelson inequality).
Jay, I do not see entries like A(a)*{ B(b) + B(b’ ) } and A(a’ )*{ B(b) – B(b’ ) } in your dataset. But entries likes those are used by the followers of Bell to prove his “theorem.” Please show me where those entries are — if they are hiding somewhere — either in your description, or in the coincidence-counts recorded between the simultaneously observed detector clicks in the actual experiments (as described by Gill, for example). Thanks.
You can add these rows to the “slip dataset” if you like:
+2 0 0 -2 +2 0 0 -2 -2 0 0 +2 -2 0 0 +2
0 +2 -2 0 0 -2 +2 0 0 +2 -2 0 0 -2 +2 0
For each of the 16 slips I calculated A(a)*{ B(b) + B(b’ ) } and A(a’ )*{ B(b) – B(b’ ) }. Where, of course, “+” stands for “plus” (addition of numbers), not for “logical and” (conjunction of events).
In the “slip and selection” dataset, these new rows are hidden.
It would be useful to know which actual coincidence-count experiment has ever observed the simultaneous “clicks” +2 and -2 instead of the simultaneous “clicks” +1 and -1.
They are numbers written on pieces of paper, not “clicks”.
Then they have nothing to do with physics.
The urn metaphor is justified by Einstein’s conception of physical reality.
If you don’t care for Einstein realism, then you are not interested in Bell’s theorem anyway.
Richard, your post about frequencies for each roll of the hexadecimal die clarified that point, so I no longer assume that. Now each face of the die has its own weight. Jay
The original question about how the oft-repeated dice analogy relates to Bell’s theorem was never clearly answered. I guess that Dr. Christian means to illustrate that some physical events cannot be simultaneously realized, just in case his readers didn’t previously understand that !
The last paragraph of the above seems to indicate that he thinks that Bell’s theorem has something to do with spin and “aligning” detectors (like little telescopes, I presume). If so, that is a misconception.
The “local realism” model from which Bell’s theorem draws conclusions (that CHSH holds) says nothing about the physical means by which Alice measures *a* or * a’ *. It does not exclude that Alice could possess “detectors” which measure both at once, even if she does not happen to have such detectors.
In the local realistic model, a “detector” is represented by a random variable which we have been denoting A(a) or A(a’ ), etc. In the urn model, a detector A(a) could be implemented by a device which simply reads the part of a drawn slip which refers to *a*. Similarly for A( a’ ).
It is not logically contradictory that the part of the slip relevant to *a* could be read at the same time as the part relevant to * a’ * .
Dear Q,
Appreciating your straight-forward question, I gave you my ‘one word’ answer: Yes!
So where to from here?
PS: Click on my name for some suggestions, and/or contact me privately if you wish.
With best regards; Gordon
Thanking RW for bringing this issue to our attention, here I stand:
(a) Bell’s theorem is the basis for Annals of Physics retraction of Joy Christian’s study.
(b) Bell’s theorem is a mathematical theorem consistent with Bell’s assumptions.
(c) To the extent that Bell’s assumptions relate to physical reality, Bell’s theorem is limited to that reality.
(d) Alas, under Bell’s assumptions: Bell’s theorem is theoretically and experimentally limited to a naively classical world-view.
(e) I have therefore written an open letter to Annals of Physics, challenging their position re Bell’s theorem.
(f) My letter is consistent with common-sense, QM, modern experiments, Einstein’s principles, EPR’s belief and Bell’s hopes and expectations. In short, I replace Bell’s naive classical definition of local realism with an all-embracing definition of local realism.
(g) Click on my name to read that open letter and/or contact me privately if you wish.
My thanks again to RW.
What do you mean “It doesn’t make sense to add events”? That is what the experimenters do to calculate the correlation.
Christian wrote “While B(b) and B(b’) are counterfactually possible events in spacetime, their sum B(b)+B(b’ ) is not possible even counterfactually” and “their conjunction B(b)+B(b’) is evidently an impossible event in any possible world”.
But B(b) and B(b’) are not events, but numbers. And B(b)+B(b’) is just the sum of two numbers, not the conjunction of two events.
Neither Einstein’s nor Bell’s local realism demands simultaneous coincidence “clicks” in spacetime like A(a)*{ B(b) + B(b’ ) } and A(a’ )*{ B(b) – B(b’ ) }, but the local realism of the followers of Bell does. If the followers of Bell wish to test their local realism, then they must use coincidence-counts involving only these hybrid clicks to demonstrate that Nature violates their local realism. They can’t have it both ways. Either they must admit only coincidence-counts like A(a)*{ B(b) + B(b’ ) } and A(a’ )*{ B(b) – B(b’ ) } in their dataset, or not use such impossible spacetime events in the proofs of their supposed “theorem.”
Einstein’s local realism demands that A(a), A(a’), B(b) and B(b’) simultaneously exist as elements of reality. Hence A(a)*{ B(b) + B(b’ ) } + A(a’ )*{ B(b) – B(b’ ) } exists.
Einstein does not say that we can measure it, nor (if it could be measured) how.
Einstein’s local realism demands that A(a), A(a’), B(b) and B(b’) exist at least counterfactually as elements of reality. It by no means demands that their conjunctive sum A(a)*{ B(b) + B(b’ ) } + A(a’ )*{ B(b) – B(b’ ) } exist, even counterfactually. The latter sum in fact cannot possibly exist in any possible world, because a and a’ and b and b’ are mutually exclusive directions.
EPR criterion of reality: “If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity.”
It is quite obvious that according to this criterion there does not exist an element of reality pertaining to the impossible quantity A(a)*{ B(b) + B(b’ ) } + A(a’ )*{ B(b) – B(b’ ) }.
Nobody is claiming that A(a)*{ B(b) + B(b’ ) } + A(a’ )*{ B(b) – B(b’ ) } is an element of reality in Einstein’s sense. It exists, because A(a), A(a’), B(b) and B(b’) all exist. Whether or not anybody measures anything.
The symbols +, – and * stand for addition, subtraction and multiplication (of integers). There are no “conjunctive sums” involved. You are mixing up logic and arithmetic.
The point is that the events B(b) and B(b’ ), even if represented as pure numbers, can exist in conjunction with A(a) and A(a’ ) only counterfactually. Therefore it makes no sense to mix them up as B(b) + B(b’) and B(b) – B(b’ ) and pretend that such mixing has some physical meaning. It doesn’t. The quantity A(a)*{ B(b) + B(b’ ) } + A(a’ )*{ B(b) – B(b’ ) } is a pure fiction.
B(b) and B(b’) are not events. They are not “represented” by numbers. They *are* numbers.
B(b) and B(b’ ) are two counterfactually possible events in spacetime — such as clicks of Bob’s detector — which may be represented by pure numbers, such as +1 or -1.
Should’nt the conclusion of Bell’s Theorem stop at “when it comes to this QM property, the sum of the averages isnt the average of the sum” ?
As an experimental physicist, I wonder about this ongoing discussion. To my mind, Bell-type inequalities can be straightforward derived on base of the conception of physical reality as proposed by Einstein, Podolsky and Rosen (Phys. Rev. 47, 777-80 (1935)):
All spin components of an interacting two-particle system are simultaneous elements of reality**, viz. nature – so to speak – “knows” the values of all these quantities for any given interacting two-particle system before any measurement is performed by an observer. The values, which these elements of reality can have – in any allowed combination under a given constraint like, e.g., “total spin = 0”, hence “pre-exist” or are “pre-scribed” before any measurement is done. Measurements thus merely reveal the physical reality (Einstein’s point of view regarding local physical reality). Our inability to exactly predict the outcomes of certain measurements on members of an ensemble of identically prepared, interacting two-particle systems must therefore result from the fact that some “randomness of spin orientations” exists when the two particles separate after having interacted. This “randomness” can formally be taken into account by assuming that there is some random and to us hidden variable lambda which rules the story.
That’s all I need to know: There is an objective physical reality. That means, from the viewpoint of an experimental physicist: Thought experiments on such systems have “functionally” the same significance as real experiments. Thus, I can perform – in thought experiments – “measurements” on every member of an ensemble (constructed by taking into account the “randomness”) of identically prepared, interacting two-particle systems to find out, for example, possible expectation values for some correlations and possible physical bounds these expectation values are subjected to. When I am lucky, these thought experiments lead to some relations which can be checked by performing real experiments. Then, there remains only a “technical” question: How many real measurements have to be performed to ensure that the mean values of measured outcomes of certain correlations approach the expectation values of these correlations.
**The assumption that all spin components are simultaneous elements of reality does not automatically imply that we must be able to measure all these components simultaneously. The respective quote from Einstein, Podolsky and Rosen (Phys. Rev. 47, 777-80 (1935)) reads: “Indeed, one would not arrive at our conclusion if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. On this point of view, since either one or the other, but not both simultaneously, of the quantities P and Q can be predicted, they are not simultaneously real. This makes the reality of P and Q depend upon the process of measurement carried out on the first system, which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this.”
Joy and Richard,
Let me reply to some of the above and also ask you some questions. Here is a convenience link to the datasets we have been discussing: https://jayryablon.files.wordpress.com/2016/11/bell-dataset.pdf.
First, Richard, thank you for answer as to locality, realism and hidden variables, excerpted above. That is exactly the type of thing I was hoping for. Joy, do you agree with Richard on these answers?
Second, Richard, we are getting ahead of ourselves to talk about the statistician yet: Any statistical analysis always has to start with a kernel of sample datapoints that we have, and then extrapolate these by making certain suppositions about data we do not have. For instance, you poll a sample of the populous about who they prefer for president, and then you extrapolate to the entire populous and predict the first female president, and then you fall flat on you face. 🙂 because I tied to make a funny, 🙁 because of what happened last week. But the point is that we need to first to lay out and agree on a core data kernel from which we are extrapolating, before we can start statistical calculations, and that is what I want to first settle. You can poll 4000 household and then make a presidential prediction even if it is wrong. But you cannot poll 0 households and make a prediction. You need a core dataset. And until you and Joy agree, if possible, on the core dataset and what that means, we will end up back at he same arguments you have been having for months and years.
Third, Joy, we are getting ahead of ourselves to talk about A(a)*{ B(b) + B(b’ ) } and A(a’ )*{ B(b) – B(b’ ) } because these too are calculations and we need to first have a common understanding of the core data. Specifically, how we ought to treat the “null entries” once we start to do calculations is very important, and should be settled before doing any calculations, be they dot products and sums (CSHSH), or statistical projections.
This brings me to my excerpt above “Objectively speaking, I see one of three ways to input the slip to our calculations….” This is what Richard now calls the three urns. This is very important to discuss. I was surprised when Richard said he would choose urn 2; I thought he would say urn 1 because that contains the “realism” data and I would have thought that realism would mandate that we include that data even if you do not know that data, because it still exists as a matter of realism. Also, I myself am squeamish about treating the “0/” as a mathematical zero, unless that is an explicit predicate to use these zeros to clear the decks for the layer statistical calculation.
Further, I think that every time Joy talks about the incompatibly of adding something observed to something that was never observed as in B(b) + B(b’ ) and B(b) – B(b’) I read this as saying that we are adding a determinate number to an *indeterminate* number, which is why my betting is that Joy would choose Urn 3 where the unseen numbers are indeterminate and treated as such mathematically. And I think that when Richard talks about statistics, he is saying that even if you do not know these numbers, you can make statistical guesses about them. So Joy, I am quite deliberately not yet talking about numbers A(a)*{ B(b) + B(b’ ) } and A(a’ )*{ B(b) – B(b’ ) } which are calculated from these datasets, because there is a core question that needs to be answered before we do any calculations, and it is this:
***How do we interpret and use the null entries in the datasets at https://jayryablon.files.wordpress.com/2016/11/bell-dataset.pdf once we start doing calculations?***
For example, the “null” symbol “0/” may be hiding a -1 or it may be hiding a 1 but we do not know which it is. Just like a lottery ticket where to use a coin to scrape off a cover from a hidden lottery number. If we feed the -1 or +1 into our calculations by treating “0/” as a -1 or a +1 we will get one answer. If feed a 0 into our calculations by treating “0/” as a 0 (which Richard is implying when he selects urn 2) then we will get a second answer. If we feed “indeterminate” onto our calculations by treating “0/” as an indeterminate number, then we will get a third answer, or perhaps conclude that the answer is nonsense, which is what Joy is doing. And that does not even get us to taking the incomplete information owing to these “0/” entries and using statistics to try to extrapolate something from these “0/” entries, which is also predicated on how we treat these “0/” in the underlying dataset.
On top of that, my instinct is to use urn #1 because this encodes realism that we must use if we are trying to develop a theorem about realism in physics, Richard says use urn #2 and I think part of that is because this sets up the statistical argument, and I think Joy is saying use urn #3 because that sets up the incompatibility / indeterminacy / meaninglessness argument. Which means a two-position argument may well now be a three-position argument.
So again:
***How do we interpret and use the null entries in the datasets at https://jayryablon.files.wordpress.com/2016/11/bell-dataset.pdf once we start doing calculations?***
I am not yet convinced that there is necessarily a “mathematically” correct answer to this. So far, it seems to me that there are judgement calls about the treatment of the “0/” data entries being made and Joy and Richard and I want to have these be clearly articulated before starting any calculations, be they writing scalar products or using statistics on unknown data rooted in a known data kernel.
Jay
To me A(a)*{ B(b) + B(b’ ) } and A(a’ )*{ B(b) – B(b’ ) } ARE the core data (albeit impossible), because that is what is used to prove Bell’s theorem. The data you have constructed in your table has nothing to do with Bell’s theorem, because that is not what is used to prove the theorem.
The “Slip and Selection Dataset” contains the data available to an experimenter. It tells us, for each trial, what settings were chosen, and what outcomes were observed when measurements were done at those settings. If you want to know what is the correlation between, say, A(a’) and B(b), you select just those trials where settings a’ and b were used together (that’s 25% of all the trials).
To say the same thing in a different way: null entries in the “Slip and Selection Dataset”are not to be interpreted as zero’s, but as “not available”. Each of the four correlations is calculated using the 25% of trials for which the two relevant variables are both available. No guessing, no statistical extrapolation.
Jay, I think that the very nice way you have structured the tables in you pdf-file makes Bell’s proof practically jump out of the page. The ogic of the proof will go like this:
1) Show that if you use the slip and selection dataset to compute the CHSH-expression, the result will be the same as if you used the slip dataset (after all, the slip and selection dataset is simply the slip dataset distributed into four tables).
2) Show that for the slip dataset, the CHSH expression is mathematically bounded between -2 and +2.
3) Use 1) to conclude that the bounds will also apply for the slip and selection dataset.
To clarify this, by saying “compute the CHSH expression on the slip and selection dataset”, I mean that for each of the four terms in the CHSH expression, you use the values in the corresponding tables in the slip and selection dataset, just as experimenters would do. The Ø data entries play no role in this.
It seems to me that the only mathematical fact that may be agreed by all is the following:
-2 < { Int_K A( B + B’ ) rho(k) dk + Int_K A'( B – B’ ) rho(k) dk } < +2 ,
where K is the space of the hidden variables k. Everything else is open to interpretation.
That particular mathematical fact does not play any role whatsoever in Bell’s proof. Why do you bring it up?
The first step in Bell’s proof is to show that the quantity
Int_K { A B + A B’ + A’ B – A’ B’ } rho(k) dk
lies between -2 and +2.
What I have written above is the same as
Int_K { A B + A B’ + A’ B – A’ B’ } rho(k) dk lies between -2 and +2.
I have no problem with this mathematical fact because it does not require any interpretive baggage the followers of Bell would like to attached to it. To me it is clear that since the integrand of the above “average” is physically meaningless, the bounds of -2 and +2 on the “average” are also physical meaningless. They are simply mathematical curiosities, without any relevance for the question of local realism, or for physics in general.
I would say, this is also the same as saying
Int_K A B rho(k) dk + Int_K A B’ rho(k) dk + Int_K A’ B rho(k) dk – Int_K A’ B’ rho(k) dk
lies between -2 and +2.
And you have yourself said that this “seems to me the only mathematical fact that may be agreed by all”.
It’s a mathematical fact which requires no interpretative baggage. The integrands are now physically meaningful, so the bounds are now physically meaningful.
On the contrary.
I agree that
Int_K A B rho(k) dk + Int_K A B’ rho(k) dk + Int_K A’ B rho(k) dk – Int_K A’ B’ rho(k) dk
is mathematically — and hence physically — identical to
Int_K { A B + A B’ + A’ B – A’ B’ } rho(k) dk ,
which in turn is mathematically — and hence physically — identical to
Int_K A( B + B’ ) rho(k) dk + Int_K A'( B – B’ ) rho(k) dk } .
Moreover, all three expressions lie between the values -2 and +2. Therefore there is no harm in concentrating on the last one. The last expression involves fictitious quantities like A( B + B’ ) and A( B – B’ ). These quantities do not belong to the hidden variable space, because that space is not closed under addition, just as the set D = {1, 2, 3, 4, 5, 6} of all possible outcomes of a die throw is not closed under addition (for example, 3+6 is not in the set D).
To put differently, the quantity A( B + B’ ) does not represent any meaningful element of any possible physical world, classical or quantum. This is because B and B’ can coexist with A only counterfactually. If B is realized with A, then B’ is NOT realized with A, and vice versa.
Consequently, since the two integrands of the last integral expression are physically meaningless in the sense just noted, the bounds of -2 and +2 on that expression are also physical meaningless. They are simply mathematical curiosities, without any relevance for the question of local realism, or for physics in general.
Nevertheless, I am willing to see what Jay comes up with in his calculations, even though his representation of local realism still does not fully address my worry brought out above.
It seems that Jay is now focusing on an urn model, where the model can pick one of 16 slips for each run. What correlations such a model can produce seems to be a purely mathematical question, with no openings for interpretation.
A few days ago I posted some datasets in https://jayryablon.files.wordpress.com/2016/11/bell-dataset.pdf which I have tried to use to provide a common base of data for discussing Bell’s Theorem. Based on the discussion these last several days, I have modified that presentation of that dataset in several important ways, and posted a new dataset file at https://jayryablon.files.wordpress.com/2016/11/bell-dataset-2.pdf. Let me explain the important differences:
The earlier dataset simply placed a “null” entry to represent measurement directions not chosen. That was because I did not want to prejudge the meaning of those unchosen directions or how they are used in downstream calculations such as A(a)*{ B(b) + B(b’ ) } and A(a’ )*{ B(b) – B(b’ ) } which are of particular concern to Joy. But now I feel I have learned enough to make my own judgments about how to properly take the next steps, and so I have done so in these new datasets.
First, I continue to show the same “slip dataset” as before, which represents the 16=2^4 possible +/- combinations for A, A’, B, B’ using the “slips in urn” metaphor that Stephen first recommended and that all seem to have agreed to use as a stand-in for the EPR-Bell-CHSH experiment. However, for any one trial, based on what their separate, independent, *local* coin flips tell them to do, Alice and Bob only choose one or the other of A or A’, and B or B’ respectively. So the key issue I have tried to get a handle on via the discussion in recent days is how we talk about the measurements not chosen. And I do think this is where Joy diverges from others here. Now I will say what I think, again, with the caveat that I can always stand corrected and wish to be corrected if I am amiss in any way.
In my humble opinion, the next step that needs to be taken based on my earlier datasets is to postulate “local realism.” Specifically, we postulate that all of the 4 values A, A’, B, B’ for any given slip in the “slip dataset” *actually do exist in reality*, even though the slip is like a lottery ticket with the four values covered over. Then, the separate coins are flipped telling each of Alice and Bob which *two* of the four covers they can scrape off the ticket, and they do so. Now, I emphasize: this is a *postulate*. It is NOT an assertion that the observed universe with quantum mechanics is local realistic. We make this postulate, and see what it leads to. If it leads to a contradiction with something that is observed, for example, QM correlations, then we disprove the local realism postulate. If it does not lead to any contradiction, then the postulate is not proven, but rather, is allowable, i.e., not disproved. Then Joy’s S^3 model enters as allowable, subject to proof that it reproduces the QM correlation data.
But the important point is that by making the realism postulate, we must then be allowed to talk about all four values on the lottery slip as *actually existing*, even though two of them must remain covered. Based on this realism postulate, I have modified my representation of the earlier “slip and selection datasets.” I am no longer using the “null” symbol, but rather have used alpha, alpha’, beta and beta’ to represent the values +/-1 which are on the slip but not scraped off, based on the realism postulate which says they are still under the covering. You will see four such “slip and selection datasets,” one each for the A, B; A, B’; A’, B’; and A’, B’ measurement choices. Based on the values I have also shown assigned to alpha, alpha’, beta and beta’, at https://jayryablon.files.wordpress.com/2016/11/bell-dataset-2.pdf, you will see that these precisely replicate the original slip dataset. These have indices running from 1 to 64. In representing the data this way, alpha, alpha’, beta and beta’ represent the values that are “hidden” on the slip and not scraped off. These are the “hidden variables” and emerge naturally and inexorably once we introduce the realism hypothesis. (I am using alpha and beta rather than lambda because I want to distinguish Alice’s from Bob’s hidden variables and I am using the primes or not as a reminder of which measurement directions are not chosen.) By the realism hypothesis, we are now forbidden from supposing that these hidden +/-1 values do not exist or have no meaning. Rather, we do not know what they are in any given case, but can say for certain that they have values of +/-1. This means — eliminating some of my earlier options — we cannot call them zero, and we cannot call them indeterminate in the sense of causing a mathematical expression to blow up. We must simply call them +/-1 based on realism, with values unknown to us.
Finally, the “slip and selection datasets,” with 64 possible combinations, are a bit cumbersome. So we squeeze those down to a “consolidated slip and selection dataset.” The index numbering for alpha, alpha’, beta and beta’ now is redefined and runs from 1 to 16, and rather than give each alpha, alpha’, beta and beta’ a definite value, we simply value them at +/-1.
Now we are in a position to start calculating with local realism built in as a starting hypothesis, and hidden variables naturally emergent from this hypothesis. This is how we may now start to use expressions like A(a,alpha’), A'(a’,alpha), B(b,beta’) and B'(b’,beta).
The next step, it seems to me, is to assign “frequencies” to each of the slips for any given experiment, then use the consolidated slip and selection dataset as a guide to specify the vectors A(a,alpha’), A'(a’,alpha), B(b,beta’) and B'(b’,beta) at the selected frequencies with the understanding that alpha, alpha’, beta and beta’ are each equal to +/-1 and independently so for each slip. Then, we finally use these vectors in the CHSH calculations to see what comes out, and by assuming fair coins, we weight AB, AB’, A’B and A’B’ equally at 25% irrespective of the frequency weights.
I honestly do not know at this point what will come out of my own CHSH calculation based on these new datasets, but I do feel comfortable that I can have confidence in whatever results do emerge, because the data foundation is sound, and I will calculate this through very carefully with all of you looking over my shoulder.
If I am badly off the mark in any of what I have laid out here, please let me know.
My participation here at RW may be less intensive over the next 10 days, because I have a patent to write this week, and then am traveling for several days encompassing next weekend. But I will keep an eye on RW and at least be thinking about all this if not writing as much.
Jay
Yes, this is a far better version than what you had before. It seems to address my central worry, at least partially. I will think about it some more to see whether there is anything unaccounted for, or over-accounted for.
The only thing that separates two different urn models from each other, are the frequencies used for each of the 16 slips. Anyone who claim they have a local hidden variable that reproduces the quantum correlations, could then just simply publish the 16 weights that correspond to those frequencies from their model. In that way, anyone could check (by brute force) if the claim was correct. No need to even mention Bell’s theorem. Only publish 16 numbers.
I have had some time to reflect on Jay’s latest installment and now I know what is wrong with it. The definition of realism he has implicitly used — which is essentially the one assumed by the followers of Bell — is unjustifiably more restrictive than the one envisaged by Einstein.
Let me try to bring out what is wrong with it as transparently as I can. To this end, let me write the CHSH string of expectation values in the following form:
Int_K A B rho(k) dk + Int_K A B’ rho(k) dk + Int_K A’ B rho(k) dk – Int_K A’ B’ rho(k) dk . … (1)
This expression is both mathematically and physically equivalent to the expression
Int_K { A( B + B’ ) + A'( B – B’ ) } rho(k) dk , …………………. (2)
where Int_K stands for an integration over the space K of the hidden variables k.
Since the above two integral expressions are identical to each other, there is no loss of generality if we use the second expression to bring out the unjustified assumption in Jay’s definition of realism (which, as I have stressed, is essentially that of the followers of Bell).
To begin with, the second expression, equation (2), involves an integration over fictitious quantities like A( B + B’ ) and A( B – B’ ). These quantities are not parts of the space of all possible measurement outcomes A, A’, B, B’, etc., because that space is not closed under addition. This is analogous to the fact that the set D = {1, 2, 3, 4, 5, 6} of all possible outcomes of a die throw is not closed under addition. 3+6, for example, is not a part of D.
But there is also a much more serious physical problem with the definition of realism assume by Jay. As I have stressed before, the quantity A( B + B’ ) does not represent any meaningful element of any possible physical world, classical or quantum. This is because B and B’ can coexist with A only counterfactually. If B coexists with A, then B’ cannot coexist with A, and vice versa. But in his analysis Jay implicitly assumes that B and B’ can both coexist with A simultaneously. This would be analogous to my being in New York and Miami at exactly the same time. But no reasonable definition of realism can justify such an unphysical demand. The notion of realism envisaged by Einstein most certainly does not demand any such thing.
Finally, note that Jay does not have to actually use expression (2) in his analysis for it to be wrong. He can restrict his analysis to expression (1) only and it would still be wrong, because, as noted, (1) and (2) are both mathematically and physically identical.
I have tried to trim Christian’s post, but finally decided that to avoid distorting his meaning, I should include the whole thing.
He has repeated these arguments many times, apparently without convincing anyone in this forum except for SciPhysicsFoundations folks. Many have pointed out that Bell’s hypothesis of “local realism (and Jay’s correct implementation of it) sets up a mathematical model which may or may not be capable of describing what we see in the laboratory. The urn model *can* reproduce the correlations predicted by *some* quantum states and some measurement settings. It *cannot* reproduce the correlations for those states and settings for which the CHSH sum is not between -2 and +2. The *conclusion* is that the urn model cannot describe everything that we see in the laboratory.
I wish that Dr. Christian would clearly state whether or not he agrees that Jay’s urn model cannot reproduce all quantum correlations. If so, there is no disagreement about the facts
(at least between Christian and mainstream physics). The only disagreement would be about the *definition* of “local realistic”. If Dr. Christian has a different definition and wants to use it, that is his privilege. The definition assumed by Jay’s model is the one used by just about all physicists when then talk about Bell’s theorem.
I also have a remark about Christian’s apparent insistence that there is a flaw in the proof of Bell’s theorem. Assume for the sake of discussion that there really is such a flaw. That would make its *proof* incorrect, but would not have any implication concerning the correctness of its *conclusion*. It is logically possible for the conclusion of a theorem to be correct even if some claimed proof is incorrect.
At the risk of seeming repetitive, I state again that the *conclusion* of Bell’s theorem is that the urn model cannot reproduce quantum correlations for which the CHSH sums are not between -2 and +2. Does Christian think that this conclusion is actually false, or just as yet unproven ?
The use of those components simultaneously in the expression as Bell did amounts to an assumption that they are simultaneously measured. That is the point of the discussion.
No, it does not. In a hidden variable theory quantities can exist and have definite value without being measurble. That’s why they are called “hidden”.
OK, I want to avoid getting too excited prematurely, but I believe I have figured out in these last 2.5 hours how to reconcile Christian and Bell, which I wrote up immediately. Each is correct, in their own ways. I have put all of the development for this into https://jayryablon.files.wordpress.com/2016/11/chsh-and-quaternions.pdf.
The bottom line is that Bell applies as always when the hidden variables are treated as ordinary numbers. But if we are willing to give up some small amount of realism while keeping full locality, and the realism we forego is the same exact realism we forego when we we detect a quaternionic spin along one axis and thereby disturb the spin along the other two axes which happens to be the same thing that must physically occur in EPR, then we can have a completely local theory consistent with quantum mechanics, and thereby no longer need “spooky action at a distance,” so long as our realism is also consistent with quantum mechanics based on the degree of spin realism that is lost via rooting in Heisenberg. I have sketched out this approach. Because he has been talking about quaternions and S^3 forever, I have a feeling Joy will be able to fill in some details that I still need to sleep on. Which I shall now do. 🙂 Jay
While I agree with Jay’s mathematical analysis in his latest PDF, I am afraid I cannot agree with his use of the notion of “realism”, which he has borrowed from Bell and his followers. As I have explained in my post of “November 14, 2016 at 6:05 pm”, that notion is fundamentally misguided. It is not the notion envisaged by either Einstein or EPR. It presumes, for instance, that it is physically possible for me to be in New York and Miami at exactly the same time.
I should have added the EPR criterion of reality in my previous post. Let me do that now:
EPR criterion of reality: “If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity.”
Moreover, in my paper (which was withdrawn from Annals of Physics) nowhere do I use any operators to represent the actual measurement results of spins. I use only ordinary numbers, +1 or -1, and strictly local-realistic functions like A(a, lambda) = +/-1 and B(b, lambda) = +/-1, precisely as defined and required by John Bell: https://arxiv.org/abs/1405.2355 .
To reiterate, while I agree with most of Jay’s analysis in his latest PDF, conceptually we still have a long way to go in recognizing that Bell simply used a misguided idea of realism to prove his theorem, as I have tried to explain in my post of “November 14, 2016 at 6:05 pm.”
Bell’s theorem is a straightforward No-go theorem when one accepts the key assumptions it is based on. Maybe, the following quote from John R. Boccio’s textbook “In Search of Quantum Reality” clarifies what is meant:
“Within the framework of a theory that is realist, deterministic, and separable,
we can describe the photon pair in detail. Realism leads us to believe that
polarization is an objective property of each member of the pair, independent
of any measurements that may be made later. Determinism leads us to believe
that the polarizations are uniquely determined by the decay cascade, and that
they are fully specified by the hidden variable “lambda”, which governs the correlation
of the polarizations in A and B. Finally, separability leads us to believe that the
measurements in A and B do not influence each other, which means in particular
that the response of detector A is independent of the orientation of detector B.”
The terms A(a), B(b), A(a’) and B(b’) in Bell’s integrand A(a)* B(b) + A(a)*B(b’) + A(a’)* B(b) – A(a’ )*B(b’) must thus be understood as “codings” for the possible values these objective properties can have. In principle, there is no need for any measurement at all.
If one accepts these key assumptions, Bell’s theorem cannot be disproved. Tinkering around with these key assumptions is a completely different story as one is then leaving the basis of Bell’s reasoning.
This is a good time to again take stock of the point to which we have progressed in this discussion:
We started with Joy’s S^3 paper which was retracted by AOP. Because this paper presumed a problem with Bell’s Theorem, at Stephen Parrott’s good suggestion we focused on Bell. We got to the point of diagnosing that Joy believes the CHSH limits of +/- 2 to be too narrow. All have agreed on the Bell “dataset,” and that these limits are obtained by encoding certain definitions of locality and realism into the Bell dataset. Locality is encoded by two “fair coin” tosses separated by spacelike intervals which tell Alice and Bob how to independently align their detectors. Nobody disagrees. Realism is encoded by using the datum *not* observed following detector alignment for a given trial as “hidden variables,” and assuming these are still “realistic” data even if not seen, and assuming that these hidden variables have the undeterminable but still realistic values given either by +1 or by -1, just like the observed data. Based on those definitions to encode locality and realism, we have shown that the CHSH limits are in fact +/-2 as the Bell proponents have maintained, and as I show in (10) of https://jayryablon.files.wordpress.com/2016/11/chsh-and-quaternions.pdf. And in (9) we see that this limit comes about by certain terms B + beta’ and B – beta’ in which a hidden variable is added to an unhidden variable. Further, Joy has agreed that if this encoding of realism is used, then +/-2 will in fact be the CHSH limits.
So now, we have isolated Joy’s disagreement to his belief that the definition of realism as encoded into Bell’s theorem is too restrictive, whch comes to bear in how we regard these B + beta’ and B – beta’ term in my (9). So the next stage of this discussion must embark upon a full-fledged discussion of “realism” as a physical concept, and of how this physical concept is to be mathematically encoded into the datasets used in connection with Bell’s theorem and the CHSH identity.
Personally, I am much more bothered by the loss of locality and the spooky action at a distance that this would imply, than by having to yield on realism so long as locality can be preserved. And I am comfortable that if we use a modified version of realism in how we handle the Bell data, then we can and do avoid “spooky action at a distance.”
I have suggested that we must now “abandon” realism, to which Joy has objected. But I think we are now down to linguistics, and then implementation. So let me use a use the softer wording “modify” realism, and let me put it this way:
We can preserve locality and avoid spooky action at a distance, even in view of Bell’s Theorem, if we *modify* how we encode realism in the Bell dataset. And in particular, we may no longer assume that once Alice and Bob have made a detection, the hidden variables remain in their undisturbed +/-1 configuration. And in saying that he disagrees with the encoding of realism into Bell, but does not abandon realism, Joy is also saying that we need to adopt a modified view of realism and how it gets encoded into Bell. And he points to the EPR defintion “If, *without in any way disturbing a system*, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity.”
To me, the key phrase here is “without in any way disturbing a system.” If we use “D” to designate disturb and “~” for the logical “not”, then “without in any way disturbing a system” is logically represented by ~D which then is followed by an implication arrow. Or, “D = false implies…” This needs to be the starting point for developing an inverse definition of realism when “D=true”. And so:
*What is to be our definition of realism for when a system *is* disturbed, and how is this to be encoded into Bell? *
Specifically, when Alice and Bob do detect a spin sgn(s.a) and sgn(s.b), they “do” disturb the system and so the “disturbed system” definition of realism must be brought to bear. Because spins once detected only have a definite value along the z axis and are disturbed into the “cone indefiniteness” along the x and y axes as seen in https://en.wikipedia.org/wiki/Spin-%C2%BD#/media/File:Spin_half_angular_momentum.svg, it seem to me that a discussion of “realism” in this spin disturbance / spin cone situation is a good place to start. So here is are my further questions:
When we observe a spin up or spin down along the z axis and thereby *disturb* the observed particle, how do we discuss, and mathematically encode, the notion of “realism” as it relates to the spin along the x and y axes? We know that one way of discussing this is to say that we can only diagonalize one of three spin matrices simultaneously, so that only one direction of spin is observable even in principle. So what does this imply for the “realism” of the spin along the other two axes? And what does this tell us about the hidden variables once Alice and Bob do detect a spin direction and thereby disturb the doublet system? And how does this all get encoded into the mathematics of Bell and the CHSH sums and their limits?
Jay
While this might turn into an interesting discussion, I’m not sure that the comment section of retractionwatch.com is the right place for it. The original question here was whether it was correct to retract Christian’s paper. None of the considerations in your post is addressed in that paper. The editors cannot base their retraction decisions on what a paper the author might have written; they must make their decision based on the paper at hand. An if I understand you correctly, you agree that his claim that he has disproved Bell’s theorem, as this theorem is understood and agreed upon by the community, was wrong.
But Bell’s theorem can be and has been disproved. Let me do that for you right now, right here. I will not use any conceptual arguments, or question Bell’s assumptions.
It has also been the driving force behind Jay’s enormous efforts here to prove either me or Bell wrong, or right, purely mathematically, if possible. So this post is for you too, Jay.
My purely mathematical disproof of Bell’s theorem goes as follows. Let us write the Bell-CHSH string of expectation values in the following form:
Int_K A B rho(k) dk + Int_K A B’ rho(k) dk + Int_K A’ B rho(k) dk – Int_K A’ B’ rho(k) dk . … (1)
This expression is mathematically equivalent to the expression
Int_K { A( B + B’ ) + A'( B – B’ ) } rho(k) dk , ………………… (2)
where Int_K stands for an integration over the space K of the hidden variables k.
Since the above two integral expressions are mathematically identical to each other, there is no loss of generality if we use the second expression to disprove Bell’s theorem. Evidently the second expression involves an integration over quantities like A( B + B’ ) and A( B – B’ ). These quantities are not parts of the space of all possible measurement results such as A, A’, A”, A”’, … , B, B’, B”, B”’, etc., because that space is not closed under addition. This can be easily proved. Since each function B is by definition either +1 or -1, the sums such as B + B, or B – B, can be +2, or 0, or -2. Thus the sums like B + B are not parts of the space of all possible measurement results. Therefore the often repeated claim that “since A(a), B(b), A(a’) and B(b’) all exist according to realism, the quantity A(a)*{ B(b) + B(b’) } + A(a’)*{ B(b) – B(b’) } appearing in the integrand of (2) must also exist” is simply false. It is not a part of the space of all possible measurement results, and therefore it simply does not exist even as a purely mathematically valid expression within Bell’s assumptions. Consequently, even if we accept that A, A’, A”, A”’, … , B, B’, B”, B”’, etc. all exist as counterfactually possible results regardless of whether they are actually observed, Bell’s theorem is mathematically false. QED
Why should the quantity A(a)*{ B(b) + B(b’) } + A(a’)*{ B(b) – B(b’) } be a part of the space of all possible measurement results? It is obviously not meant to be the value of a measurement .
Joy, let me fine tune this. You have objected all along to “quantities like A( B + B’ ) and A( B – B’ ),” and especially the sums B+B’ and B-B’, using terms like counterfactual and the like. I have shown that in (9) of https://jayryablon.files.wordpress.com/2016/11/chsh-and-quaternions.pdf that these sums start with an observed variable B, and then add or subtract a *hidden* variable beta’ (or vice versa). In other words, the operations you are complaining about always involve addition and subtraction between *observed* variables and *hidden* variables. You regard these as illegal operations, violating some definition of realism. So, assuming you still believe my (9) is correct based on the Bell realism definitions that you disagree with:
1) Please provide an alternatve definition of “realism” which you believe ensures that this operation of additively combining observed variables with hidden variables, is illegal. That seem to be the core of what you are objecting to.
2) Is it your view that the hidden variables should be regarded as +/-1 even after an observation has been made? And if not, then what comes of these hidden variables, i.e., what should be their proper treatment?
Jay
IMHO, you can’t say anything about the spins before detection. The polarizers (Stern-Gerlach device in the EPR-Bohm scenario) actually “guide” the spins into alignment with the angle of the polarizer. And you can have either spin up or spin down at that point. So even a LHV model can’t predict the A and B results because of the very nature of EPR-Bohm.
Jay, I am neither inclined nor at liberty to provide my own definition of realism. But what I will do is remind everyone, as I have done before, of the carefully thought-out definition of realism by Einstein and two his collaborators. I will then explain, or simplify, this definition in terms of our current concerns and notations. So, first, here is Einstein’s definition of realism:
EPR criterion of reality: “If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity.”
What does this criterion mean within our current context? Well, at least Gill seems to have understood what it means in the context of EPR versus Bell debate. It simply means that measurement results such as A(a), B(b), A(a’) and B(b’) all “exist” at least counterfactually in the space of all possible measurement results, whether or not they are actually realized.
Incidentally, your alpha and beta are not hidden variables. Someone from the Bell camp should have corrected this long ago. Your alpha and beta are simply counterfactually possible measurement results, not hidden variables. Hidden variables specify the initial states of the singlet system at the source, whereas your alpha and beta specify possible final states, or possible end results, at the detectors. The variables A(a), B(b), A(a’) and B(b’), on the other hand, could be definite final results, or they could also be only counterfactually possible results — i.e., not yet settled results, like alpha and beta.
In any case, the EPR criterion of reality amounts to all A(a), B(b), A(a’) and B(b’), whether they are actual results or merely counterfactually possible results, “exist” in the space of all possible results. This space is evidently closed under multiplication, but not closed under addition. As such any quantity such as B + B’ does not belong to this space. Moreover, a physically quantity like A( B + B’ ) is an impossibility in any possible world. It is therefore not surprising that it is not a part of the space of all possible results, because it is not possible to measure an impossible quantity in any possible world. Now you have constructed quantities like B + beta. But physically they are equally nonsensical, and for the same reason. And mathematically they too do not belong to the space of all possible results.
I don’t quite understand your question (2), but I think you are using the phrase “hidden variables” in it incorrectly. But if I may replace that phrase with “alpha or beta”, then the answer is that you cannot possibly make a measurement on alpha or beta while having a definite result for A and B. Just because you can mathematically write an expression like B + beta does not make that expression physically meaningful. It is subject to the same objection I have raised for the expression A( B + B’ ) above. To quote myself, “the quantity A( B + B’ ) does not represent any meaningful element of any possible physical world, classical or quantum. This is because B and B’ can coexist with A only counterfactually. If B coexists with A, then B’ cannot coexist with A, and vice versa. But in his analysis Jay implicitly assumes that B and B’ can both coexist with A simultaneously. This would be analogous to my being in New York and Miami at exactly the same time. But no reasonable definition of realism can justify such an unphysical demand. The notion of realism envisaged by Einstein most certainly does not demand any such thing.”
Are the physical quantities – spin components of the spin ½- particle#1 along direction a and a’ and spin components of the spin ½- particle#2 along direction b and b’ elements of reality or not?
Depends on what you mean by “spin components.” If you mean by them Alice’s measurement results A(a, lambda) = +/-1 and A(a’, lambda) = +/-1, and similarly for Bob’s measurement results B(b, lambda) = +/-1 and B(b’, lambda) = +/-1, then yes, they are all elements of physical reality according to the EPR criterion of reality.
But B(b, lambda) + B(b’, lambda) is not an element of physical reality in the EPR sense.
OK, Joy, let’s focus on this above for a moment. When we have a a physical quantity A which we do measure with certainty to be +1, or to be -1, then there exists an “element of reality” corresponding to that quantity, because probability = 100%. It is “realistic.” So far so good?
When we have a second physical quantity A’ which may be +1 or -1 but we will never know which it is because we measured A and thereby excluded the possibility of ever measuring A’, then we *cannot* “predict with certainty (i.e., with probability equal to unity) the value of” A’, because +1 only has 50% probability and -1 the other 50%, so therefore. . . what? What is the relation of that 50%-50% “half baked” quantity to reality? You clearly do not think that one can add “reality” to “half-baked reality.” What is meant by “*element* of reality”? Einstein was not known to be sloppy about his choice of language for scientific concepts. Why the word “element”? These are the questions I am trying to zero in on when I ask you to really focus on “realism” and how that is defined and applied in EPR-Bell, and generally. Not to mention “disturbing a system” which I talked about yesterday, also part of EPR. How does “system disturbance” work in practice? How does it work when we deal with Heisenberg, or spin cones? Repeating “counterfactual” over and over does not seem to be impressing the jury.
That is what this whole ball game has boiled down to in the 10th inning of World Series Game 7. If Bell-CHSH uses an inappropriate definition of reality (which, fairly, you did argue in your Appendix D, though it is still facing stiff headwinds), then your S^3 model must be admitted and AOP erred to exclude it from consideration. If Bell-CHSH uses an appropriate definition of reality, then local realism is a cooked goose and we all go home. (Nutshell answer to HR.)
So, Joy, I am not trying to give you a hard time. On philosophical grounds I would love nothing more than to see local realism be once again viable in physics, if good science will support it. I am trying to push you to articulate in depth, your view of realism, or Einstein’s view of realism, or somebody other than the Bell-world’s view of realism which ends the ball game, and apply it to the CHSH derivation. It appears that everything you are doing pivots around how realism in physics is defined and implemented, and around convincing others of a fatal flaw *in the way realism is defined and used in Bell*. If I were in your shoes, I might write a whole learned paper dedicated to the subject of realism in general and in Bell’s theory. It is that important to what you are trying to do, to get this done correctly and convincingly if you can.
Jay
The EPR-Bohm scenario actually can’t say anything about realism due to the very nature of it. So Bell erred by forming his model around it as far as realism goes. But you got it right when you say the only prediction we can have about the experiment not performed is 50-50 chance + or -1. But it is not “half-baked” realism. The simple fact is that you can’t determine realism that way. We cannot predict with “certainty” so the EPR criteria doesn’t even apply.
As you can imagine, Jay, there are literally tens of thousands of learned papers written and debated on the kind of questions you have raised about the EPR criterion of reality and related issues. I myself have written on the subject occasionally. Let me paraphrase from one of my papers which I think may address some of your questions (cf. https://arxiv.org/abs/0904.4259 ):
The singlet state has two remarkable properties. First, it happens to be rotationally invariant. That is to say, it remains the same for all directions in space, denoted by the unit vector n. Second, it entails perfect spin correlations: If the component of spin along direction n is found to be “up” for particle 1, then with certainty it will be found to be “down” for particle 2, and vice versa. Consequently, one can predict with certainty the result of measuring any component of spin of particle 2 by previously measuring the same component of spin of particle 1. However, locality demands that measurements performed on particle 1 cannot bring about any real change in the remotely situated particle 2. Therefore, according to the EPR criterion of physical reality, the chosen spin component of particle 2 is an element of physical reality. But this argument goes through for any component of spin, and hence all infinitely many of the spin components of particle 2 are elements of physical reality, which may be represented by +1 or -1 points of a unit 2-sphere. However, many of these elements of physical reality have no counterparts in the quantum mechanical description of the system, because there is no quantum state of particle 2 in which all components of its spin have definite values. Consequently, by the completeness criterion of EPR — which states that “every element of the physical reality must have a counterpart in the physical theory”, quantum theory cannot be a complete theory, because at least in the present example it does not provide a complete description of the physical reality. That is to say, the notion of quantum entanglement merely conceals our lack of knowledge.
I don’t know whether the above helps or not. But at least it may provide a good background for my central concern, which involves just three points of the 2-sphere mentioned above. We can represent these three points as A, B and B’, as we have been doing. Now the central issue for me is this: Given that we have agreement that all three of these points, A, B, and B’, are elements of the physical reality (EPR called them “elements”, because we are not concerned here about the whole of reality but about only a very tiny part of it), is their combination such as A*( B + B’ ) also an element of physical reality? If it is, then we have to accept that Bell’s mathematical argument is also physically complete in the sense discussed in the previous paragraph, and then my objections (and those of others) are not worth a penny. But if their combination A*( B + B’ ) is not an element of physical reality [that is to say, if A*( B + B’ ) is physically a meaningless quantity], then Bell’s argument is subject to the same criticism that Einstein leveled against quantum mechanics — i.e., that the reality considered by Bell in his local-realistic framework is just as incomplete as that underlying the quantum mechanical reality.
By now it should be obvious that the quantity A*( B + B’ ) cannot possibly be an element of reality in any possible physical world, classical or quantum, in the sense discussed above. As you have put it, A*( B + B’ ) is a combination of “reality” and “half-baked reality.” To me A*( B + B’ ) is simply a gobbledygook and Bell was simply sloppy in his reasoning to even consider it. As a result, in my view, and in view of many others, Bell-CHSH argument is fatally flawed.
The discussion has been about correctness but more is needed for acceptability. Even if Bell’s theorem were actually disproven in the paper, equally important is that the proof is presented so that its trivial to verify its correctness or identify an error. There must be nothing obscure in the proof. E.g., implicit arguments of functions should be avoided or at least made clear in the text. I would reject Dr. Christian’s paper for bad presentation.
This is actually very simple. Bell’s theorem can be put as “no urn model can generate all the quantum correlations.” Either Christian’s model can be reduced to an urn model, in which case he can just provide us with the 16 frequencies used in his urn model, so we can check the correlations ourselves.
Or his model cannot be reduced to an urn model, in which case it has nothing to do with Bell’s theorem.
I would be happy to reduce my FRW 3-sphere model to an urn model if someone can first demonstrate that an urn model is a solution of Einstein’s field equations of general relativity:
G_ab = 8pi T_ab.
Until then the urn model is physically as fictitious as Bell’s theorem (even if we accept your dubious claim that Bell’s theorem is the same as “no urn model can generate all the quantum correlations.” Needless to remind everyone that the local-realistic Universe is governed by Einstein’s field equations of general relativity: https://arxiv.org/abs/1405.2355 .
That, needless to say, is a responsibility that lies with you, since it is you who claim to have disproved Bell’s theorem.
The burden of proof is actually on you on both counts: (1) you must convince the physics community that the urn model is the same as Bell’s theorem as you claim; and (2) that the urn model has anything to do with the laws governing the physical Universe. I am under absolutely no obligation to pay any attention whatsoever to the urn model. If I have not objected to it so far, it is because of my respect for Jay, who seems to have taken it seriously.
As to (1), the physics community does not have to be convinced about that, since this is how they understand Bell’s theorem already. Regarding (2), of course you are not under any obligation to pay any attention whatsoever to the urn model. But then you should not get grumpy when someone decides not to pay any attention whatsoever to your paper, like the AoP editorial board.
To all:
I believe that I have finally gotten to understand, and can now articulate, the argument that Joy Christian has been attempting to make regarding Bell’ Theorem. In a new 5-page note at https://jayryablon.files.wordpress.com/2016/11/jc-realism-argument.pdf, I have a) laid out the derivation of the CHSH bounds of +/-2 using the urn and slip metaphor in a way that I believe will be agreed upon by the Bell proponents. Then, I have b) tried to present what I believe is the argument Joy has been trying to make in his Appendix D of https://arxiv.org/abs/1501.03393, which IMHO does center around the use of “realism” in Bell’s theorem.
I have tried a couple of times before to represent Joy’s argument on this point of realism, but he has not agreed with my representation. Maybe the third time is the charm. 🙂
First, to Joy: do you agree that this fairly represents your Appendix D argument?
Then, if Joy agrees, to others: does this argument make sense to you when represented in this way?
Thanks
Jay
Your eq. (6) is wrong. It is impossible. Quantum mechanics cannot produce those bounds while strictly adhering to Bell-CHSH. I challenge anyone to prove that your eq. (6) could possibly be true. As long as the A_j in the first term is the same as the A_j in the second term and likewise for B_j, etc., you cannot exceed the bound of |2|.
You are quite right, Fred. Jay’s eq. (6) is wrong as long as the subscripts j on A_j etc. in the four averages are kept the same. But that does not affect the rest of Jay’s analysis.
One way to correct eq. (6) is to use four separate indices like p, q, r and s for the four averages. But then Bell’s argument does not even get off the ground, because (1) in that case the EPR criterion of reality is not satisfied from the start, and (2) the “sum of averages = average of sums” step does not go through, because A_p is then not the same as A_q etc.
To get around this problem the followers of Bell like Gill argue that one must use “the law of large numbers” and “the continuous hidden variables hypothesis.” In other words, they argue that one must use the continuum limit on the four averages in Jay’s eq. (6) and use calculus and probability distribution of hidden variables to calculate the four expectation values.
But that does not help them in fending off the rest of the analysis by Jay, or in fending off my original argument in my Appendix D. The law of large numbers and the continuous hidden variables only obscure their misuse of the EPR criterion of reality, do not eliminate it. But now both my original argument and Jay’s probabilistic analysis has brought out their misuse of the EPR criterion very clearly.
Jay is just saying that for an urn model, the limits on the CHSH expression is between -2 and +2. If anyone think that can be disproved, they can simply provide us with a counterexample of 16 frequencies that makes the expression compute to a number outside those bounds. No need to argue about anything.
You seem to be reading a different post. What I see in Jay’s latest post are these conclusions: “local realism can be admitted consistently with quantum mechanics, we are permitted to proceed to study Joy’s S3 model, and AOP made an error by retracting Joy’s paper.”
@HR
Prove Jay’s eq. (6) is true and we will be finished here.
Thank you very much Jay. Your latest attempt to understand my argument has come a long way, and it gets my main point correctly. I did not phrase the issue as charitably to Bell as you have done, in probabilistic terms. As I have noted before, for me the quantities like A*( B + B’ ) and A*( B – B’ ) appearing in Bell’s proof are utterly meaningless howlers, at least as big as the mistake von Neumann had made in his debunked theorem against hidden variables.
Bell started out with an intention to implement the realism criterion of EPR in his argument against local realism. But along his way to proving his final result, he unwittingly abandoned this criterion, or compromised it very seriously. As a result his final conclusion — the tighter bounds of -2 and +2 on the CHSH correlator — have no relevance for the question of viability of a local-realistic underpinning of quantum correlations.
Apart from this minor caveat (namely, that I would rather bestow a much harsher verdict on Bell’s theorem), I completely agree with your final conclusions: “local realism can be admitted consistently with quantum mechanics, we are permitted to proceed to study Joy’s S3 model, and AOP made an error by retracting Joy’s paper.”
We’re reading the same post. I was commenting on the critisism of Jay’s eq. (6) in that post.
Prior to heading out soon for a long weekend to see my son and daughter-in-law (first grandchild expected in February, could not be happier! :-)) where I will not be in a position to write much, I would like to add a few more reflections about my post yesterday in https://jayryablon.files.wordpress.com/2016/11/jc-realism-argument.pdf.
Over the many years I was aware of Bell from a distance, and knew that the questions Bell was trying to answer related to locality and realism, I always thought more about locality than realism, because without locality one would have to admit what Einstein called “spooky action at a distance.” And in view of Heisenberg, I, like most others, had already given up the ghost on probabilities. So it has only been in the past week as I have drilled down into divergent opinions of Joy and Richard et al., that I have come to see that the root of their disagreement lies not in locality, but in *realism* and how that is applied in Bell. And as I said a few days ago, this means realism deserves a very close look.
So as Joy has done, I too will take as my starting point the following from the first page of the EPR paper which is at http://journals.aps.org/pr/pdf/10.1103/PhysRev.47.777:
“If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity.”
EPR then say:
“It seems to us that this criterion, while far from exhausting all possible ways of recognizing a physical reality, at least provides us with one such way, whenever the conditions set down in it occur.”
I shall take a few moments here to try to advance this reality criterion, as I have now grown to understand it. And to give you the punch line at the start rather than at the end, I will say that just as there is a philosophical yin-yang between “locality” and “spooky action at a distance,” there is also a tension between “realism” and what I shall now coin as “probabilism.” This is to say, in the absence of physical locality, then we must have spooky action at a distance which in the modern era seems to have acquired the less pejorative label “entanglement.” And, given that “realism” is defined by EPR in relation to quantities that can be predicted with certainty “with probability equal to unity,” in the absence of physical realism we define “probablism” by making the following definition to mirror the one in EPR:
“If, without in any way disturbing a system, we *cannot* predict or know with certainty the value of a physical quantity (i.e., if the probability is less than unity or is unknown or is unknowable entirely), then there exists an element of probabilism corresponding to that quantity.”
In juxtaposing probabilism as the counterpoint to realism just as spooky action is the counterpoint to locality, we are actually making a statement about the way in which humans experience nature at the most elemental level, and are inexorably introducing well-known physical notions about past and future. Let me explain:
At any given moment, the future, to the extent we know it, is always probabilistic. Setting aside debates about determinism versus free will, etc. which do not appear to be necessary to approach the questions at hand, the human experience of the natural world seems to objectively teach that there is nothing which has not yet happened to which we can ascribe a “probability equal to unity.” Conversely, once an event is in the past, and it has been measured and quantified, then we may ascribe to that event quantification a “probability equal to unity,” and so that event passes from probablism to reality.
The words here may be fancy, but this objectively describes the flow of every moment of human life. I am about to toss a coin, and I know it has a 50%-50% probability of heads or tails. This is probabilism. I have now tossed the coin and it came up tails. Now, tails is realism, and heads is a probability that never happened and is not an element of reality. Time is an ever-present fork in the road which sorts probabilism into realism (what did happen, such as electing a 4 year nightmare to the White House), and unrealism (what could have been but did not happen, sigh!). Likewise, any physical result which is unknowable in principle, also contains an element of probabilism.
So if we can all agree that probabilism is the counterpoint to realism, then the deduction in https://jayryablon.files.wordpress.com/2016/11/jc-realism-argument.pdf that the CHSH bounds of +/-2 are obtained by unwittingly relinquishing realism and using probabilism as between (3) and (4) [note, typo, it should say beta’_1*1 = -1 just before (4)] then this means that Bell’s Theorem is actually not a test of local realism, because that hypothesis was relinquished by the probabilistic treatment of (3) and (4). So now the question becomes: what does Bell’s Theorem test for if not local realism?
In fact, by subtly switching to probabilism at (3) and (4), we have now deduced that +/-2 are the CHSH correlator bounds, not for local realism, but rather for local *probabilism*, because this is the stealth hypothesis that makes is way into Bell via (3) and (4). That is, Bell tests for *probablism*, not realism, with the +/-2 correlator bounds. And because quantum correlations go outside those bounds, this means that wholly opposite to what has long been believed, Bell’s theorem is actually a disproof of local probabilism!
To sum up, it has long been said based on Bell that:
“No locally realistic theory can ever reproduce all of the predictions of quantum mechanics.”
But what actually now seem to be the case is that:
*No locally *probabilistic* theory can ever reproduce all of the predictions of quantum mechanics.*
This is somewhat astounding to me, and entirely unexpected, as I too became cowed long ago by Heisenberg. But if probabilism sneaks into Bell and this is what gives us the correlator bounds, then Bell disproves probabilism rather than realism.
But there is an even simpler statement of all this, and Einstein may have had it right all along:
God does not play dice! Nor does God employ spooky action at a distance!
And that is what Bell’s theorem actually appears to prove. And I really want to see how this backwashes to the uncertainty principle.
I know Joy will agree with these conclusions. And I always stand to be corrected. Is there anybody who can talk me down from these conclusions?
Jay
“*No locally *probabilistic* theory can ever reproduce all of the predictions of quantum mechanics.*”
You are making another false assumption that quantum mechanics goes out of the bounds of |2|. It doesn’t. Try to prove your eq. (6).
Jay,
Your equation (6) is terribly wrong. You should not have the same subscripts on all terms.
Another thing:
Note that the relationships expressed in equations (3) and (4) apply to actual and counterfactual outcomes obtained from the exact same particle pair. Those relationships don’t exist between outcomes from different particle pairs.
In any case, as Joy has explained, equation (5) might be correct as a mathematical expression but is completely meaningless as far as physics is concerned, since no experiment can ever measure it. Perhaps the reason you make the error in expression (6) originates from not recognizing the difference. The physically meaningful expression is the one with 4 different subscripts (i,j,k,l) in expression (6).
Don’t want to discourage your probabilism exercise but IMHO that is going down a rabbit trail. The correct approach would be to first fix subscripts of expression (6) to obtain a physically meaningful expression which agrees with QM, then find all the assumptions you need to make to reduce the bounds to (+/-2).
To me the key assumption needed to reduce the bounds from |4| to |2| is the following:
If A, A’, B and B’ are real in the sense espoused by EPR, then so are A( B + B’ ) and A'( B – B’ ).
Until this assumption enters his argument, I am willing to go along with Bell’s reasoning. But of course the assumption is simply false, and therefore the bounds of +/-2 derived using the assumption are fictitious. They have nothing whatsoever to do with any kind of physics.
“At any given moment, the future, to the extent we know it, is always probabilistic. Setting aside debates about determinism versus free will, etc. which do not appear to be necessary to approach the questions at hand, the human experience of the natural world seems to objectively teach that there is nothing which has not yet happened to which we can ascribe a “probability equal to unity.””
On the contrary, the commonsense assumption of “probability equal to unity” is explicitly derived from our ordinary experience, now hardwired into solid expectation, as living creatures in the macroscopic physical world. For example, only an immature individual assumes the slightest likelihood that gravitation can suddenly be obviated or fire not possess the power to destroy flesh. A minor quibble perhaps.
The details of whatever “proof” of Bell’s “theorem” Gill may have claimed are completely irrelevant as long as a CHSH-like sum is involved in the proof in some way. That sum is all that is needed to disprove Bell’s theorem, because that sum is both mathematically and physically identical to the single “average” presented in the expression (D16) of the latest version of my Appendix D: https://arxiv.org/abs/1501.03393 ; in current notation it is
Int_K { A( B + B’ ) + A'( B – B’ ) } rho(k) dk .
From my (D16) it immediately follows that the bonds of -2 and +2 on CHSH are entirely fictitious, because they arise from averaging over an entirely fictitious quantity. Once it is recognized that the bounds of -2 and +2 on CHSH arise from a quantity that does not even respect the EPR criterion of reality, it is completely irrelevant how many other ways one can claim to have arrived at the same bounds.
I have produced a self-contained 3-page document that summarizes my argument against Bell’s theorem, quite independently of my 3-sphere model for the EPR-Bohm correlations. The first two pages of this document (linked below) are the same as Appendix D discussed here. The last page is new, and it summarizes what I have been arguing here in recent posts.
The upshot of my argument is that Bell badly blundered in implementing the EPR criterion of reality while attempting prove his theorem against local realism. The sad consequence of his blunder is the fact that the much discussed bounds of -2 and +2 on the CHSH correlator are simply mathematical curiosities. They have no relevance for the question of local realism:
http://libertesphilosophica.info/blog/wp-content/uploads/2016/11/Fatal.pdf
So what the followers of Bell have yet to realize is that the problem is not with one supposed proof or another of Bell’s “theorem”, but with the CHSH sum itself. That sum has always been promoted as implementing the EPR criterion of reality. But that is a lie. The truth is exactly the opposite. As soon as one writes down the CHSH sum it violates the EPR criterion of reality, as I have explained in considerable detail on the last page of this paper:
http://libertesphilosophica.info/blog/wp-content/uploads/2016/11/Fatal.pdf .
There are 4 expressions involved in this discussion
(1) ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ ≤ 2
(2) ⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ ≤ 2√2
(3) ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ ≤ 2√2
(4) ⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ ≤ 2
Two of those expressions (1) and (2) are correct, the other two (3) and (4) are incorrect. Bell’s theorem amounts to the claim that expression (2), the QM prediction is a violation of expression (1), the claimed “local realistic prediction”. But look closely at the LHS of what is being compared. It is apples and oranges. Bell and proponents mistakenly think they are comparing expression (3) with expression (1) so the comparison appears justified. But it is completely wrong, expression (3) is NOT the QM prediction, expression (2) is.
Bell proponents repeatedly perform experiments in which they measure
⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩
Yet they claim their results violate
⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩
Of course if I drop all the subscripts, you might think I was not making any sense. But that is why Joy’s argument is so powerful — it shows that no experiment can ever measure
⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩
Because it amounts to simultaneously performing mutually exclusive experiments since the derivation of expression (1), which is absolutely correct by the way, involves terms of the form
⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ = ⟨B₁(A₁ + A’₁)⟩ + ⟨B’₁(A₁ – A’₁)⟩
Which makes it absolutely clear that it involves simultaneously performing of mutually exclusive experiments, even after dropping the subscripts.
But note that an expression such as
⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩
represents a valid perform-able experiment, which cannot be factorized as above. This is why I asked Jay to fix equation (6) because this error has crept up in his equations too.
To conclude,
* expressions (1) and (2) cannot be compared with each other as they represent completely different physical scenarios, hence the disagreement on the RHS
* expressions (1) and (3) can be compared the only reason they disagree on the RHS is because expression (3) is wrong.
* the same applies for expressions (2) and (4)
* expressions (3) and (4) are wrong. Anyone who thinks they are correct, should provide the proofs.
Bell’s inequalities are mathematically correct. We do not need more proofs of the inequalities. Bell’s theorem is wrong, not because the inequality (1) is wrong but because it compares the inequality (1) with the inequality (2) which represents a completely different physical situation from (1). The difference being that the LHS of expression (1) is an impossible experiment, while the LHS of expression (2) is a feasible experiment.
This is an excellent post by MF. But just to add some points which may not be obvious to everyone, Jay’s eq. (6) is actually problematic in several ways. It is supposed to express the quantum mechanical prediction of the bounds of -2√2 and +2√2 on the CHSH-type sum of averages, but in quantum mechanics there are no variables like A₁ and B₁. In fact, quantum mechanical prediction of the correlations E(a, b) = -a.b for the EPR-Bohm type experiments involves calculating four separate expectation values using Pauli matrices, as I have done in eq. (4) of my very first paper on Bell’s theorem: https://arxiv.org/abs/quant-ph/0703179 .
There are also other ways to derive the quantum mechanical bounds of -2√2 and +2√2, e.g. using Clifford or Geometric algebra (as I have done later in the above paper), but let us stick to eq. (4) of my paper, which gives the quantum mechanical prediction of E(a, b) = -a.b. The bounds on the sum E(a, b) + E(a, b’ ) + E(a’, b) – E(a’, b’ ) can then be quite easily derived to be:
-2√2 ≤ -a.b – a.b’ – a’.b + a’.b’ ≤ +2√2 .
Since there are no local-realistic variables like A₁ and B₁ in quantum mechanics, we can’t even write an equation like Jay’s eq. (6) and claim that it expresses a quantum mechanical prediction of the above bounds.
Moreover, in the actual experiments, as MF has so clearly explained, the four terms of Jay’s eq. (6) are computed using four sets of separate, independent variables:
⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩.
The reason for this, as MF has explained, is that the pairs like (a, b), (a, b’), (a’, b) and (a’, b’) are mutually exclusive pairs of detector directions, corresponding to four incompatible experiments. It therefore makes no sense to expresses the experimentally observed bounds as
-2√2 ≤ ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ ≤ +2√2 .
As several people have noted, the above equation is simply wrong. The sum of four averages in the above equation is actually bounded by -2 and +2, and nothing can ever break those bounds. The often made claim that something (e.g., quantum mechanics) violates the bounds of -2 and +2 on the above sum of four averages is mathematically nothing but gibberish.
Is there anyone following this discussion who is *not* affiliated with SciPhysicsFoundations and who thinks that the above is a correct analysis? If so, I will explain what is wrong with it. If not, I won’t bother. This discussion has become tedious and probably fruitless. The same arguments are repeated and refuted over and over.
If your question is whether there is anybody who has not been a partisan in the Bell discussions and comes to this with an open mind trying to learn the truth, or at least learn the truth about where the intractable disagreements reside and accurately understand and reproduce the divergent points of view, then yes, I am interested in knowing.
The impression I have gathered, which I would like to have confirmed or corrected, is that this analysis by MF and JC is correct in the first instance, but is then overcome by appeal to statistical arguments and the law of large numbers. And my impression is that the use of large numbers parallels what I laid out several weeks ago in what Richard referred to as a “novel proof of Bell’s Theorem.” Also, when I wrote my equation 6 which has been disputed in this discussion, it was my impression that everybody agreed about that equation, and that the point of dispute rested in the interpretation of the CHSH inequality itself, as derived with the bounds of 2. That point of disagreement about CHSH on its own terms still appears to remain; this discussion about equation 6 flags to me a second flashpoint of disagreement, and I need to confirm the roots of this second disagreement.
I am still traveling for another day, and not in a position to write much, but have been following the discussion. But I did go back and reread the EPR paper several times the past couple of days while sitting in taxicabs or waiting for my wife while she shopped in the malls. I had read this a number of years ago, but have a much deeper background today as a result of the discussions here for the last few weeks.
Also, before I left I made a final post which I probably would not have made had I not been planning to be away for several days. And Stephen, you were kind enough to reply to that already. I am happy to defer any discussion that can be characterized as “philosophy,” because at the outset I want to know whether I am on solid ground with the CHSH derivation I put together in equations 1 through 5 of https://jayryablon.files.wordpress.com/2016/11/jc-realism-argument.pdf. Always, the mathematics has to be correct before we can start to discuss interpretations. It is with these equations 1 through 5 that I plan to pick up my own RG and JC “mediation” once I return, now also being refreshed on the details of EPR and not merely a few sentences excerpted from that paper.
Thanks,
Jay
” If so, I will explain what is wrong with it. If not, I won’t bother.”
You should try to explain what is wrong with MF’s analysis for the benefit of lurkers.
Yes, you are correct. It is as the following: I suspect that a coin is not fair, so I toss it 1000 times, and it lands heads up in 800 of the tosses. I conclude it is a biased coin, but then someone objects that 800 heads are logically possible even with a fair coin (in fact 1000 are possible), so I have no logical argument for my conclusion. Point being, I have a very strong *statistical* argument for my conclusion.
While Richard Gill seems to have recognized his mistake after my reply to him on “November 14, 2016 at 12:36 pm”, I see that HR and Stephen Parrott are still resisting the inevitable conclusion I have presented in http://libertesphilosophica.info/blog/wp-content/uploads/2016/11/Fatal.pdf .
Let me try to explain my conclusion in a different way. Consider the following hypothesis:
“We live in a world where it is possible — at least momentarily — to be in two places as once — for example, in New York and Miami — at exactly the same time.”
From this hypothesis it follows that in such a world it is possible for Bob to detect a spin along both directions b and b’ at exactly the same time as Alice detects her spin along the direction a, or a’. If we denote the measurement functions of Alice and Bob as A(a, h) and B(b, h), respectively, then we can say that in this world it is possible for the measurement event such as A(a, h) of Alice to simultaneously coexist with both the measurement events B(b, h) and B(b’, h) observed by Bob, where h is the initial state (or hidden variable) of the singlet spin system. Consequently, we can write the “coincidence click” (a single click) observed by both Alice and Bob as
A(a, h) { B(b, h) + B(b’, h) }
(notwithstanding the fact that there are only two particles available to Alice and Bob for each run of the EPR-Bohm type experiment). It is also worth stressing here that in our familiar macroscopic world ( after all a and b are macroscopic directions ) such a bizarre spacetime event is never observed. Because the measurement directions a and b freely chosen by Alice and Bob are mutually exclusive macroscopic measurement directions in the physical 3-space.
Similarly, nothing prevents Alice and Bob in such a bizarre world to simultaneously observe the following event:
A(a’, h) { B(b, h) – B(b’, h) } .
And — yes, you guessed it — nothing prevents Alice and Bob in such a bizarre world to simultaneously observe the sum of the above two events as a single event
A(a, h) { B(b, h) + B(b’, h) } + A(a’, h) { B(b, h) – B(b’, h) }.
Consider now a very large (effectively infinite) number of the initial states (or hidden variables) h and the corresponding simultaneous events like the last one above. We can then calculate the expected value of such an event, occurring in this bizarre world, by means of the following integral (naturally respecting the “large N limit”):
Int_H [ A(a, h) { B(b, h) + B(b’, h) } + A(a’, h) { B(b, h) – B(b’, h) } ] rho(h) dh ,
where Int_H represents an integral over the space H of all possible hidden variables h, and rho(h) is the corresponding normalized probability distribution of h.
Note that I am assuming nothing about the hidden variables h. They may be functions of a and b, h = h(a, b), in which case we could be dealing with a highly non-local model.
Next we ask: What are the upper and lower bounds on the above expected value? You will find the answer to this question fully worked out in the paper linked above; which is:
-2 < Int_H [ A(a, h) { B(b, h) + B(b', h) } + A(a', h) { B(b, h) – B(b', h) } ] rho(h) dh < +2 .
Now it is a trivial mathematical fact that the above expected value, together with its bounds of -2 and +2, can be written as a sum of four expected values as follows:
-2 < Int_H [ A(a, h) B(b, h) ] rho(h) dh + Int_H [ A(a, h) B(b', h) ] rho(h) dh + Int_H [ A(a', h) B(b, h) ] rho(h) dh – Int_H [ A(a', h) B(b', h) ] rho(h) dh < +2 .
This can be rewritten in a more recognizable form as
-2 < E(a, b) + E(a, b' ) + E(a', b) – E(a', b' ) < +2 .
Now we conduct the EPR-Bohm experiments and interpret their results as exceeding the bounds of -2 and +2 on the above sum of four independent expected values:
-2\/2 < E(a, b) + E(a, b' ) + E(a', b) – E(a', b' ) < +2\/2 .
Consequently, we conclude that the hypothesis we started out with must be false. In other words, we do not actually live in a bizarre world in which it is possible, even momentarily, to be in New York and Miami at exactly the same time. This is what Bell proved. He proved that we do not live in such a bizarre world. But EPR never demanded, or hoped that we do.
Note that the ONLY hypothesis used to derive the bounds of |2| on the CHSH correlator is the one stated above: "It is possible, at least momentarily, to be at two places at once!" Locality was never assumed, nor was realism of Einstein and EPR ever compromised in any way.
“Let me try to explain my conclusion in a different way. Consider the following hypothesis:
“We live in a world where it is possible — at least momentarily — to be in two places as once — for example, in New York and Miami — at exactly the same time.””
Joy, I will have more to say about this when I am back home at my computer rather than talking into my iPhone in an airport. As you know I have been consistently uncomfortable with the New York and Miami statement, as have others. But if you change the hypothesis to “it is possible to be uncertain whether I am located in New York or Miami” which is a discrete problem, and then backtrack to the EPR paper through equation 6 of that paper for continuous positions, then there is a prospect that in fact Bell has actually disproved the uncertainty principle rather than disproving local realism. Keep in mind, when the EPR was first written, uncertainty, I.e., no simultaneous reality between momentum and position, is exactly what the authors were hoping to disprove. I simply offer this right now as a tentative statement of what I am seeing, knowing the exceptionally high burden of proof that would be required for something of this sort to be brought to ground. But I will lay out the details of what I am seeing over the next few days over Thanksgiving, and then give everybody a chance to have some fun beating up on it. 😀 Jay
Jay, you are very mistaken on several counts. Neither Einstein nor EPR ever tried to disprove the uncertainly principle, or any part of quantum mechanics for that matter. What Einstein aimed was to prove that quantum mechanics is an *incomplete description of realty*, while fully accepting every aspect of the statistical predictions of quantum mechanics. Poor Einstein! His views on quantum mechanics have been so grossly distorted by the semi-popular science journalists. Einstein by no means questioned the impossibility of the simultaneous observability between position and momentum. If you think that that is what he and his collaborators are arguing in the EPR paper, then you have not understood their argument at all. What really concerned Einstein was the apparent violation of local causality entailed by quantum entanglement. He was not at all committed to determinism as the popular press would have us believe. So I would be very careful with where you are going with this.
I am also very puzzled why anyone would be uncomfortable with my New York and Miami statement. In the EPR-Bohm experiment two counterfactually possible *macroscopic* measurement directions such as b and b’ are involved. Their mutual exclusiveness have nothing whatsoever to do with the uncertainty principle. The counterfactually possible choice between b and b’ is exactly like the counterfactually possible choice between my being in New York and Miami at the same time. The point of New York and Miami example is that Bob cannot possibly measure his spin component along both b and b’ at the same time. And likewise Alice cannot possibly measure her spin component along a and a’ at the same time.
Joy, you probably are right about this, but I do not want to leave any stone unturned.
For now, let me ask you the same question I asked Stephen:
Specifically, do you agree: a) that the mathematics is done correctly through equation 5? b) that the issues I raise about probability after equation 4 are correct? and c) that these probability issues may provide an alternative albeit equivalent may of discussing the counterfactualism that you have raised?
Jay
My main worry so far is that neither A’s and B’s nor alpha’s and beta’s in your equations have to be actually observed values. They are all only counterfactually possible values.
You are still confusing events and random variables.
A(a, h) { B(b, h) + B(b’, h) } + A(a’, h) { B(b, h) – B(b’, h) } is just a number. Nobody is assuming that it is a measurement outcome of some actual experiment.
No, I am not confusing events and random variables.
In my example of a “bizarre world” I considered above
A(a, h) { B(b, h) + B(b’, h) } + A(a’, h) { B(b, h) – B(b’, h) }
is an actual event in spacetime, not just a random variable. That example is evidently a fiction, considered only to illustrate my point that neither locality or realism are at risk from the observed “violations” of the bounds |2| on CHSH.
In the actual world one is free to put any combination of the elements of realities such as A(a, h), A(a’, h), B(b, h) and B(b’, h) together, to cook up any random variable one likes, such as the following:
A(a, h) { B(b, h) + B(b’, h) } + A(a’, h) { B(b, h) – B(b’, h) } .
However, the above random variable is not a part of the space of all possible elements of reality, and therefore its expected value calculated using a large number of such random variables has nothing to do with the correlations among the actual elements of reality.
This is most clearly seen from my simple example of the set D = {1, 2, 3, 4, 5, 6} of all possible outcomes of a die throw, which also happens not to be closed under addition. Clearly, the outcome 3+6 is not in the set D, and hence it is not an element of reality. No one would claim that an expected value of a large number of “outcomes” of a die throw such as 3+6 has any meaning, or relevance to reality. Physical reality is not made up of fictitious quantities like 3+ 6, or like
A(a, h) { B(b, h) + B(b’, h) } + A(a’, h) { B(b, h) – B(b’, h) } . That is my point.
Nobody is claiming that A(a, h) { B(b, h) + B(b’, h) } + A(a’, h) { B(b, h) – B(b’, h) } is an “element of reality”.
The assumption of local hidden variables does make A(a, h), A(a’, h), B(b, h), B(b’, h) all exist. They are numbers +/-1. Hence the quantity A(a, h) { B(b, h) + B(b’, h) } + A(a’, h) { B(b, h) – B(b’, h) } also exists. Its expected value (its average value over very many independent drawings of the hidden variable h) exists too.
I agree that the assumption of local hidden variables does make A(a, h), A(a’, h), B(b, h), B(b’, h) all exist as counterfactually possible elements of reality, with possible values +/-1.
But since the quantity A(a, h) { B(b, h) + B(b’, h) } + A(a’, h) { B(b, h) – B(b’, h) } — which can certainly be written down and its values evaluated — is not an element of reality (or is not a member of the space of all possible measurement outcomes), its expected value has no relevance for physics. Its expected value is simply a mathematical curiosity.
JC: “Since the quantity … is not an element of reality, its expected value has no relevance for physics. Its expected value is simply a mathematical curiosity.”
That is a matter of opinion, not a logical deduction. In this case it is an opinion which is very wrong. The “mathematical curiosity” leads to empirically testable consequences.
RG: “That is a matter of opinion, not a logical deduction.”
Hardly. I have provided a very transparent logical deduction, both in my post above and in this paper: http://libertesphilosophica.info/blog/wp-content/uploads/2016/11/Fatal.pdf
RG: “The “mathematical curiosity” leads to empirically testable consequences.”
Sure. It proves that you can’t be at two places at once, just as the pre-Socratics thought.
Bell’s reasoning was based – following the analysis of the EPR-paradox – on the pre-assumption of local realism as outlined by Einstein, Podolsky and Rosen (Phys. Rev. 47, 777-80 (1935)). Bell’s reasoning is a “before the measurements”-reasoning. Local realism says: All spin components of a particle’s spin represent simultaneous elements of reality and can therefore be ascribed independent, measurable values, even before any measurement is performed on the particles. Measurements on one particle do not influence the outcomes of measurements on the other particle (no “spooky” interactions). The values of these spin components are uniquely determined by the decay cascade, and are fully specified by a local, hidden random variable “h”.
Neither the terms A(a,h), B(b,h), A(a’,h) and B(b’,h) nor Bell’s integrand A(a,h)* B(b,h) + A(a,h)*B(b’,h) + A(a’,h)* B(b,h) – A(a’,h )*B(b’,h) are measurement “events”! Neither the terms A(a,h), B(b,h), A(a’,h) and B(b’,h) nor Bell’s integrand A(a,h)* B(b,h) + A(a,h)*B(b’,h) + A(a’,h)* B(b,h) – A(a’,h )*B(b’,h) are elements of reality! What a confusion! The terms A(a,h), B(b,h), A(a’,h) and B(b’,h) in Bell’s integrand are mere “predictions” of numbers, viz. the possible values one would definitely find or reveal when one would perform measurements on a particle’s spin along given directions.
Bell’s proof is a bottom-up approach: From considerations about physical quantities which can be ascribed independent, measurable values to the predictions of the averages of measurement outcomes. It rigorously follows Einstein’s classical “recipes” how to look at physical reality. Bell’s theorem is thus an inevitable consequence: No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.
Or, to put it in a nutshell: Einstein was wrong!
What you are saying is the standard folklore. It resolutely ignores the arguments MF and I have presented in our recent posts.
Moreover, all of the statistical predictions of quantum mechanics have already been reproduced by a strictly local hidden variables model. You will find that model on pages 12 to 16 of this paper: https://arxiv.org/abs/1201.0775 .
In a nutshell: Bell was wrong, and Einstein couldn’t have been more right.
@Lord Jestocost
If that is the case (Einstein was wrong), then you should be able to prove that MF’s eq. (3) is true.
The last remaining defense of Bell’s theorem by Richard Gill concerns only semantics. He prefers to call the fictitious quantity A*( B + B’ ) involved in the derivation of the Bell-CHSH inequality a “random variable.” But what’s in a name? That which we call a rose by any other name would smell as sweet. Therefore I have obliged, and using the name “random variable” for the quantity A*( B + B’ ) I have added a short appendix to my 3-page paper linked above:
http://libertesphilosophica.info/blog/wp-content/uploads/2016/11/Fatal.pdf .
In the above paper I have derived the Bell-CHSH inequality form a hypothesis that “one can be in two places at once”, without assuming local causality. Since the Bell-CHSH bounds are exceeded in actual experiments, it follows from Bell’s logic that we cannot be in two places at once.
One of the things I always notice when reading famous seminal papers is a remarkable clarity, simplicity and precision in which new ideas are put forward. The ongoing discussion here on RW, however, shows again that all attempts to “disentangle” quantum entanglement lead to more and more confusion.
Bell’s “original” theorem (J. S. Bell, Physics Vol. 1, No. 3, pp. 195-290, 1964) needs, to my mind, not a bit of defense. It says:
“The paradox of Einstein, Podolsky and Rosen [1] was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables. These additional variables were to restore to the theory causality and locality [2]. In this note that idea will be formulated mathematically and shown to be incompatible with the statistical predictions of quantum mechanics. It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty……………..”
It is sometimes quite cumbersome to explain Bell’s statement because some underlying assumptions and notations appear rather abstract. Therefore, there exist alternative presentations of Bell’s theorem using different approaches. All these presentations show: Bell’s theorem is an inevitable conclusion when one consequently follows Einstein’s concepts of causality and locality.
Thus, every paper that tries to undermine Bell’s theorem must be characterized by a remarkable clarity, simplicity and precision in which the disproof is put forward.
A number of things have grown clear for me over the past week. First, one of the lurkers in this discussion pointed out to me offline that Joy and Richard et al. have reached an impasse here at RW over what Joy calls “the fictitious quantity A*( B + B’ )” and what Richard, Stephen, HR etc. consider a perfectly logical and meaningful quantity; and that this is the same impasse which has been reached before in other forums. So we are all going round and round on a hamster wheel. Second, despite some offline communications with Joy a few days ago, I still am troubled by his his view that the allegedly-fictitious nature of A*( B + B’ ) is the same as the fiction of being “in two places at one time.” On this score, I have misgivings as do Richard and Stephen and HR. Third, I agree completely with Lord Jestocost (LJ) that “clarity, simplicity and precision” are required of any disproof of Bell. Here, a never ending “yes it is fiction / no it is not fiction” argument is anything but clear and simple and precise, primarily because there seems to be no clear way to ascribe a precise physical meaning or a litmus test to the term “fictitious” in the way that we can do for “locality” and “realism.” And yet, finally, notwithstanding all of the foregoing, my gut tells me that Joy is onto something important, and that his problem is not that he is wrong, but that he has simply failed so far to articulate what he is sensing about Bell in a convincing way that is clear, and simple, and precise.
In trying to reconcile all of the above in my own mind, I believe I have been able to finally put my finger on what bothers me about Joy’s “two places at one time” argument, and in so doing, have in mind a way to understand what he is trying to drive at with a clarity, simplicity and precision rightly demanded by LJ that to date has been lacking in this discussion. Let me lay this out conceptually here, with quantified details to follow over the coming week.
The puzzle presented by the quantity A*( B + B’ ) is that the sum B + B’ adds a measurement that was taken to a measurement that was not taken. On the one hand, if we assume “realism” as we are required to do, then this does not matter: If Bob measures B and not B’, realism says that B’ is still a real measurement, and so I am permitted to add a real number B=+1 or B=-1 that was measured to a real real number B’=+1 or B’=-1 that was not measured. The taking or not taking of the measurement does nothing to the reality of B versus B’. By the realism hypothesis, B’ still has a certain value, with probability equal to unity in the EPR sense, *even if it was never measured*. And this, I believe, is why Richard et al. have not bought Joy’s arguments.
Yet, trying to provide “clarity, simplicity and precision” in the sense of LJ, it is also relevant to introduce a third concept to go along with “locality” and “realism.” You can choose whatever name you want for this, and the candidate I have been thinking about is “destruction of information”: Before Bob measures B or B’ but not both, each of B and B’ is real. Each can be measured. Each has an element of reality. Once Bob measures B, then B’ is still “real,” but the disturbance Bob introduces to the system by having measured B not B’ makes it impossible to ever again measure B’ with certainty. The information about B’ — although it is still real — has been destroyed and can never be recovered. That real information has now been trapped in nature’s vault of “unknowable realities” and can never again be known with certainty. For here on, we can never know if B’=+1 or B’=-1 with certainty; we can only talk about the *probability* P that B’=+1 versus the probability 1-P that B’=-1. That is, measuring and then knowing B with certainty forces us into a position where we can thereafter only know B’ probabilistically, because measuring B destroyed our access to the humanly-accessible information — other than probabilistic — about B’.
Both Richard and Stephen have previously said that even if humans cannot know this information, God does. And so in this sense, we are weighing what God knows, against what humans can know, and cannot necessary assume a priori that these are equivalent. Because physics must *by definition* be rooted in what can be measured and quantified by humans, we must take seriously the prospect that some elements of reality known to God cannot be measured by and therefore known to humans, except in an actuarial sense. And therefore, we must ponder with great seriousness — as Joy has done — the question of what to do when presented with a mathematical construct which adds information that is knowable with certainty by humans, to information which humans cannot know with certainty and can only know probabilistically.
The problem I believe Joy is sensing in his discomfort with the quantity A*( B + B’ ) (assuming for this discussion that Bob measures B not B’) is that the construct B + B’ adds a realistic, certain number B which has been measured to a realistic number B’ about which certainty has been destroyed, so that the sum is “certain quantity + probabilistic quantity.” Or “available information + destroyed information.” While there may or may not be something wrong with this, we do have to ask ourselves if there is something we are overlooking by simply adding B + B’ without pausing to consider that once we know B with certainty, the information about B’ is now destroyed and can only be known probabilistically.
This is not an unreasonable question to ask. Indeed, there is very strong precedent for this in the uncertainty principle, and the EPR paper explores this very directly: If we take a definite measurement of a particle’s momentum, we disturb the particle and destroy all information about its position. If we definitely measure its position, we again disturb the particle and destroy all information about it momentum. In recognizing this phenomena, per EPR, we do not adopt the “restrictive” view that “two or more physical quantities [P and Q] can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted.” This would make “the reality of P and Q dependent upon the process of measurement…” and “no reasonable definition of reality could be expected to permit this.” Rather, we maintain a view that P and Q are both elements of reality, but that there is an information loss about Q which occurs whenever we measure P and vice versa, which information loss has definitive profound consequences if one asks the physics question “what can be measured with certainty, not just probability, by humans?” The simultaneous reality of P and Q may be known to God, but it is unmeasurable and unknowable to humans. And in physics, the question of what we measure and *what we can measure in principle* is of central importance.
With all this in mind, the problem I have with Joy’s analogy between “the fictitious quantity A*( B + B’ )” and being in New York and Miami at the same time is this: When I measure B with certainty, I destroy information about the certainty of B’, and render B’ a number that now only has an actuarial, probabilistic value. It is not just that B and B’ can have no more simultaneous reality than being in New York and Miami can have. It is the consequence that *measuring B destroys certain information about B’* and vice versa, *as the very consequence of measuring B*. Although being in New York cannot have simultaneous reality with being in Miami, ***there is no information that is destroyed in Miami, owing to my being in New York.*** This is why Joy’s analogy does not hold for me: because the knowable, measurable information loss is what registers as important to physics, because physics centers on what can be measured. Measuring B destroys or renders probabilistic, information about B’. Unless I am missing something that Joy would need to explain, being in New York does no such thing to information about Miami.
But notwithstanding what I still think is a flawed analogy, the important issue to which Joy does bring a spotlight, is the destruction of information about B’ which occurs when we measure B. When we measure B, certainty (probability = 1) information about B’ becomes lost into nature’s vault of reality elements unknowable to humans. We can then only know these elements of reality probabilistically, as actuaries. And this does matter to physics, which is based on what we measure and what we can in principle measure. When we take an action which changes what can be measured in principle, we change the physics.
It will likely take me several days to do so, but I will write up and post a careful quantitative analysis of all the foregoing, which I believe may be able to finally break the impasse and even deals with the QM CHSH bound of 2 sqrt(2) as a natural byproduct. But this does require assent to the notion that measuring one physical quantity with certainty can destroy or render probabilistic, information about a second physical quantity. And, it requires assent to the notion that this destruction or probabilistic rendering of information is important to physics, because physics must account for and be built upon quantities which are measured and measurable, independently of which quantities are realistic and not realistic. Separating out the concept of “information loss / measurability” from the concepts of “locality” and “realism,” which at another level introduced previously by Richard and Stephen separates what God knows from what humans can know and measure, may provide the clarity, simplicity and precision that LJ rightly demands, and may point the way out of this long-standing impasse.
Jay
I’m impressed that this still goes on. Watching from the outside, it is *still* clear that the disagreement comes down to whether or not something that is mathematically legitimate but corresponds to nothing in the real world, is legitimate. The answer is, of course, yes. This is kind of fundamental to mathematics, that we can infer things through logic.
The arguments of Joy sound to me like someone arguing that since nothing in the real world can have a length that is the square root of two, then any argument involving such a number would be flawed. Or that since complex numbers are purely an algebraic nicety, then all of the conclusions that we can draw about electrical systems are flawed, because we had to pass through to complex numbers to get there. Probability distributions, regardless of whether or not they are attached to measurements, are perfectly rigorous mathematical constructs like complex or irrational numbers, and as such using them to prove something mathematically is not only perfectly legitimate, if it weren’t then all of mathematics would be fundamentally flawed.
Reading this, I believe that at one point Jay was convinced by Joy, and is now trying to make a reasonable attempt to understand. So I say this to you Jay, try to think about this as a purely mathematical theorem, because after all, that’s what it is. The consequences for physics can be considered after the theorem is understood.
The following are my answers to Jay’s latest post above:
(1) The argument I have put forward in the following four pages is clear, simple and precise:
http://libertesphilosophica.info/blog/wp-content/uploads/2016/11/Fatal.pdf .
(2) New York and Miami have nothing to do with B and B’; they are analogous to b and b’.
(3) Neither B + B’, nor P + Q are EPR elements of reality in any possible world, even for the God of Spinoza. B + B’ is physically as meaninglessness a quantity as P + Q is.
(4) Information and probabilities have nothing to do with my argument, only ontology does.
The following is my response to Q’s latest post above:
In an equation X = Y, if Y on its RHS is a physical absurdity, then so is X on its LHS.
Joy, in response:
1) My calculations in (9) of https://jayryablon.files.wordpress.com/2016/11/chsh-and-quaternions.pdf and in https://jayryablon.files.wordpress.com/2016/11/jc-realism-argument.pdf in the form of B +/- beta’, do not at all use the sum of averages = average of sums logic, and still produce the B + B’ terms in the form of B +/- beta’.
2) Agreed. Misspoken. But still: measuring in the b direction destroys information about what would happen were we to measure in the b’ direction analogously to how measuring position destroys information about momentum and vice versa. However, being in New York does not destroy any information about Miami. That is why the analogy does not work for me. I believe your path to progress beyond the present impasse involves recognizing and utilizing the fact that B +/- beta’ etc. involves adding a certain (i.e., probability = unity) quantity B with a quantity beta for which certainty valuation is destroyed. You quoted that probability = unity passage from EPR over and over; I would think you would use it when it comes time to bring your nine years of research across the goal line.
3) I am confused. Earlier, I argued that I thought you were saying that Bell — which is a disproof by contradiction — was making a hypothesis about local realism, but then violating that through terms like B +/- beta’ which are not elements of reality, thereby voiding any conclusions about that hypothesis being contradicted. Then you seemed to say that B and beta’ are still elements of reality even if the latter is never observed and indeed is destroyed by the measurement of B. Now you seem to argue against these being simultaneous elements of reality. Where are you? Realism hypothesis abandoned along the way, or not? Also: what is your view of EPR’s statement below? Agree or disagree?:
In a “restrictive” view, “two or more physical quantities [P and Q] can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted.” This would make “the reality of P and Q dependent upon the process of measurement…” and “no reasonable definition of reality could be expected to permit this.”
Jay
Jay, here are my responses to your latest comments above:
1) The fact that you do not use “sum of averages = average of sums” is irrelevant for my argument. As long as you use “sum of averages” in your argument as Bell and CHSH did, you are committed to the “average of sums”, because the two are mathematically identical, and therefore physically identical. Your not using this identity only obscures the fact that you are doing something unphysical from the start, just as Bell and CHSH did. My using the identity exposes the fact that the very LHS of the identity, the CHSH sum, is absurd. The identity helps to expose this absurdity, and brings out the problem for all to see.
2) No. your view does not fit with the EPR argument. Measuring along b direction does nothing to the element of reality along b’ direction. According to the EPR criterion of reality, had we measured along b’ instead of along b, we would still have obtained the same value, +1 or -1, that exists counterfactually. Your invoking the uncertainty principle and position-momentum analogy here is misplaced. All A(a), A(a’ ), B(b) and B(b’ ) exist as counterfactually possible elements of reality, whether or not we actually measure them. None are ever destroyed as counterfactual possibilities even if we do end up measuring some of them. This is exactly like the fact that the possibility of our being in Miami still exists as a counterfactual possibility even if we actually end up in New York. We could have still been in Miami had we not been in New York, or been in either of the two places.
3) I completely agree with EPR’s statement you have quoted. The elements of reality are not dependent on the process of measurement, nor is one of them, say B(b’ ), destroyed if we measured A(a ). Because the God of Spinoza can re-run the entire universe but for the fact that this time we measured A(a’ ) instead of A(a). Thus, in your notation, both B and beta are elements of reality, even if neither is ever measured. But beta is never destroyed by the measurement of B, and vice versa. Your introduction of alpha and beta are thus redundant.
To reiterate my argument, both B and B’ are elements of reality. They will always be as such whether or not one of them is actually measured. The other one does not get destroyed by the process. However, the hybrid quantities like A*( B + B’ ) are entirely meaningless in any possible world because (1) A cannot simultaneously coexist with both B and B’, and (2) B + B’ is not an element of reality at all, even though both B and B’ are elements of reality.
The problem arises only with the quantities like A*( B + B’ ). You are using such quantities even though you say that you are not using the “sum of averages = average of sums” logic. By not using that logic you are simply hiding the quantities like A*( B + B’ ) from sight. They are still there because “sum of averages = average of sums” is a mathematical identity.
I hope this clarifies my position.
Joy,
In quantum theory:
A=(Q+iP)/sqrt(2)
is the annihilation operator. It is a linear combination of P and Q, albeit with a factor of i. Query: is A an element of reality? In your view would this change if we removed the i which leads to the i in the canonical commutator
[Q,P]=i h-bar?
I agree that measuring along b does nothing to the element of reality along b’, and that the God of Spinoza still knows the element of reality along b’. But, measuring along b does create an in-principle limitation on the ability of any human to measure B(b’) with certainty. For example, say I measure B(b)=+1. I can thereafter never know for certain what B(b’) would have been, even if Spinoza’s God does. But, if b and b’ are 90 degrees apart, I can know that there is a 50% chance the measurement would have been B(b’)=+1 and 50% for B(b’)=-1. In other words, measuring B(b) limits my ability to know B(b’), and for human beings, turns our possible knowledge of B(b’) from something certain, to something probabilistic. And, measuring B(b) does “disturb” the system in the EPR sense. I called this “information destruction” but if you do not like that label give me one that you do like. Do you disagree with any of this?
I agree. And I think everyone else would agree also.
But the ability of a human to measure B(b’) once B(b) is measured is lost, is it not? Only the God of Spinoza (reality) can know B(b’) with certainty once B(b) is measured. But measuring B(b) disturbed the system, and so removed some information from the human, experimental, physics purview. We humans can only know B(b’) with probability. This is the human reality of this experiment, objectively stated, IMHO. Am I missing something?
That is an interesting argument which reminds me of complementarity. Can you bring accepted views about complementarity to bear directly on this?
So does this mean that we are violating the realism hypothesis by having B + B’ in the expression for the CHSH limits being +/-2? And is this the specific reason why you believe any contradiction with local realism asserted by Bell’s Theorem is nonsense?
Jay
Quantum theory is not a local-realistic theory. The raison d’être of the EPR argument and Einstein’s misgivings about quantum theory was that quantum theory is seriously missing many elements of reality. Your example vividly exhibits this fact. Neither Q+iP, nor [Q,P] has definite values similar to their classical counterparts, and consequently they do not have all the elements of reality that classical physics would assign to these variables.
Abilities of humans or advance aliens was of no concern to Einstein or EPR. That is something Bohr tried to bring-in while arguing against the EPR argument. Einstein regarded such a view anthropocentric and maintained that physical systems have intrinsic properties whether they are observed or not. Bohr thus entirely missed Einstein’s point. And you are missing it too.
Yes, as I noted above you are missing Einstein’s point. You are in a good company, however.
But this is the point I have been making for the past month or so. Complementarity has nothing to do with this. I think your stumbling block is that you are trying to bring-in ideas from quantum theory when we are not doing quantum theory at all. The fact that A(a) cannot simultaneously coexist with both B(b) and B(b’ ) is a classical, macroscopic fact (hbar = 0).
Absolutely. As we all agree, both B and B’ are elements of reality. But B + B’ is definitely not. So at the least Bell and CHSH overstretched, or rather abused EPR’s notion of realism when they surreptitiously smuggled-in the sum B + B’ into the CHSH expression. The bounds of +/-2 on CHSH are simply a consequence of this abuse. This is indeed the main reason why any contradiction with EPR’s notion of local realism asserted by Bell’s Theorem is nonsense.
I had more time than I expected the last day or two, and so have written up the quantitative analysis I promised above. You may read this at https://jayryablon.files.wordpress.com/2016/11/bell-impasse.pdf. My title for this is rather clear: An Attempt to Break the Impasse between Joy et al. and Richard et al. over Bell’s Theorem
I kept this clean. This is to say that although I have some ideas of my own regarding the 2 sqrt(2) QM bounds and also have already talked the past few days about measurement issues in terms of how a detection of B(b) in the b direction forecloses the opportunity for a human experimenter to detect B(b’) in b’ with certainty and only leaves probabilistic possibilities for characterizing what can thereafter be known about B(b’) given that measuring along b disturbs the system, ***I have not relied upon any of these personal views to arrive at the results in the attached***. I have kept this clean and focused on the re-framing impasse between Joy et al. and Richard et al. so that it can perhaps be resolved.
Specifically, I have used a direct, simple mathematical analysis to reformulate the derivation of the the CHSH bounds of +/-2 from the “slip and urn” model, into a “coin toss” representation based on the probabilistic completeness relation which states that for any single coin toss, we know with certainty that the result will be heads or tails but not both. This coin toss analysis leads to two alternative representations for the CHSH inequality which are (25) and (37). Perhaps importantly, perhaps not, I leave that to the readers, (25) and (37) make clear that the upper and lower CHSH bounds are the result of probabilistic completeness relations for coin tosses which do not care what the particular results of those coin tosses actually are.
The questions I pose at the end to *everybody who has been involved in these discussions* are quite simple:
Joy et al.: Do the representations (25) and (37) of the CHSH inequality help advance your argument against Bell?
Richard et al.: Do the representations (25) and (37) of the CHSH inequality help strengthen your argument in favor of Bell?
Jay
I usually would side with Einstein in this sort of argument, but here I find myself favoring Bohr. Whether or not human experimentation was of interest to Einstein or EPR, physics requires experiments by humans. If a physical act (e.g., measuring B(b)) makes it impossible to measure something else (e.g., B(b’)) because it creates a physical disturbance in a system being observed, then our observational data will reflect that disturbance and we will lose the opportunity to conduct experiments to detect what was disturbed as if it had not been disturbed. Put differently, if measuring B(b) affects my ability to measure B(b’), then what I am able to measure will be different in a post-B(b) world than in a pre-B(b) world. And to the degree that physics must describe what we measure and observe, this disturbance is a physical consequence which must be of concern and accounted for.
I have been grilling Joy quite a lot lately. But now I need to put Richard and Stephen and HR and LJ and maybe a few others up on the griddle:
The foregoing statement by Joy is now crystal clear: He says that the sum B+B’ (and the difference B-B’) is not an “element of reality” as that term is defined in EPR. Do you agree or disagree? Are this sum and this difference “elements of reality,” or are they not? And why?
Jay
The above is indeed THE key issue for me. To remove any ambiguity from the questions raised by Jay, let me make clear what is meant by the “elements of reality” in the present context of spin by quoting myself from an earlier post:
“The singlet state has two remarkable properties. First, it happens to be rotationally invariant. That is to say, it remains the same for all directions in space, denoted by the unit vector n. Second, it entails perfect spin correlations: If the component of spin along direction n is found to be “up” for particle 1, then with certainty it will be found to be “down” for particle 2, and vice versa. Consequently, one can predict with certainty the result of measuring any component of spin of particle 2 by previously measuring the same component of spin of particle 1. However, locality demands that measurements performed on particle 1 cannot bring about any real change in the remotely situated particle 2. Therefore, according to the EPR criterion of physical reality, the chosen spin component of particle 2 is an element of physical reality. But this argument goes through for any component of spin, and hence all infinitely many of the spin components of particle 2 are elements of physical reality, which may be represented by +1 or -1 points of a unit 2-sphere.”
More details about my claim that B + B’ is not an element of reality in the above sense, which is extracted from EPR (cf. the famous paper by GHSZ), can be found in my latest paper:
http://libertesphilosophica.info/blog/wp-content/uploads/2016/11/Fatal.pdf
It seems that Joy Christian maintains that B and B’ both are elements of reality, but somehow the sum B + B’ is not. Well, it’s just a sum of two numbers. If we follow Joy’s logic, it means that the average of two elements of reality is a meaningless concept, since it can be written as 0.5*(B + B’). But of course the average of two numbers is a very well defined concept, and in physics one uses them all the time.
If B + B’ is just a sum of two numbers, then it — and hence Bell’s argument — has nothing to do with the question of local realism. If, on the other hand, B + B’ is indeed an element of reality, then you should be able to demonstrate that using the well-established method of EPR I have just facilitated in my reply to Jay:
Bell’s argument is simply that a 16 slip urn model cannot reproduce all the correlations that QM predicts (in particular, with Bell’s argument it can be shown that the CHSH expression is bounded between -2 and 2 in the limit of N->inf). If you disagree, you can end the discussion by providing us with a counterexample of 16 frequencies for the 16 slips, that actually reproduce the QM correlations. It’s not harder than that. Then there is no need to discuss any proofs, because the case will be settled.
So What is do you predict the measured value of B + B’ will be in an experiment? Has any such experiment ever been done, and what did they observe the value B + B’ to be?
Earlier, HR, you and I agreed on a statement of Bell’s Theorem and its contrapositive. Let me now try to broaden that statement. Also, relatedly, Richard emailed me offline and in response to my question if B + B’ is an element of reality, his answer was “who cares?” I do not know yet if B + b’ is an element of reality, but let me explain why we have to care.
For all of its mystique, Bell’s theorem via the CHSH correlator is mathematically no more and no less than a very clever proof by external contradiction. Such proofs have four main steps:
1) Assume a hypothesis H to be true.
2) Deduce downstream consequences of H.
3) Test the downstream consequences against external (independent) knowledge and determine if there is a contradiction.
4) If there is a contradiction, then H is disproved.
For Bell, step 1 is to hypothesize the truth of H = local realism, which is nicely done via the urn and slip model. Step 2 is to use this to deduce the CHSH inequality with the bounds of +/-2. Step 3 is to show based on QM theory and experimental data that the CHSH sum can actually reach the bounds of +/-2sqrt (2), which is a contradiction. (And I will agree for this discussion with using the law of large numbers as part of establishing the +/-2sqrt (2) bounds.) Step 4, because of this contradiction, is to declare that H = local realism is a false hypothesis in relation to the external knowledge. Thus: quantum mechanics (the external knowledge which provided the contradiction) cannot be explained using local realism. Can we all agree that this is a fair characterization of Bell?
If this is the case, then it is very important that H=local realism and remain so throughout the entire proof, and that nothing be “smuggled in” along the way to change the hypothesis to:
H=something other than local realism.
So: if B+B’ is NOT an element of reality, but it is still used to derive the correlator limits of +/-2, then the CHSH inequality derived in step 2, when combined with the 2 sqrt(2) QM comparison in step 3, only allows us to conclude that
H=something other than local realism
is a false hypothesis in relation to QM, rather than:
H = local realism
is a false hypothesis in relation to QM.
So we must care, and must ascertain if B+B’ and B-B’ is or is not an element of reality, because if it is not, then Bell does not disprove local realism as it purports to do. And this is a SEPARATE QUESTION from whether B+B’ and B-B’ are valid sums and differences of numbers. It only has to do with whether these sums and differences are or are not elements of reality.
We can drill down even further here, because we are positing and are all agreed that B and B’ separately, ARE elements of reality. So the underlying question is this:
Is the the sum or difference of two elements of reality, itself ALWAYS an element of reality? Or, are there are situations in which the the sum or difference of two elements of reality may itself be NOT an element of reality?
If there are situations where the sum or difference of two elements of reality is NOT itself an element of reality, then the question is whether B+B’ and B-B’ are examples of such situations. If they ARE examples, then Bell fails as a disproof of local realism. If they are not examples, or if
element of reality 1 + element of reality 2 = another element of reality, always,
then Bell succeeds and is sustained as a disproof of local realism.
Final point:
If you go back to EPR which is here: http://journals.aps.org/pr/pdf/10.1103/PhysRev.47.777, you will see that in equation (1) they used the eigenvalue equation:
psi’ = A psi = a psi
to establish that when the eigenvalue a is a number and the “physical quantity A has with certainty the value a whenever the particle is in the state given by psi…[then] there is an element of physical reality corresponding to the physical quantity A.”
EPR came out in 1935, but in modern parlance, we would call the operator A an OBSERVABLE, and would say that a is one of its observed eigenvalues. So all of the opaque discussion here about elements of reality, really appears to boil down to a discussion about what in the modern era we call observable operators.
So the question that Joy really poses, is about the circumstances under which we can and cannot add or subtract two observable operators to yield a third observable operator, and how this relates to the sums B+B’ and B-B’ which are the eigenvalues of these operators.
Jay
1 universe + 1 universe = 2 universes ?
Anyone who claims to have ruled out the viability of local realism must care whether or not B + B’ is an element of the physical reality. This is not a matter of choice. It is a matter of logical necessity for any would-be eliminators of the viability of Einstein’s local realism.
The CHSH sum of expected values written in its traditional form, namely
Int_H AB rho(h) dh + Int_H AB’ rho(h) dh + Int_H A’B rho(h) dh – Int_H A’B’ rho(h) dh , …. (1)
is both mathematically and physically identical to the expression
Int_H A( B + B’ ) rho(h) dh + Int_H A'( B – B’ ) rho(h) dh , …. (2)
where Int_H stands for an integration over the space H of the hidden variables h.
Since the above two integral expressions are absolutely identical to each other, (1) = (2), it would be highly irresponsible to evade the question whether or not the quantities B + B’ and B – B ‘ appearing in (2) ( in respective products with A and A’ ) are elements of reality.
To appreciate the question, note that the second expression involves an integration over fictitious quantities like A( B + B’ ) and A'( B – B’ ). These quantities are not parts of the space of all possible measurement outcomes A, A’, B, B’, etc., — i.e., the space of all possible elements of reality — because that space is not closed under addition. This is analogous to the fact that the set D = {1, 2, 3, 4, 5, 6} of all possible outcomes of a die throw is not closed under addition. For example the impossible “outcome” 3+6 is not a part of the set D.
But more importantly, the quantities such as A( B + B’ ) do not represent any meaningful physical element of any possible world, classical or quantum. This is because B and B’ can coexist with A only counterfactually. If B coexists with A, then B’ cannot coexist with A, and vice versa. But in their hypothesis (2) Bell and CHSH implicitly assume that both B and B’ can coexist with A simultaneously. This would be analogous to my being in New York and Miami at exactly the same time. But no definition of realism can justify such an unphysical demand. The notion of realism envisaged by Einstein most certainly does not demand any such thing.
In my opinion the answer “who cares?” by Richard Gill is an admission that B + B’ is not an element of reality, and moreover both B and B’ cannot possibly coexist with A, and therefore the claim by Bell and his followers of having ruled out Einstein’s local realism is simply false.
Bell starts with the assumption that the data is generated by an urn model. An urn model is a mathematical construct. He then shows by a purely mathematical proof that for such a model, the value for the CHSH expression must lie within -2 and +2 in the limit for large N. This is a mathematical result about a mathematical model, and any discussion about “elements of reality” is irrelevant at this stage. Have you ever seen someone fuss about “elements of reality” when proving the Pythagorean theorem? And even someone who doesn’t understand Bell’s proof should be able to convince himself of the correctness of Bell’s claim by simply searching for a counterexample. That would involve 16 frequencies for the 16 slips that (in a simulation) consistently generates a CHSH value outside the bounds of -2, 2. Such a counterexample should be easy to find, if it exists. (A brute search by computer would find it in the blink of an eye).
Much of the confusion here stems from not realizing that Bell’s result is a mathematical theorem, not a physical theory. The only things left to discuss are: Is Bell’s theorem interesting? Is the urn model a good representation of whatever is meant by “local realism”?
This claim by HR is blatantly false. Not in a single paper of his Bell ever mentions anything about “urn model”, which by the way has absolutely nothing to do with physics.
If anyone disagrees with this, then they can provide a published reference where Bell or any of his followers like Clauser and Shimony mentions “urn model” in any of their papers.
The title of Bell’s famous paper is “On the Einstein-Podolsky-Rosen paradox”, which refers to the famous EPR paper, which in turn is about local realism. In neither of these papers the word “urn” is ever mentioned. The title of Bell’s second most famous paper is “On the problem of hidden variables in quantum mechanics.” In it Bell is again concerned about physics — in particular, about the lack of local realism in orthodox quantum mechanics.
I challenge HR to provide a single published paper by a respectable physicist who mentions anything about “urn model” within the context of Bell’s theorem and local realism.
He called it a “hidden variable interpretation” instead (it’s even mentioned in the abstract). Same thing as an urn model (the hidden variables correspond to the slips).
These are your misinterpretations. You have failed to provide a single published reference.
On this point, with all due respect to everybody else, I must absolutely agree with Joy. If Bell’s Theorem purports to disprove local realism by contradiction with QM, then you cannot be agnostic about whether or not the disproved hypothesis of local realism has turned into some other hypothesis along the way. That IMHO is an illegitimate attempt to have one’s cake and eat it too.
As to the above, here I am willing to say “who cares?” I am happy to assume for sake of argument that Bell’s theorem is an urn model, whether or not Bell actually ever said that. I have in fact accepted the urn model at face value in https://jayryablon.files.wordpress.com/2016/11/bell-impasse.pdf and shown starting in (9) that one still obtains the B+B’ style terms to which Joy has objected. Once we have those terms, it is legitimate and indeed ESSENTIAL to ask if those terms are elements of reality, because if they are not, then what we prove is that the slip model which assumes and encodes reality as a hypothesis leads to objects which are not elements of reality and so surreptitiously undermines its own realism hypothesis, i.e., is *internally self-contradictory*. Just like the classic statement:
“This statement is false”
Sorry my friend, but that is trying to have your cake and eat it too. You cannot say that Bell is just a mathematical theorem, then claim that it proves some fundamental point about physics.
OK, HR, here you redeem your argument, and now you and I are saying the same thing, just coming at it from opposite directions. If the urn model leads on its own terms to B+B’ style terms which are NOT good representations of “whatever is meant by ‘local realism’,” then Bell’s mathematical theorem has to be sealed off as just mathematics, and cannot be used to reach any conclusions about the physics of local realism.
So I come back to the question I asked last night, which everybody MUST care about, which is whether terms such as B+B’ are the themselves proper representations of local realism, on the assumption that B and B’ separately ARE proper representations of local realism. That is, does:
Realistic element 1 + Realistic element 2 = Realistic element 3, always? (1)
The reason I wrote https://jayryablon.files.wordpress.com/2016/11/bell-impasse.pdf, is so that if we cannot agree on (1), we can shift the question to whether in my (25), the CHSH upper bound:
.5 (H+T) = 2 (2)
is an element of reality. Or if that is inconclusive, whether in my (37), the CHSH upper bound:
.5 tau_j^*T tau_j = 2 (3)
is an element of reality.
Not only is Joy owed a serious answer to this question rather than its dismissal, so too is the entire scientific enterprise. And the central subject of this Retraction Watch thread, whether Joy’s paper was properly or improperly retracted by AOP, hinges on whether these sums and differences of elements of reality are themselves necessarily elements of reality, or not.
I know people are tired of this discussion. But fatigue is not a good excuse to allow what may be a flawed premise to continue to dominate scientific thinking. Pursuing sound science is always exhausting and consuming. People do it because it is also exhilarating and noble.
Jay
Thank you, Jay. At least we have managed to boil it down to this very precise question.
Actually, I would be quite happy to have an explicit answer to a much more modest question than your question (1). All I ask is: Given that B(b) and B(b’ ) are elements of reality as we all seem to agree, is B(b) + B(b’ ) also an element of reality? That is all I would like to know. Of course, if anyone answers “Yes”, then I would want to see a precise proof of that assertion.
Bell has set up a theoretical physical model for “an external local reality independent of our observations”, rigorously following Einstein’s assumptions and “recipes”. We are talking about nothing more than a theoretical physical model for the considered system, constructed in such a way that we dismiss any quantum mechanical features. To say it again: A physical model for “an external local reality independent of our observations”, viz., an alternative to quantum mechanics which tells us nothing but the probabilities to get certain measurement results.
Using this model, one can evaluate expectation values for physical quantities and their correlations, for example – when considering an entangled system of two spin-1/2 particles -, E{A(a)* B(b)}, E{(A(a)*B(b’)}, E{(A(a’)* B(b)} and E{A(a’ )*B(b’)}. All this can be done within the scope of the model, nobody has to bother about possible or impossible measure experiments. It doesn’t matter that we know, for example, that there is experimentally no direct access to a quantity like A(a)*(B(b)+B(b’)). We are talking about nothing else than a physical model, so one merely passes over to an evaluation of E{A(a)* B(b)} +E{A(a)*B(b’)}.
That’s all. Experiments serve only to compare model predictions with experimentally accessible average values (we have only to assure statistical significance). In case there are contradictions, we must then conclude that our physical model is wrong.
Well, I have never said that Bell’s theorem proves some fundamental point about physics. Quite the contrary, Bell’s theorem is completely useless for the practicing physicist.
There is only one group that should care about Bell’s theorem: If you try to find a theory that explains quantum mechanics at a more fundamental level (local realistic?), then you should stay far away from 16 slip urn models (aka local hidden variables), or anything that can be reduced to one, because if you don’t you will surely fail.
No Jay, the paper should never have been published independent of the applicability of Bell to physics solely on the basis of the math errors committed. S^3 is not closed for addition, his use of the triangle inequality when the domain was restricted to S^3 was improper. In a sequence of dependent equations, one cannot perform a simplification by assigning equality between two variables then in a later dependent equation allow the two variables to converge “in the limit” to different values. The two orientations for S^3 are represented by isomorphic algebras, when chiral right(left) is mapped to left(right) such that one may legitimately add right:A*B to left:A*B, the bivector result does not subtract out as Christian claims.
A local hidden variables model posits the existence of mathematical objects rho, A and B where A and B are functions taking values +/-1 and rho is a probability measure. The model says: nature draws a value lambda at random according to the probability distribution rho; the experimenter independently thereof picks settings a and b; and the measurement devices output measurement results A(a, lambda) and B(b, lambda).
Thinking of an experiment where the experimenter only uses settings a or a’, and b or b’, the model reduces to an urn-and-slips model: fill the urn by writing the four numbers A(a, lambda), A(a’, lambda), B(b, lambda), B(b’, lambda) on a slip of paper, one for each of many, many drawings of lambda. Now we can simulate the model by repeatedly choosing settings for Alice and Bob, drawing a slip of paper from the urn, and reporting the corresponding measurement outcomes already written on the slip of paper.
Bell-CHSH inequality is an elementary true result for the urn-and-slips model which was already known to the famous logician George Boole in the 19th century. There are numerous ways to prove it. The proofs are pure mathematics. Metaphysical concepts like “element of reality” play no role in them whatsoever.
Nobody can forbid the mathematician from studying the range of sets of correlations which the urn model can generate, by objecting to use of an expression like AB + AB’ + A’B – A’B’. Physical objections to this quantity come down to physical objections to local hidden variables, in the first place. If you a priori (on physical grounds) reject local hidden variables models, then you are not interested in Bell’s theorem. If on the other hand you want to promote some local hidden variable model, then you had better take account of Bell’s theorem in one way or another.
Christian’s present paper claims to contain a counter-example to Bell’s theorem. As far as I can see, his concept of local hidden variables is the conventional one. His concept of “correlation” is the conventional one. Hence Bell-CHSH is unavoidable. Hence his counter-example must contain errors. So far, plenty of serious errors have been pinpointed; and no *mathematical* objection to Bell’s theorem has been raised. The situation seems to me to be pretty clear.
Christian objects that the concept of “urn-model” does not belong in physics in general, nor in the literature on Bell’s theorem. Elementary probability theory does belong in physics and was moreover used by Bell. The urn model is a pedagogical device for making abstract probability concepts more tangible for those unfamiliar with probability theory.
Personally I like to think of Bell’s theorem as an elementary theorem from computer science. It tells us that it is impossible to simulate the EPR-B correlations on a network of computers (source, measurement station 1, measurement station 2) when communication between the two measurement stations and any kind of post-selection is forbidden. The measurement stations each receive a realisation of the hidden variable, and each receive a setting, and are obliged to output a measurement outcome. Repeat many times …
Richard,
Is it fair to say that what you are saying is that terms like B+B’ can be considered in the CHSH limits, irrespective of whether they are “elements of reality” according to some definition (and what would that definition be?), precisely because Bell’s input hypothesis also includes that of local *hidden variables*?
And are you then saying that the urn model encodes local hidden variables together with local realism and therefore permits you to use B+B’ etc. the obtain the CHSH limits of +/-2 and have them stick whether or not those B+B’ constructs are themselves elements of reality?
If that is so, I would be interested as a purely academic matter in understanding how that works, i.e., how the hidden variables free us from concern about the realism of B+B’.
And I would still like to know whether my (25) and (37) in https://jayryablon.files.wordpress.com/2016/11/bell-impasse.pdf, which I believe you would agree with as mathematical results, provide a way for you to strengthen your argument, just as I have asked Joy if they provide a way to strengthen his.
I will add one more question, just for some fun in case we are not having enough fun yet: 🙂 If Einstein were to come back to life, what would he say about all of this? Be serious; get into his shoes and lay out how he would approach and try to resolve this.
Jay
Because we are assuming local hidden variables, B + B’ exists. It’s just some numerical function of lambda.
Jay, I am still waiting for the answer to our simple question about the “reality” of B(b) + B(b’ ).
If an honest answer is not forthcoming, then we have to conclude that Bell’s mathematical theorem has no relevance for physics. As I have already detailed in my recent paper,
http://libertesphilosophica.info/blog/wp-content/uploads/2016/11/Fatal.pdf ,
the bounds of -2 and +2 on the CHSH correlator are purely mathematical curiosities, without any relevance for the question of local realism. Consequently there is no logical, physical, or mathematical obstacle against reproducing the quantum correlations E(a, b) = -a.b purely local-realistically. Indeed I have successfully reproduced the correlations in several different ways, in the paper that was peer-reviewed for 7 months and published in Annals of Physics:
https://arxiv.org/abs/1405.2355 .
Any competent physicist and/or mathematician can easily verify and confirm my results, as many have done already. Those who have not yet tried, I would urge that they do not fall for the fallacious online claims of errors in my paper. There are no errors in my paper, period.
The honest answer to your question is yes: if A(a), A(a’), B(b) and B(b’) are simultaneously existing definite numerical properties of some physical system, then the value taken by the vector (A(a), A(a’), B(b), B(b’)) is also a definite property of the system, and the value taken by any given numerical function thereof exists in reality also.
It does not have to be accessible to experiment. It does not have to have any particular physical interest.
Einstein never *defined* element of reality. He gave a sufficient condition for something to be an element of reality, but not a necessary condition.
The question I asked was the following:
Given that B(b) and B(b’ ) are elements of reality, is B(b) + B(b’ ) also an element of reality?
If the answer to this question is “yes”, then where is an explicit proof of that answer?
For convenience, here is how EPR would specify “elements of reality” for the singlet state:
The singlet state has two remarkable properties. First, it happens to be rotationally invariant. That is to say, it remains the same for all directions in space, denoted by the unit vector n. Second, it entails perfect spin correlations: If the component of spin along direction n is found to be “up” for particle 1, then with certainty it will be found to be “down” for particle 2, and vice versa. Consequently, one can predict with certainty the result of measuring any component of spin of particle 2 by previously measuring the same component of spin of particle 1. However, locality demands that measurements performed on particle 1 cannot bring about any real change in the remotely situated particle 2. Therefore, according to the criterion of reality, the chosen spin component of particle 2 is an element of physical reality. But this argument goes through for any component of spin, and hence all infinitely many of the spin components of particle 2 are elements of physical reality (cf. the paper by GHSZ).
It is evident to me that B(b) + B(b’ ) is NOT an element of reality according to EPR’s criterion.
An element of reality is a property of a physical system which takes on a definite numerical value independently of whether or not anyone observes that system. If B(b) and B(b’) are elements of reality, then the value of B(b) + the value of B(b’) exists independently of observation of the system. It’s a physical property of the physical system which takes on a definite numerical value. This seems to me to be good enough reason to call it an “element of reality” too.
Note that EPR did not define “element of reality”. They only gave a criterion which can sometimes be used to determine that some variables are elements of reality. It seems to me that they would agree that if X and Y are elements of reality, then (X, Y) is an element of reality too; and if Z is an element of reality, and g is some function, then g(Z) is an element of reality too. They are all physical properties of the system which take on definite values completely independently of any external observer.
I disagree. According to the sufficient criterion of reality specified by EPR, if there is no way to predict the value of a physical quantity with certainty EVEN IN PRINCIPLE, then there is no element of reality corresponding to that physical quantity. Since there is no experiment that can ever be performed to predict the value B(b) + B(b’) with certainty even in principle (in any possible world), B(b) + B(b’) is not an element of reality even if B(b) and B(b’) individually are.
Rick, fair point, so let me rephrase:
“And the central subject of this Retraction Watch thread, whether Joy’s paper was properly or improperly retracted by AOP, ***to the extent the retraction was based on Bell’s Theorem’s exclusion of local realism as a basis for explaining quantum mechanics***, hinges on whether these sums and differences of elements of reality are themselves necessarily elements of reality, or not.”
Of course, if Bell is removed as a “no go” obstacle, any real or perceived flaws in Joy’s S^3 model then do become completely fair game.
I took my emphasized addition above to be implicit, going without saying. This is because the RW discussion for the past few weeks based on Stephen Parrott’s good suggestion has been focused solely on Bell’s Theorem, with the understanding that we would turn to the specifics of Joy’s S^3 model if and only if it could be shown and agreed that this is permitted notwithstanding Bell. That question about Bell is now tightly focused on the quantities B+B’ and a) whether these are “elements of reality” according to definitions also not agreed upon by all the participants, and b) whether it matters whether these are “elements of reality,” no matter what these definitions may be.
I also reproduce what was stated by the AOP editors in the retraction:
“After extensive review, the Editors unanimously concluded that the results are in obvious conflict with a proven scientific fact, i.e., violation of local realism that has been demonstrated not only theoretically but experimentally in recent experiments. On this basis, the Editors decided to withdraw the paper.”
So it seems to me that Bell’s Theorem, which everyone has been discussing passionately here, was in fact the basis for this retraction, and that no mind was paid by the editors to the specifics of Joy’s S^3 model because they concluded that Bell did not permit them to get that far.
So whether there is or is not a lurking flaw with the realism hypothesis in Bell expressed through the B+B’ variables, about which there is clearly vehement disagreement, is properly the focus here.
Jay
For those who claim that B(b) + B(b’) is an element of reality, I have a question: Can you predict with certainty the result of the measurement B(b) + B(b’)? Has any such measurement ever been done for the EPRB Experiment, and what was the observed result? If not, is the absence of such experimental proof evidence that B(b), B(b’) independently do not correspond to elements of reality?
Good questions, MF.
From those who claim that B(b) + B(b’) is an element of reality satisfying the EPR criterion of reality, I would like to know precisely what experiment Alice can perform with a result A(a) — at least in principle — to predict with certainty the result of Bob’s measurement of B(b) + B(b’), given the fact that b and b’ are mutually exclusive measurement directions.
I don’t understand how this whole discussion can get so muddled and complicated.
Here is the claim: “No 16 slip urn model can produce a value for the CHSH expression greater than 2 (in the limit N->inf).”
If the claim is false, there must be a counterexample. Since such a counterexample would amount to finding just 16 numbers (frequencies) for the slips that produced a value greater than 2, someone would have found it long ago.
We are discussing physics. urn model has nothing to do with physics.
Your claim is obviously true but you missed how it connects to the physics of the EPR-Bohm scenario. Please provide that connection.
OK, HR, I am game.
I believe I have found a way to produce a value for the CHSH expression greater than 2, and in fact, it will be 2 sqrt(2), exactly. I stayed away from going there because I wanted to stay focused directly on the impasse at hand. But maybe the best way to resolve this impasse is to show a derivation outside the bounds that neither Richard nor Joy will expect, and then let y’all beat up on that if you can.
Before I do that, however, please review what I have posted to https://jayryablon.files.wordpress.com/2016/11/bell-impasse-2.pdf, and let me know if you see any problem with my “coin toss” rendering of the CHSH inequality in (25) and (37), because my counterexample will proceed from (25) and (37). I fixed this up slightly from what I posted yesterday, because I had overlooked a normalization that needs to take place for a larger number of trials than the minimal subset of 4 trials needed to represent locality.
The derivation does require, however, that we draw a distinction between what goes on in the purely local realistic universe of “Spinoza’s God,” and what humans are able to measure and observe which is only a subset of that reality and is necessarily probabilistic owing to the inherent physical limitations of being an observer living inside of the physical universe. Since I know that this “Bohr argument” will draw controversy from both sides, I would like to know first if (25) and (37) are “so far, so good.” 🙂
Thanks,
Jay
For your eq. (37), as long as you have all the indices as j on each expectation term, you will not be able to exceed the bound of |2|. It is mathematically impossible.
Fred, I always like a challenge. I am now thoroughly aware of why the unbreakable bound is |2|, and yes, j is the index on all the terms and I am thoroughly aware of what that does to restrict the bound and how it does that.
But there is a probabilistic line of development that I believe will do just what you say is impossible — get beyond the bound out to |2 sqrt(2)| — and it does rely on an argument that Einstein would never have made but that Bohr would have about inherent natural limits on what human can *predict* about individual tosses of fair coins or biased coins (toward Richard and HR’s points about slip frequencies). And it is not the usual QM derivation of 2 sqtr(2). And it does not make any of the arguments that has caused the impasse between Joy and Richard. But it does use the urn and slips. And it does allow the QM correlations to be explained consistent with local realism, using actual physics limits on what humans can *predict*. And I have twice emphasized the word *predict*, because with all else that has been discussed here, this is also a key word — “predict,” not “measure” — that appears in EPR, which has not yet drawn what I believe is its proper share of attention here. So all in all, I think I may have finally learned enough here about EPR and Bell to risk flying solo without a net. 🙂
But first I will wait to see if someone from the Bell side of the debate agrees with (25) and (37), and if so, then I’ll proceed.
Jay
I am sorry Jay, but you are on a fool’s errand as it is already proven that it is mathematically impossible for anything to exceed the bound of |2| probabilistic or not. Believe me as Joy and I spent quite a bit of time on the same fool’s errand unknowingly many times.
I don’t agree with (25) and (37). I agree with HR that all this is unnecessarily muddled and complicated.
Richard, You must agree with (17). And (25) says the exact same thing in a different way. So if you disagree, please point out exactly what I did wrong. Thank you. Jay
The EPR Criterion of reality:
“If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.”
With the above background of various opinions, the crucial question is now quite clear:
If B(b) + B(b’) is claimed to be an element of reality satisfying the above EPR criterion of physical reality, then precisely what experiment Alice (or anyone else) can perform — at least in principle — to predict with certainty the result of Bob’s measurement of B(b) + B(b’), given the fact that b and b’ are mutually exclusive counterfactual measurement directions of Bob.
I claim that no experiment can ever be performed, by anyone at all, even in principle, in any possible world, to predict with certainty the value of the quantity B(b) + B(b’), and therefore B(b) + B(b’) is not an element of reality. The implications of this claim are far reaching.
The EPR sentence says “predict”, not “measure”. If you can predict B(b), and you can predict B(b’), you can also predict B(b) + B(b’).
EPR explicitly requires us to specify an experimental arrangement that can be performed, at least in principle, to predict a quantity with certainty (see footnote 10 of the GHSZ paper).
Please describe an experimental arrangement that can be used to predict the quantity B(b) + B(b’) with certainty. You can’t. Because it is impossible. Therefore B(b) + B(b’) is not an element of reality.
Nobody is claiming that B(b) + B(b’) satisfies EPR’s criterion.
EPR propose that if X satisfies their criterion then there exists an element of physical reality corresponding to X.
They did not argue the converse: that if there exists an element of physical reality corresponding to X, then X satisfies their criterion.
They did not even give a definition of “element of reality”. Their criterion is a sufficient, but not a necessary condition! http://plato.stanford.edu/entries/qt-epr/
So: B(b) + B(b’) can be called an element of physical reality, even if there is no way of predicting its value with certainty without disturbing the system.
But what is an element of physical reality, then? Einstein has not given us a definition; nor has anyone else, as far as I know.
We can deduce that it is a property of a physical system which takes a definite value independently of measurement of the system. If B(b) and B(b’) are both elements of reality of a physical system, this means that they take definite values whether or not we observe the system. This means that their sum takes a definite value whether or not we observe the system. For me, this makes their sum just as “real” as each component of the sum.
Well, in the EPR-Bohm scenario the best that can be predicted is that you will have a 50-50 chance of a +1 or -1 outcome at A or B for any theory. It is because the Stern-Gerlach polarizer aligns the spins with it’s angle and at that point, you will have either up (+1) or down (-1) detected. So since Bell based his model on that, it can’t say anything about realism. There can never be a prediction with 100 percent certainty.
The error made by Bell and his followers is now very clearly brought out. The bottom line is that the quantity B(b) + B(b’) appearing in the CHSH sum is not an element of physical reality, because there is no way of predicting its value with certainty without disturbing the system.
But it has to be an element of reality for the claim of Bell and his followers (and of the editors of Annals of Physics) of having ruled out local realism to be valid. There should not be any difficulty in understanding the significance of this elementary point. But let me explain:
To begin with, the CHSH sum of expected values written in its traditional form, namely
Int_H AB rho(h) dh + Int_H AB’ rho(h) dh + Int_H A’B rho(h) dh – Int_H A’B’ rho(h) dh , …. (1)
is both mathematically and physically identical to the expression
Int_H A( B + B’ ) rho(h) dh + Int_H A'( B – B’ ) rho(h) dh , …. (2)
where Int_H stands for an integration over the space H of the hidden variables h. Since the above two integral expressions are identical to each other, it is vital that the quantity B + B’ appearing in (2) is an element of reality for the CHSH sum to have any relevance at all for the question of local realism. There is no point in denying or evading this simple observation.
Now the first requirement for anything to be an element of reality is that it is a physically meaningful quantity. But B(b) + B(b’) is not even that. Physically it is an impossibility — an absurdity, let alone being an element of reality. If we allow such a meaningless quantity in (2), then any consequences derived from (2), such as the bounds of -2 and +2, are equally meaningless, and therefore cannot possibly have any relevance for physics, or local realism.
It is not difficult to see that B + B’ is a physically meaningless quantity. For example, it is no different from an impossible “outcome” like 3+6 of a die throw with only possible outcomes being 1, 2, 3, 4, 5 and 6. Nothing more sophisticated is needed to see that B + B’ too is an impossible outcome. What is more, B and B’ in (2) can coexist with A only counterfactually. If B coexists with A, then B’ cannot coexist with A, and vice versa. But in (2) both B and B’ are assumed to coexist with A simultaneously. Since both B and B’ cannot coexist with A, the claim by Bell et al. of having ruled out Einstein’s local realism is clearly in gross error.
Jay,
Just one comment. What does it mean for a.coin to be biased? Is it a property of the coin, or is it a property of the toss? Say you have a coin reading machine into which you toss your coin. You set it in advance which side you want it to read, every time the coin comes up with the chosen side, the bell rings. You now perform the experiment with the machine set to H. After 50 tosses, you get 40 rings. P(H)i = 0.8,. You set it to T and after 50 tosses, you get 35 rings. P(T)j = 0.7. But P(T)j + P(H)i = 1.5 =/= 1. Our machine is always biased towards the side we pick to measure.
Because we necessarily performed the experiment in two passes(I,j), because it was impossible for the machine to do both measurements simultaneously we arrived at a sum that is absurd. But due to symmetry, we could have inferred the counterfactual probability in each experiment as P(H)i = 1 – P(T)i. In that case, the sum of the observed and counterfactual probability, P(H)i + P(T)i= 1 same for (j).
Now you see that both P(H)i + P(T)i= 1, and P(H)i + P(T)j = 1.5 are correct, for the same experiment. The former is unmeasurable since it is a sum of actual and counterfactual. The latter is the actual result. The former sum is not an element of reality, in the world of the specific experimental situation.
That is why you can’t ignore the subscripts.
MF,
I am assuming the coin or the toss has the same odds no matter how many runs you do. I know some people do not like the slip and urn, but using that anyway, if I put 10 million slips in the urn and 9 million have B’=+1 (heads) and 1 million have B=-1 (tails) then my probability on any single draw is P(H)=.9 and P(T)=.1, and they always sum to 1. That is:
P(H) + P(T) = 1 (1)
What I will point out is that we can use the mathematical identity
cos^2 X + sin^2 X = 1 (2)
to parameterize to some theoretical angle X (that we hope to learn more about):
P(H) == cos^2 X (3)
P(T) == sin^2 X.
This gets very interesting once you do that, because if rather than saying that H=1 and T=0 or H=0 and T=1 always, which states the *result* of each coin toss after the fact, we instead talk about the *predicted* probability beforehand (or about what the probability of what have occurred if we can never know a result) and set H=P(H) and T=P(H), and then use the wavefunction representation:
H+T = P(H) + P(T) = tau^t tau = 1 (4)
Then, we can represent these probabilities by setting:
tau^t = (cos X sin X) (5)
So an actual, certain result of heads or tails can be parameterized by:
Heads: X=0 Tails: X=pi/2 (6)
Then, from (5), cos x and sin x become the expansion coefficients c (think Dirac bras and kets) for the orthogonal (counterfactual?) base states:
Heads: tau^t = (1 0); Tails: tau^t = (0 1) (8)
That is, the expansion coefficients c which combine the base states (8) into (5) will be:
c_H = cos X; c_T = sin X (9)
So in general the sum:
c_H + c_T = cos X + sin X (10)
So when X=0 or X=pi/2 which is a special case of this general case, we have
c_H + c_T = 1 + 0 = 1 or (11)
c_H + c_T = 0 + 1 = 1
and the CHSH correlator upper bound can be written as the prison we all know and love (or not):
|CHSH| le 2(c_H + c_T) = 2. (12)
But for the general case, looking for the maximum value of (12), we go to X=pi/4, and find what we know is the the maximum permitted QM and observational value for this upper bound:
|CHSH| le 2(c_H + c_T) = 2(cos X + sin X) = 2(cos pi/4 + sin pi/4) = 2 sqrt(2) (13)
Voila! We know what that number means. Fred told me last night how impossible this was and I told him I love a challenge. And we know, for integer n, that pi/4 + n pi/2 are also the a dot b angles at which the quantum correlations reach their most extreme values as seen in the Wiki graph at https://en.wikipedia.org/wiki/Bell's_theorem#/media/File:Bell.svg by our very own Richard Gill. So setting (may need sign check on aXb)
cos X = -a.b (14)
sin X = -aXb
it looks like perhaps:
2(c_H + c_T) = -2(a.b + aXb) (15)
and the unknown parameter X that we started using at (2) and (3) may directly represent the angle between the Alice and Bob detectors.
I am still pondering how to interpret the calculation I just did abobve, but now you see my endgame with the heads and tails representation of the CHSH bounds. There is indubitably a mixing of realism and probability going on here that needs to be brought to ground to help us better understand what EPR was originally designed to help us better understand.
Now I am now fearlessly walking on the highwire without a net, and not advocating for anybody’s view of Bell. I am simply revealing a possible new calculation that has dawned on me the past few days as I was trying to reconcile Richard and Joy’s views, and am myself presently working my way through its interpretation. So y’all may fire away! 🙂
Jay
How do you justify eq. (12)? That is not Bell-CHSH. You would have to have
cos^2 (x) + sin^2 (x) = 1 for it to be valid Bell-CHSH.
Fred, you are absolutely correct about that. At (12) I am finding a general case which is equal to a special case and equating those. Then afterwards I generalize back to the general case. I am effectively setting 1 = sqrt 1 for the probability sum. I do not have at this time a justification for (12) and would need to find one. Right now the justification is that it leads to a result that we know is independently true. Joy may very well be correct that this is just another way of doing the usual quantum mechanical derivation. But if that is so, it never hurts to have more than one way to derive certain things that we know to be true. The most justification I can give at the moment is that in a skeletal sense, 1=sqrt(1) represents a singular circumstance under which realism coincides with probability in the world. But it is also possible to be more mundane and say that this is the place where the classical correlations correspond to the quantum correlations. Jay
PS: Do let me add one other note that is also behind my thinking. Maybe it has legs, maybe not. I did say I am on a highwire with this but willing to take the risk:
EPR provides a “sufficient” definition of reality based on operators with definite eigenvalues, their equitation (1), A psi = a psi. Once I take a coin toss and represent this as a 2 component column vector, I implicitly need to introduce operators to go along with that. The quaternions of SU(2) –> S^3 seem to arise from this calculation. In SU(2) for spin there are two observable operators: the +/-1/2 eigenvaues of the diagonalized sigma_3 which is linear, and the Casimir operator s=1/2 defined as s(s+1)=sigma_1^2 + sigma_2^2 + sigma_3^2. The “coin toss group” seems to have an analogous structure and if that is so, then it too has a linear operator and a non-linear Casimir operator. My sense is that formally speaking, Bell-CHSH is built upon a Casimir operator of the “coin toss group” (which relates to cos^2 (x) + sin^2 (x) = 1) and the QM outer bounds of 2 sqrt(2) is built upon the linear operator which is at the square root level from of the non-linear operator, and that this is where probability diverges from realism.
Can someone please provide a reference for a good, clear, concise derivation of the +/-2 sqrt(2) boundaries for the CHSH correlator from quantum mechanics? Or, lay it out here as simply as you can.
Particularly, given that
2 sqrt(2) = 4 * sin (pi/4) = 4 * cos (pi/4), (1)
and that pi/4 = a.b and 90 degree variants are the angles at which the quantum correlations deviate most extremely from the classical correlations, I am especially interested in any derivation ***in which some role is played by anything resembling (1) above***. So please check your memory banks and see if (1) rings a Bell 🙂 for anyone in any QM CHSH derivation.
Thanks,
Jay
Fred is still right. You cannot “violate” the bounds of +/-2 on the traditional CHSH correlator. It is impossible to do so by anything; QM, experiments, LHV, non-local HV, you, or “God.”
What you have done is to “exceed” the bounds of +/-2. But that is not difficult to do. For example, quantum mechanics does that quite easily. So does my local-realistic 3-sphere model. There are several ways to derive the quantum mechanical bounds of -2√2 and +2√2; e.g. using Clifford or Geometric algebra, as I have shown many times.
For the quantum mechanical prediction of
E(a, b) = -a.b ,
the bounds on the sum
E(a, b) + E(a, b’ ) + E(a’, b) – E(a’, b’ )
can be quite easily derived to be:
-2√2 ≤ -a.b – a.b’ – a’.b + a’.b’ ≤ +2√2 .
Assuming what you have done is mathematically correct and physically local-realistic, what you have done is just a variant of one of the above derivations.
Richard,
In the event that your disagreement is not really a disagreement but a “you tried but didn’t understand what I was doing,” let me simplify things here, and I will use your own paper https://arxiv.org/abs/1207.5103 to do so. And notwithstanding any objection Joy might have, I am willing for discussion here to accept the slip and urn model fully. My more extensive piece at https://jayryablon.files.wordpress.com/2016/11/bell-impasse-2.pdf was an effort to be rigorous and detailed, but the crux is simple.
Let’s go right to your FACT 1, part (1). Let us consider a single “slip” drawn from an urn, with “any four numbers A, A’, B, B’ each equal to +/-1.
Naturally, mathematically:
AB + AB’ + A’B + A’B’ = A(B+B’) +A'(B-B’) (1)
which is exactly what you have.
Now let’s ask the question, what is the largest value that one can obtain for the number in (1)? To answer this, let me choose 3 of the 4 slip values, but leave B’ unspecified, so that:
(A, A’, B, B’) = (+1, +1, +1, +/-1) (2)
For B’, I might say that the probability of +1 is 50% and the remaining probability of -1 is also 50%, so these of course sum to 100%. If I do that, I can model this with a fair coin. If I choose probabilities other than 50%-50% then I can model this with a biased coin (or biased toss). But in any event, the probabilities must sum to 100%.
So to keep things simple, let’s say we go with 50%-50%. So now I have specified a “probability” for what will be on this single slip, which you and HR have said I can do:
50%: (A, A’, B, B’) = (+1, +1, +1, +1)
50%: (A, A’, B, B’) = (+1, +1, +1, -1) (3)
0% otherwise.
Perhaps — you educate me — the better way to state this is that I have two slips in the urn, with the first two values shown in (3), and I will only draw one. Or, I have lots of slips, in the proportions shown in (3).
With this, (1) will become:
AB + AB’ + A’B +A’B’ = A(B+B’) +A'(B-B’) = (B+B’) +(B-B’) = +2B =+2 (4)
You and I and everyone else knows that this then becomes the upper bound on the CHSH correlator. Especially with Fred having now stated that he and Joy believe it is “impossible for anything to exceed the bound of |2|,” I will accept every bit of your later argument in https://arxiv.org/abs/1207.5103 that goes from the +2 in (4) above to the generalized CHSH bounds of |2| once you do large numbers and averages and expectation values.
The argument between you and Joy then boils down to whether B+B’ and B-B’ are “elements of reality,” which depends a) on how that term is defined and b) whether it even matters. And as to a) I must note as you have done before, EPR stated that their criterion for reality was “sufficient” but not “necessary.” So, IF it matters at all (which you believe it does not), the question as to what are acceptable “elements of reality” is a live question.
Also, I do not believe anyone here can make a blanket assertion that anything which does not meet what EPR laid out cannot be a proper criterion for reality, because EPR clearly stated that what they used was “sufficient” nor “necessary.” So my goal from here is drill down on the question what other “sufficient” criteria might be acceptable for “elements of reality” to once and for all resolve what I agree with you and HR has become “unnecessarily muddled and complicated.”
With all that I wrote at //jayryablon.files.wordpress.com/2016/11/bell-impasse-2.pdf, all that I am saying beyond the usual CHSH inequality is this:
If B’ has a 50%-50% coin toss chance of being +1 or -1, then as to (4) above, we will have:
50%: .5(B+B’) = 1 and .5(B-B’) = 0 (5)
50%: .5(B+B’) = 0 and .5(B-B’) = 1
Except for my using a factor of .5, and using a specific probability as an example, you make exactly the same argument in your proof (1) of https://arxiv.org/abs/1207.5103.
So just like the “urn and slip” metaphor is used for Bell’s Theorem generally, I can call .5(B+B’) a “heads counter” and .5(B-B’) a tails counter, that is, I can define:
.5(B+B’) == +H (6)
.5(B-B’) == +T
With those definitions, (5) above become the very simple:
50%: H = 1 and T = 0 (7)
50%: H = 0 and T = 1
And because these are orthogonal (mutually-exclusive) outcomes for any given toss / slip draw, I may also define a two-component vector tau (I’ll use transpose notation t because I cannot make these vertical column vectors here, as I would like to do) to represent heads and tails as such:
Heads: tau^t = (1 0); Tails: tau^t = (0 1) (8)
Then, if I use the definitions (6) in (4) I obtain:
AB + AB’ + A’B +A’B’ = A(B+B’) +A'(B-B’) = (B+B’) +(B-B’) = +2H + 2T = 2(H+T) = +2 (9)
The final equivalence to +2 arises because H+T=1, which simply states that the probability of getting one or the other of heads or tails on a single coin flip is equal to 1, irrespective of whether I actually get heads or tails. And also, this is irrespective of whether the coin is a 50%-50% fair coin, or a coin that is weighted (or tossed to be weighted) toward some other probability balance. That is where I can remove the 50%-50% limitation and call it a biased coin or a biased toss (implied reply to what recently came in from MF).
Likewise, if I use the column vectors (8), then another way to write H+T=1 for a single toss is to write:
H+T = tau^t tau = 1 (10)
so that (9) now becomes:
AB + AB’ + A’B +A’B’ = A(B+B’) +A'(B-B’) = (B+B’) +(B-B’) = 2(H+T) = 2 tau^t tau = +2 (11)
This “2” then feeds into everything else you do in https://arxiv.org/abs/1207.5103 to derive the CHSH correlator and its bounds of |2|. You can run as many trials j=1…J as you wish. Designate each toss by H_j and T_j which sum to 1 for any given toss for probability completeness, e.g., H_2+T_2=1, or using the vectors, designate each toss with tau_j so that, e.g., tau_2^t tau_2 =1, also for completeness. The point is that now the number 2 in the correlator bounds after you have done whatever derivation you please from the single slip in part (1) of your proof, will be:
2 = 2 sum_j (H_j+T_j) / J = 2 tau_j^t tau_j / J (12)
Now to the point: how does this (maybe?, hopefully? with some prayers so we can all find peace?) help settle the argument between you and Joy?
Joy says that B+B’ and B-B’ are not elements of reality. But if I turn them into coin tosses as in (6), then the question is this:
Is the true logical, mathematical fact that the sum H+T=1 for any given coin toss a sufficient element of reality? Alternatively stated, is it a sufficient element of reality to say that when I flip a coin, the probability of getting either heads or tail is 100%? Alternatively stated, if I toss a coin J times, is it a sufficient element of reality to to say that the sum of heads plus tails will always be the following:
J = sum_j (H_j+T_j) (13)
In other words, may we regards probabilistic completeness statements, which are know with certainty, such as:
“If I toss a coin J times, then the total number of times the coin will land on heads plus the total number of times it will land on tails will equal the number of tosses, J”
to contain a “sufficient element of reality”?
Or, do I also have to know about the ***individual results*** of each coin toss before I have established a sufficient criterion of reality?
I do not know, and I am asking this question to both you and Joy, and am hoping that this re-frames your so-far endless disagreement.
What I think I do know is this:
If we accept as a sufficient element of reality the statement that tossing a coin, fair or biased, will lead on either heads or tails with a probability of 1, then the debate is over and you prevail. If this is insufficient, and it is necessary to also know the particulars of the coin tosses before we have sufficiently defined reality, then Joy prevails, unless the criterion for reality does not matter.
Further, your view that the definition of elements of really does not matter, leads to the exact same result as saying:
“Who cares if the individual coin tosses are come up heads or tails, because we know that each toss will definitely land on one or the other both not both. And, that is all we need to make a sufficient definition of reality: reality is that probabilities always sum to 1.”
Put even more concisely, whether you realize it or not, you are arguing for a “looser” definition of reality which regards it as *sufficient* to make a certain statement about the complete set of probabilistic coin toss outcomes which sum to a “probability equal to unity.” Joy, whether he realizes it or not, is arguing for a “tighter” definition of reality in which each individual coin toss must be known with a “probability equal to unity,” and he is arguing that probabilistic completeness relations are no more certain about reality than saying (and this is a loose analogy not a tight analogy) “I am standing somewhere on earth with certainty.” Joy wants to know if I am in New York or Miami before he deems that to be reality. You are saying it does not matter, so long as you know with a probability equal to unity that I am somewhere on earth.
Different arena, same idea: Feynman developed a path integral to account for all possible paths of an electron through the universe, summed to a probability of 1. The looser definition of reality for which you implicitly advocate would IMHO make the path integral for any particle an element of reality, and not care about the actual path (or even believe that there is an actual path). Joy would say (or at least realism would say) that there is an actual path, even if we do not know it, and that only the actual path is reality. Pick you poison, then stick with it.
Final point: this coin toss representation of the “2” in CHSH grows naturally and directly and rigorously out of the sums B+B’ and B-B’ that you and Joy have been arguing about, and so truly does give us a rigorously-equivalent way of reframing that disagreement, which is just as rigorous as using an urn and slip representation of EPR-Bell-CHSH. Whether that breaks the impasse or moves it to a new place, is something you both will need to tell us.
And because EPR was an effort by Einstein and his proteges to come to grips with the probabilities that appear in quantum mechanics, it should come as no surprise that the disagreement between you and Joy — after it is made unmuddled and uncomplicated — boils down to to how to treat probabilities in relation to reality. Welcome back to the Einstein / Bohr debates about the nature of reality!
Jay
Physics (according to Einstein):
The spin components of a spin-1/2 particle along direction b and b’ are simultaneous elements of reality and can therefore independently be ascribed definite values, even before any measurement is performed on that particle. Einstein assumes “an external local reality independent of our observations”.
Arithmetics:
B(b, lambda) + B(b’, lambda) is the arithmetic sum of two independent variables which represent – for any given lambda – independently definite “what would be”- outcomes of conceivable measurements (there is no need for real measurements in order to say something about “an external local reality independent of our observations”). The term B(b, lambda) + B(b’, lambda) is nothing else than a mathematical expression formulated on base of what is called “Counterfactual Definiteness”.
In case Joy Christian i) insists on the puzzling question: “Given that B(b, lambda) and B(b’, lambda) are ‘elements of reality’, is {B(b, lambda) + B(b’, lambda)} also an ‘element of reality’?” and ii) turns his attention towards real measurements, I would insist on the puzzling answer: “In this case, {B(b, lambda) + B(b’, lambda)} is a hidden ‘element of reality’!”
They are not and never can be “simultaneous elements of reality” in an EPR-Bohm scenario. The polarizers align the spin to its angle before detection of up or down, +1 or -1.
I was a little hasty with the cross product, it looks like that needs to be one component of the cross product. Can’t add scalar to 3-vector.
But a possible point of hookup I am seeing is to (48) and (49) of Joy’s paper at https://arxiv.org/pdf/1405.2355v6.pdf.
But you can add a scalar to a bivector. It is called a quaternion:
ab = a.b + a /\ b = a.b + I.(axb).
Before stopping for tonight, let me make a few meta-observations about what I have posted here the past couple of days, in many cases creating as I was writing.
In my own physics work, such as my near-final paper at https://jayryablon.files.wordpress.com/2016/10/lorentz-force-geodesics-brief-4-2.pdf, when I am not trying to mediate RW disputes 🙂 , I have a golden rule that I am always reminding myself about, which is just this:
***We must always allow the mathematics to speak to us and steer our conceptions, rather than us try to dictate our preconceptions onto the mathematics.***
This is a variant on Dirac’s observation that his equation was much more intelligent than its author.
I now have a view on the ditch that Joy and Richard and all the rest of us on this ride have been stuck in, and it is this:
Joy wants us to call “timeout” at the terms B+B’ and B-B’ and declare that we must stop there because these are illegal in some way. In view of the approach I have now laid out which proceeds by using these terms as “coin tosses” and developing them much further *without* calling timeout, we are allowing the mathematics to speak to us rather than allowing imposing our views on the mathematics. And, funny thing, downstream the 2 sqrt(2) bound appears, and Joy’s own S^3 quaternion approach also appears to present itself.
For that reason alone (before the “funny thing”), my better sensibilities lead me after a tortuous path to agree with Richard et al. at least insofar as that we should allow the full Bell calculation to proceed to its penultimate logical conclusion, rather than truncate everything by throwing a flag down at B+B’ and B-B’. If there is something wrong with these terms, than whatever is wrong will still be there after further development, and may be clearer to see at that point. And if they are perfectly OK that should also become clearer as we proceed and the math has more to tell us. So I take the view of those who argue that we should develop the math as far as it will go without calling “illegality” unless we are really doing a facially-illegal math operation. And then, step back and ask: Dear math results, what can we learn from you?
This is especially so because the discussion here has devolved into a debate about reality and probability and counterfactual observations in the face of realism. Such debates are rife breeding ground upon which people can impose, and have imposed, their own preconceptions, which is the worst way to so science. The whole purpose of EPR, in a macro sense, was to give us a gedanken to allow the math that underlies physics to teach us about the reality of the world and the probabilistic nature of the world and how those interrelate. Bell then moved the ball by giving us a very concrete and Boolean way way to approach EPR. While we are all entertained by debates from time to time, ideology needs to have as little a role as possible in science. We should be using the downstream math that we obtain to *teach us about the nature of physical reality* and where and how and why probability and uncertainty enter the picture of reality, rather than engaging in ideological debate about elements of reality. EPR took great care to state that they were laying out a “sufficient” view of reality. They pointedly said their view was not “necessary,” and IMHO they were deliberately telling their posterity (us) to keep on the path of trying to learn about the true nature of physical reality, using their paradox as a tool.
So as to the applicability of my own golden rule here: We ought not impose a preconceived view of reality on the EPR-Bell-CHSH math. We should develop the EPR-Bell-CHSH math as far downstream as we can, and then use those results to post-facto inform our conceptions of reality. That is how we end the impasse here.
Jay
Jay, with all due respect to you (and I truly mean it, because you have devoted much of your time and effort here), I disagree with your philosophical (and dare I say diplomatic) approach above. It is self-contradictory. You are arguing against preconceptions and arguing for “just follow the math” philosophy, while you are yourself putting forward a philosophical position that does not make sense as far as our central problem is concerned, which is the following:
The gross mistake made by Bell and his followers was to unwittingly smuggle-in something mathematically illegitimate and physically impossible ( namely the quantity B + B’ ) as THE essential part of their derivation of the bounds of -2 and +2 on the CHSH correlator in the name of Einstein’s local-realism, and then they declared that the experimentally observed “violations” of those bounds is a proof of a “violation” of Einstein’s local-realism. But what they illegally smuggled-in — namely the quantity B + B’ — has nothing to do with Einstein’s local-realism which they used to define B and B’ individually. Any precocious schoolchild can see the logical fallacy here. Your mathematics simply helps to obfuscate this logical fallacy.
Quoting Jay R. Yablon: “We must always allow the mathematics to speak to us and steer our conceptions, rather than us try to dictate our preconceptions onto the mathematics.”
Physics doesn’t work this way! Quite the contrary!
You start with definitions and pre-assumptions and then you have to do the **correct** math. In case your predictions are in conflict with experimental observations, you are forced to reflect on your definitions and pre-assumptions. That’s all!
Well it looks like I hit the trifecta, because I managed to get JC to agree with LJ that they disagree with me! 🙂
I had a feeling that my statement needed more predicate, because I would never suggest that physics can be built on free-floating mathematics. So let’s do it this way.
I said “We must always allow the mathematics to speak to us and steer our conceptions.”
LJ says: ” In case your predictions are in conflict with experimental observations, you are forced to reflect on your definitions and pre-assumptions.”
Same intention, but I will go with LJ’s way of saying that. Any mathematics used to describe physics has to be rooted in a set of definitions and pre-assumptions and hopefully established knowledge. But these roots must always been seen as provisional hypotheses that we must be prepared to reexamine if a conflict arises. And if a conflict arises, we must listen to the results and change our preconceptions, rather than ignore the results to preserve our preconceptions.
Joy, because I am trying to channel you to Richard and Richard to you since that is the only good way I know to to mediate a dispute, let me say what I think Richard is saying, without making any judgments:
Realism says that the universe is there, and that certain events happen in the universe whether or not they get observed or even can in principle be observed, which is the whole basis for the realism hypothesis. (Of course, we then test this hypothesis, and as LJ says, have to reexamine if we run into conflict.)
EPR say (which you fully agreed with) that if we “insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted” we would make reality “dependent upon the process of measurement” and “no reasonable definition of reality could be expected to permit this.” And I know they have Heisenberg in mind when they say that, not New York and Miami.
What I hear Richard saying is that you are hinging the reality of B + B’ upon the fact that you can measure one but not the other simultaneously and so are making this reality “dependent upon the process of measurement” which “no reasonable definition of reality” could permit. I recall that you pointed out that we are talking about classical and not quantum simultaneity, but if that makes a difference, why?
So, continuing with how I hear Richard, might realism actually say that as long as B and B’ are both real, it does not matter that I can only observe one or the other but not both, because they are still real? Then, like we do any time we apply mathematics, I can manipulate and combine those real elements wherever the math provides a pathway to do so? Put differently, why are you caring what was observed and what was not observed cannot be, when the the realism hypothesis says that events in the universe are real and what gets observed and not observed in practice or even in principle, i.e., our “process of measurement,” does not or should not make any difference?
To be clear, I personally am finding myself more comfortable with a Bohr view which is unusual for me, because while I do lean toward the view of a reality that exists whether or not we measure it or even can measure it on a simultaneous basis, there are clearly things about reality that we are unable to measure even in principle, and these do affect what we do measure.
Put differently: either 1) there is no reality other than what we can measure which we take to be a *definition* or reality, or 2) there is a reality that we cannot measure in which case there are two different realities: reality, and the reality we can measure. Odd as it feels, I find myself leaning toward the latter.
These are all words. I ask Gill for a precise criterion — analogous to that by EPR — which dictates that B is real, and B’ is real. Then, using the same criterion, prove that B + B’ is real.
“Put differently: either 1) there is no reality other than what we can measure which we take to be a *definition* or reality, or 2) there is a reality that we cannot measure in which case there are two different realities: reality, and the reality we can measure. Odd as it feels, I find myself leaning toward the latter.”
So the moon is not there if nobody is looking at it? 🙂
Hi Fred:
I said “there is a reality that we ***cannot*** measure.” I did not say there is a reality that we *did not* measure. So the world is still safe the moon. 🙂
… So the moon is not there if nobody is looking at it?”
You’re flipping us a metaphysical Straw Moon here just like ES presented us a deliberately obfuscatory Straw Cat. Good rule of thumb from Carl Friedrich von Weizsäcker:
“What is observed certainly exists; about what is not observed we are still free to make suitable assumptions. We use that freedom to avoid paradoxes.” Also reference the excellent term of art “intersubjective agreement.”
Again I ask a simple question:
For the coin-toss reading machine analogy, I gave earlier,
Is P(H)i + P(H)j = 1.5 a violation of the P(H)i + P(H)i = 1 relation?
I haven’t seen any of you address this simple example (Jay?). This goes to the heart of the paradoxes created by reasoning incorrectly about counterfactuals.
If you can understand how P(H)i + P(H)j =/= P(H)i + P(H)i can be valid for the same experiment, without any spookiness, then you are home.
“After 50 tosses, you get 40 rings. P(H)i = 0.8. You set it to T and after 50 tosses, you get 35 rings. P(T)j = 0.7. ”
Then after 100 tosses you had 55 heads so P(H) = .55 and 45 tails so P(H) .45. But I never said you can switch the probability in the middle. Wouldn’t that be like using A, A’, B, B’ in one CHSH set of trials, then different A”,A”’,B”,B”’ in a second set of trial, giving a different premise to the whole experiment?
The issue I am driving at with the coin toss is different. Maybe I have not been clear enough:
If you accept the premise that B+B’ and B-B’ are elements of reality *as a hypothesis* (not as an article of truth), then that allows you to proceed to the next step of using coin tosses to represent these hypothesized elements of reality. Then you set up a disproof by contradiction, just like Bell did.
When you then do the math in that second-tier disproof, the CHSH bounds become 2*1=2, where 1 is the sum over all possible outcomes of coin tosses. So the consequence of the hypothesis B+B’ and B-B’ are elements of reality is that 1 as a completeness sum of coin tosses is ***also an element of reality***. That is, the consequence we deduce is that:
Probability (Heads) + Probability (Tails) = 1
is an element of reality. If I open that door, then more generally I must accept that:
“sum over probabilities = 1”
is an element of reality, and I am on the slippery slope: On this slope, a path integral that says a particle took some unknown path to get from A to B is an element of reality. And the 100% chance that a dice roll will be 1 or 2 or 3 or 4 or 5 or 6 is an element of reality. And any other like statements.
If you are prepared to generally accept that sum over probabilities = 1 is an element of reality, then you have not contradicted the hypothesis that that B+B’ and B-B’ are elements of reality. But if you do not accept that sum over probabilities = 1 is an element of reality and can convince others of that position, then you have contradicted the hypothesis that that B+B’ and B-B’ are elements of reality. And once you do that, you expose an internal logical contradiction in the CHSH derivation on based on the local realism premise, and that undermines Bell’s theorem as a no-go for local realism.
Some may have thought that my downstream calculations into coin tosses are obscuring, but I see them as clarifying. And they surely cannot be any more obscuring that the “yes they are, not they aren’t ad infinitum” discussions about B+B’ and B-B’ being elements of reality.
So, is the 1 in “sum over probabilities = 1” an element of reality? Why or why not?
“Then after 100 tosses you had 55 heads so P(H) = .55 and 45 tails so P(H) .45. But I never said you can switch the probability in the middle. Wouldn’t that be like using A, A’, B, B’ in one CHSH set of trials, then different A”,A”’,B”,B”’ in a second set of trial, giving a different premise to the whole experiment?”
Bingo! That is MF’s whole point. The premise is being changed between Bell-CHSH and how QM or the experiments are applied to it.
Jay, what MF is driving at is nicely captured by Fred in his reply above. But what you are saying is something different.
I must admit that I had missed your point previously. It is not obfuscating as I previously said but rather illuminating. To claim that B + B’ is real — or is an element of reality with certainty in the sense of EPR — is the same as saying that
Probability (Heads) + Probability (Tails)
is an element of reality with certainty for a coin toss. In other words,
Probability (Heads) + Probability (Tails) = 1.
That indeed puts the main issue in a different light, with clear-cut implications.
Let’s denote the spin component of a spin-1/2 particle along direction b by s(b) and along direction b’ by s(b’).
If one dismisses “spooky interactions”, one can prove that s(b) is an element of reality, viz. it can be ascribed a measurable value, even before any measurement is performed. Whether I find s(b) = “up” or s(b) = ”down” in the course of a measurement has thus nothing to do with the measurement itself. The spin component along direction b must be either “up” or “down” before the measurement. It’s our lack of knowledge that we cannot predict measurement outcomes, so there must some hidden, random variable “lambda” which determines whether the spin component along direction b is “up” or is “down”.
Casually formulated, Einstein’s “classical particle state” – before any measurement! – might thus formally be described as
{ s(b, lambda) = ”up” OR s(b, lambda) = ”down” }
when one is interested in the particle’s spin component along direction b.
Additionally, one can also prove that s(b’) is an element of reality, viz. it can be ascribed a measurable value, even before any measurement is performed. Einstein’s “classical particle state” – before any measurement! – might thus formally be described as
{ s(b’, lambda) = ”up” OR s(b’, lambda) = ”down” }
when one is interested in the particle’s spin component along direction b’.
Moreover, as both s(b) and s(b’) are simultaneous elements of reality, a complete theory must be able to tell us something about all theses “values” independent of what specific measurement one is choosing to perform. The “total classical state” of the spin-1/2 particle might thus formally be described as:
[ { s(b, lambda) = ”up” OR s(b, lambda) = ”down” } AND { s(b’, lambda) = ”up” OR s(b’, lambda) = ”down” } ]
Take now all combinations of being “up” or “down”, use the codings B(b, lambda) = +1 when s(b) = ”up”, B(b lambda) = -1 when s(b) = ”down”, B(b’ lambda) = +1 when s(b’) = ”up” and B(b’, lambda) = -1 when s(b’) = ”down” and you will get for any given lambda:
B(b’ lambda) + B(b’, lambda) is either equal to 2, to 0, to 0 or to -2.
There is nothing mathematically illegitimate and physically impossible. It’s merely a formal description of “an external reality which is absolutely independent of whether we perform measurements or not”.
Correction for the last sections:
Take now all combinations of being “up” or “down”, use the codings B(b, lambda) = +1 when s(b) = ”up”, B(b, lambda) = -1 when s(b) = ”down”, B(b’, lambda) = +1 when s(b’) = ”up” and B(b’, lambda) = -1 when s(b’) = ”down” and you will get for any given lambda:
B(b, lambda) + B(b’, lambda) is either equal to 2, to 0, to 0 or to -2.
There is nothing mathematically illegitimate and physically impossible. It’s merely a formal description of “an external reality which is absolutely independent of whether we perform measurements or not”.
Jay, “sum over probabilities=1” is valid only within the same experiment. I thought I just gave you a ptetty solid counter-example.
What might be real to Spinoza might no be the same as what is real for an experimeter. There is a risk of missing the point altogether by considering “elements of reality” separately from the specific experimental situation.
(1) ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ ≤ 2
(2) ⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ ≤ 2√2
(a) P(H)₁ + P(T)₁ = 1
(b) P(H)₁ + P(T)₂ = 1.5
The relationship between (1) and (2), is not unlike the relationship between (a) and (b). All those statements are valid for the relevant experiments. But mix and match them and you can generate paradoxes at will.
In other words, if Bell’s realism assumption is equivalent to the simultaneous existence of A,A’,B,B’, how can the rejection of that assumption be based on experiments which deliberately do not measure them simultaneously? Isn’t it a bit curious to you that the main feature required to derive the CHSH — ie the simultaneous existence of B,B’ is the biggest limitations of the experiment?
Is the CHSH Spinoza’s inequality, or the experimentalists? Give onto Spinoza what belongs to Spinoza,and to the experimentalists what belongs to them. The biggest problem is when you derive an inequality from Spinoza’s POV, and try to test it from the POV of experimentalists who do not know anything about the unmeasurable.
I wonder if Christian’s hidden variable lambda is an element of reality. If not, then (according to his own reasoning) his hidden variable models are mere mathematical games, with no relevance to physics. If on the other hand yes, then B(b, lambda) + B(b’, lambda) is just one particular numerical function of lambda, whose value is equally real as lambda itself.
In the appendix of the paper linked below I have derive the Bell inequality,
-2 ≤ E(a, b) + E(a, b’) + E(a’, b) – E(a’, b’) ≤ +2 ,
only using the assumption of the realities of B + B’ and B – B’, without assuming locality. Since experiments “violate” this inequality, they prove that B + B’ and B – B’ are not real:
http://libertesphilosophica.info/blog/wp-content/uploads/2016/11/Fatal.pdf .
You didn’t answer the question, whether or not the hidden variable lambda in your own hidden variable models is an element of reality. But your answer suggests that lambda is not real, and hence your own hidden variable models are mere mathematical games, with no relevance to physics.
I will answer your question when you answer mine, which was posed yesterday:
According to your interpretation then what setting Bob could have used to see the measurement outcome B(b, k) + B(b’, k) ?
Here is a simple, classical method for producing so-called “quantum correlations”: http://vixra.org/pdf/1609.0129v1.pdf
So what is the bad assumption in Bell’s theorem, that has enabled the false claim that no classical system can behave this way? It is the assumption that there must be a one-to-one correspondence between multiple measurements made on a particle, and multiple attributes of the particle. Single bits of information do not have multiple attributes or components.
LJ, I agree with everything you say in the above. So let me take what you said a few steps further, and see what you think.
Everybody will agree that B(b, lambda) and B(b’, lambda) are each elements of reality (EORs). Joy has argued, so far without winning any converts, that B(b, lambda) + B(b’, lambda) and B(b, lambda) – B(b’, lambda), however, are NOT elements of reality. In other words, he argues that we cannot assume without giving the matter some reflection, that:
EOR1 + EOR2 = EOR3, necessarily. (1)
Or, put plainly, he maintains that elements of reality are not necessarily additive and so is making a statement about the additive properties of EORs. Some here agree, most disagree, and all are dug in. So the question I have asked is whether there is some objective quantitative way to judge who is right, which is more precise than what we have had on the table so far.
Bell itself is a disproof by contradiction. So now, let’s set up a second disproof by contradiction within the first disproof by contradiction. I will start by assuming, as a hypothesis to be disproved or not disproved, that B(b, lambda) + B(b’, lambda) and B(b, lambda) – B(b’, lambda) ARE elements of reality. If I may use the notation X ( Y to mean “X is an element of the set Y,” so that “X ( reality” means “X is an element of reality,” what I am postulating is that:
B(b, lambda) + B(b’, lambda) (reality (2)
B(b, lambda) – B(b’, lambda) (reality
There is a good case to be made that these postulates are true, and certainly everybody except Joy et al. believe them to be true. For example, suppose we measure along b not b’. Since we all agree that B(b’, lambda) is nonetheless an EOR, and since we all agree that its reality should be unaffected by that fact that we did not measure it just like the moon is still real even if the night is cloudy, (2) seems to make perfect sense.
But here, I am not judging the truth of (2), I am making its truth a postulate to be disproved or not. And, LJ, when you say as to the former in (2) that it is “equal to 2, to 0, to 0 or to -2,” we all know that this 2 and -2 end up being the CHSH bounds and so are very important numbers.
Because B(b, lambda) and B(b’, lambda) are EORs with universal agreement, let’s take the special case where we actually measure along b, and we detect that B(b, lambda) = +1. Now (2) becomes:
1 + B(b’, lambda) (reality (3)
1 – B(b’, lambda) (reality
We have not and now can no longer measure along b’, but that does not deter us, because B(b’, lambda) is still an EOR, and we know that B(b’, lambda)=+1 with some probability P, and B(b’, lambda)=-1 with some related probability 1-P. We can say that P=.5 which makes this a fair coin toss, or we can sat that P is something other than .5 which makes this a biased coin toss. But to keep life easy, let’s just say for now that P=.5 and 1-P=5.
Now if both of (3) are EORs, then we can certainly divide through by a factors of 2 and also say that:
.5*(1 + B(b’, lambda)) (reality (4)
.5*(1 – B(b’, lambda)) (reality
That is, if some quantity Q is an EOR, than half that quantity .5Q should also be an EOR.
Now let’s look at the contingencies for B(b’, lambda) which are 50% to be +1 and 50% to be -1. Because B(b’, lambda) is common to both equations in (4), their values are linked. So in the 50% chance that B(b’, lambda) = +1 we will have:
50%: .5*(1 + B(b’, lambda)) = +1 and .5*(1 – B(b’, lambda)) = 0 (5a)
And in the 50% chance that B(b’, lambda) = -1 we will have:
50%: .5*(1 + B(b’, lambda)) = 0 and .5*(1 – B(b’, lambda)) = +1 (5b)
The important point in (5a) and in (5b) is that these are mutually-orthogonal states and analogize perfectly to a given coin toss landing on either heads or tails but not both. And, we can represent these with the two-dimensional state vectors (1 0) in (5a) and (0 1) in (5b) and refer to (5a) as the “heads” state and (5b) as the “tails” state. And we can use (5a) and (5b) as “counters” for heads and tails so that if we toss the coin multiple times, we can keep track of how many heads and how many tails results we obtain. And all of this, we continue to hypothesize, describes elements of reality.
So now, let me define a “heads counter” H (b, b’, lambda) and a “tails counter” T (b, b’, lambda) where I include the arguments (b, b’, lambda) as a reminder of the variables of which these are a function, as such:
H == .5*(1 + B(b’, lambda)) (6)
T == .5*(1 – B(b’, lambda))
In view of the definitions (6), we may rewrite (5a) and (5b) respectively as:
50%: H = +1 and T = 0 (7a)
50%: H = 0 and T = +1 (7b)
Although we cannot know with certain what outcome we will obtain for any given coin toss, we do know with certainty that:
H + T = 1 (8)
which is the completeness relation for coin tosses. This says that the probability of landing on heads or on tails is equal to 1. We just do not know which one of heads or tails we will get until we toss the coin. It is basic to probability theory, whether for discrete random variables or continuous random variables, to have all of the *possible* probabilistic results sum to 1.
Now, of course, the CHSH limits are calculated from a combined sum which we also postulate to be an EOR, namely:
(A(a, lambda)*[(B(b, lambda) + B(b’, lambda))] + (A(a’, lambda)*[B(b, lambda) – B(b’, lambda)] (reality (9)
Everyone knows how to get this from the correlator skeleton AB+AB’+A’B-A’B’, so I will not spend any time on that.
Because our momentary interest to to derive the upper CHSH bound +2 knowing we can do the same type of calculation for the lower bound -2, and because Alice’s measurements are also EORs, let us suppose that (A(a, lambda)=+1 is measured by Alice along a. And, although she cannot then measure along a’, because (A(a’, lambda) is also an EOR, let us regard this to also be (A(a’, lambda)=+1. So with these values for Alice’s elements of reality (9) becomes:
[(B(b, lambda) + B(b’, lambda))] + [B(b, lambda) – B(b’, lambda)] (reality (10)
But back at (3) we also set B(b, lambda) = +1, so (10) now simplifies to:
[1 + B(b’, lambda))] + [1 – B(b’, lambda)] (reality (11)
Now, all we have left are the heads and tails counters in (6). So if we place (6) into (11) we obtain:
2H +2T = 2(H+T) = 2*1 = 2 (reality (12)
The most important number in (12) is the number 1 in 2*1 because this 1=H+T comes from the completeness relation for coin tosses. The resultant 2 is of course the upper CHSH bound, which we have postulated to be an EOR. But for this all-important number to 2 to truly be an EOR, (12) requires that that:
1=H+T (reality. (13)
In other words, the completeness relation number 1=H+T MUST be an element of reality if the upper bound of 2 in the CHSH correlator is to be an element of reality. And while we have used a .5 probability for each of H and T in (13), it should be clear that now we can relax that and assign any probability P to H and the remaining 1-P to T which is what I have called a “biased coin” for P ne .5.
SO: Our having postulated back in (2) that B(b, lambda) + B(b’, lambda) and B(b, lambda) – B(b’, lambda) ARE elements of reality, has led us to the deductive consequence that the number 1 in the completeness sum 1=H+T is an element of reality. And this is where we look for contradiction as part of our disproof by contradiction. Recall again, this is a second disproof by contradiction set up within the first disproof by contradiction that is Bell’s Theorem.
If in fact one can be convinced that that 1=H+T is an element of reality, then our postulate at (2) is not disproved. However, if one can be convinced that 1=H+T is NOT an element of reality, then that conviction will operate to disprove the postulate at (2) that B(b, lambda) + B(b’, lambda) and B(b, lambda) – B(b’, lambda) are elements of reality.
In other words, were we to become convinced that 1=H+T is NOT an element of reality, this would PROVE by contradiction that B(b, lambda) + B(b’, lambda) and B(b, lambda) – B(b’, lambda) are also NOT elements of reality, as Joy has been asserting.
To be clear: 1=H+T is certainly a true statement. It is logically and mathematically unassailable. And it does describe a certain “reality” that a coin must always land on heads or on tails but not on both, which is completeness relation that is universally central to discrete and continuous probability theory way beyond just a binary-valued coin toss. The question is whether this number 1=H+T ALSO is an ***element of reality*** in the EPR-Bell-Bohm-CHSH sense.
So the purpose of the coin toss line of development I have pursued here for several days, is to shift the focus of the debate to the following question which I ask EVERYONE to carefully consider:
Is the completeness sum below, showing all of the ordinary and hidden variables of which it may be a function to not hide anything, an EPR et al. element of reality, or is it not?
1 = H (a,a’,b,b’,lambda) + T (a,a’,b,b’,lambda) (14)
I have spent hours upon hours the past six weeks trying to pinpoint the disagreement between Joy et al, and Richard at al. over Bell’s Theorem. It is now pinpointed at the question whether (14), which adds a heads counter to a tails counter given that either heads or tails but not both will emerge from any given coin toss, is or is not an element of reality.
We will have made substantial progress in this RW discussion if we can now argue about (14) being an element of reality or not, than about B(b, lambda) + B(b’, lambda) and B(b, lambda) – B(b’, lambda) being elements of reality or not, which latter argument has gone nowhere after months if not years.
Jay
That is not how a real EPR-Bohm scenario experiment works. There is no specific “spin component” along directions b or b’ or a or a’ before the particle hits the polarizer so you can’t assign anything to those “spin components”. As I said, the best that any theory can do is predict 50-50, +1 or -1 outcomes at A and B. Therefore EPR-Bohm can’t test for realism. But if a HV model can predict the 50-50 in a realistic way, then I believe we can claim that it is realistic.
I stand corrected by MF. An EPR-Bohm experiment can test for realism when the settings of the polarizers are a = b and/or a = -b.
I remember the original question: “Given that B(b) and B(b’ ) are elements of reality as we all seem to agree, is B(b) + B(b’ ) also an element of reality?” What has changed in the meantime?
That original question was mine.
What has happened is that Richard has said yes, B(b) + B(b’) also an element of reality, and Joy has followed up by asking Richard for a reason, specifically:
In the mean time Richard has asked Joy if his:
Both questions are good questions and IMHO should be answered.
Jay
Let’s go back to the original puzzling question: “Given that B(b) and B(b’) are elements of reality as we all seem to agree, is B(b) + B(b’) also an element of reality?”
I would give the following puzzling answer (to point out that the whole discussion is trapped in a cat-and-mouse game):
One has to distinguish whether one parenthesizes the sum or not. Without brackets, viz. considering B(b) + B(b’) in a non-associated manner, we would have two “unconcealed elements of reality”. With brackets, viz. considering { B(b) and B(b’) } in an associating manner, we would have one “hidden element of reality”. “Hidden”, because we have no working operational procedure to measure this element. The same applies here as to “hidden variables”. It is assumed that they exist but no one can up to now design an operational procedure how to measure them.
Concerning this matter, Richard Gill is right. In case you insist that it is mathematically illegitimate and physically impossible to use “hidden elements of reality” within Bell-type reasoning, I would insist that it is mathematically illegitimate and physically impossible to use “hidden variables” in any physical reasoning. No hidden variable model like that presented in “Local causality in a Friedmann–Robertson–Walker spacetime” would make any sense in case you are not able to design an operational procedure of how to measure the hidden variable.
All this above might sound rather ridiculous, but my arguing round and round the subject is merely an attempt to point out: Quantum mechanics is a complete theory. It provides us with a precise mathematical formalism to make precise probabilistic predictions for **what we think are** “outcomes of measurements (observation) on something.”
Forget EPR-Bohm scenarios; forget elements of reality, hidden variables and all the related stuff. Forget the “everyday understanding” of reality, locality and causality. As long as one insists on thinking about quantum phenomena within the framework of classical ideas and concepts, one will always blunder into blind alleys (see the ongoing discussion here on RW and the myriads of discussions since decades).
One can adopt the “shut up and calculate” philosophy: Do your job and don’t worry about philosophical questions, just get the predictions.
But one can also take notice what “Quantum mechanics“ is trying to question us: Do you really think that the universe is – in the classical sense – “made of” something, that there is really some “underlying stuff.”
Try to use classical “ingredients” – as Einstein has tried – in order to “complete” Quantum Mechanics and you will always fail. That’s all, accept it as it is.
A hidden variable (whether mine or anyone else’s) is not an element of reality itself in the sense defined by EPR. A hidden variable defines an uncontrollable initial state of the system. It is not a measurement outcome, and should not be confused with one, even if the two happen to coincide numerically. A hidden variable denoted by lambda in Bell’s local model (as well as in mine) defines the “complete state” of a singlet pair of particles, specifying all of the elements of physical reality [such as A(a), A(a’), B(b), B(b’), etc.] of the pair at the moment of its creation at the source. Whereas — in the EPR sense — the elements of reality A(a), A(a’), B(b), B(b’) themselves are observed at space-like separated detectors of Alice and Bob.
You should read between the lines in order to understand what I mean.
In order to prevent any misunderstanding, it will be requisite, in the first place, to recapitulate, as clearly as possible, what our opinion is with respect to the fundamental nature of our sensuous cognition in general. We have intended, then, to say that all our intuition is nothing but the representation of phenomena; that the things which we intuite, are not in themselves the same as our representations of them in intuition, nor are their relations in themselves so constituted as they appear to us; and that if we take away the subject, or even only the subjective constitution of our senses in general, then not only the nature and relations of objects in space and time, but even space and time themselves disappear; and that these, as phenomena, cannot exist in themselves, but only in us. What may be the nature of objects considered as things in themselves and without reference to the receptivity of our sensibility is quite unknown to us.
— Kant, Critique of Pure Reason
EPR do not *define* “element of reality”. They merely give a sufficient condition. What they mean by “element of reality” is clearly a property of a physical system which exists independently of whether or not the system is observed or measured. And therefore, if you can argue that B(b) and B(b’) are elements of reality, then the pair (B(b), B(b’)) is also an element of reality, and any function thereof, such as the sum, as well. The system possesses a definite value of B(b) + B(b’) independent of whether (or how) it is observed.
MF:
I understand your example. I will let Richard and HR speak for themselves, and I am not agreeing or disagreeing, but I believe that they would try to overcome your counter-example by appeal to statistics and the law of large numbers.
Jay
But in the actual state-of-the-art experiment they were quite happy to use only 256 events!
I think you should give it your best using law of large numbers to see if you can avoid the counter example I just gave of the coin-tossing machine.
P(H)₁ + P(T)₂ = 1.5
No matter how many tosses you use. Statistics won’t help. I will like to see someone try to explain it away using statistics.
One other thing. You have to be careful about coin toss examples because you are using inferences rather than measurement results in your examples. The fact that you can only one outcome H or T can be obtained is not a good proxy for the EPR experiment in which you can’t measure more than once because you have simply used the simplicity of the coin toss example to infer that the other result was T.
That is why I ask that you think carefully about my example. Your argument does not survive it.
We may be talking about two different things. My primary concern at the moment is using the coin toss example to set up a disproof by contradiction regarding sums of elements of reality. Please take a very thorough look at my lengthy posting from earlier this morning. Jay
According to the above reasoning, since “square-root of B(b)” is a function of B(b), “square-root of B(b)” is an element of reality. But since B(b) = +1 or -1, both “square-root of +1” and “square-root of -1” are elements of reality, where “square-root of -1” = the imaginary i.
An element of reality has a name, and in any particular instance it takes on a particular value. The set of possible values can be anything we like. So for instance, provided we define “square-root” as a single valued function from the reals to the complex numbers, if X is a real-valued element of reality, then “square root of X” is the name of a well-defined complex-valued element of reality.
I completely agree with Richard. This means that we all have something of a “blank check” to discuss and better formulate what it means for some quantity to be an element of reality, or not. Which is to say, we have the freedom to postulate specific examples of quantities X, Y, Z that may or may not be observed or be observable in the physical world, and pose the question “should our definition of elements of reality define X as an element of reality? Or Y? Or Z?”
The above is where I believe the “blank check” discussion should start about what is and what is not an element of reality. And while I have heretofore tried to be a diplomat, I am now going to become a squeaky wheel until the question I raise above is discussed. Keep in mind that if a coin is tossed and lands on heads, then in the notation above this means that H=1 and T=0. Conversely, if the coin is tossed and lands on tails, then H=0 and T=1. So H and T are result counters.
Let me now break my earlier question into two parts:
Part A:
I have already tossed a coin and know the result with certainty. Should the way in which we define elements of reality define
1=H+T
as an element of reality? Here, 1 is the number of tosses I have made. I am asking for opinions. And reasons if you would be so kind as to provide them.
Part B:
I am about to toss a coin and do not yet know for certain what the result will be. But I can predict, with certainty, in advance, that the probability Pr(H) of heads plus Pr(T) of tails will sum to 1, i.e., I know for certain that the coin will land on one side or the other but not both. So, should the way in which we define elements of reality define:
1=Pr(H)+Pr(T)
as an element of reality? Again, I am asking for opinions, and reasons.
Let’s start here. There is probably no simpler probability problem than that of a simple, single coin toss. If we rocket scientists cannot even agree on how to discuss a coin toss, then there is a problem. I would like to hear opinions as to these two very simple questions, Part A and Part B, from everyone. And I will keep squeaking until I do. Joy? Richard? HR? Fred? LJ? MF? Anyone else?
Jay
I posted this last night, but it was as a reply to a post a few days ago so may have been missed. I am still very interested in this. Thanks, Jay
Tsirelson’s bound
https://en.wikipedia.org/wiki/Tsirelson%27s_bound
Wikipedia claims that to be a quantum mechanical derivation, but there is hardly any quantum mechanics in it. It is all Clifford algebra, just as on the last page of this paper:
https://arxiv.org/abs/quant-ph/0703179
An actual quantum mechanical derivation is in fact quite simple (again see the above paper):
-2√2 ≤ -a.b – a.b’ – a’.b + a’.b’ ≤ +2√2 .
Put the expressions for the quantum correlations into the CHSH formula, then solve it as a maximization problem (i.e., you now have a function of four angles, set the four partial derivatives to zero, and solve the resulting system to find the angles that maximize the value of CHSH).
On further reflection, if we are to completely enumerate all the variables that go into these H and T counters, then we should also include the spin s=s_A=s_B (using an s_A=s_B sign convention for the spins of the doublet). Thus, the above should be:
1 = H (s,a,a’,b,b’,lambda) + T (s,a,a’,b,b’,lambda)
This is important, because s is an ordinary variable dealt strictly by nature and we cannot control that. That is most akin to the coin flying and twisting after it is tossed but before it lands on the surface. We can control the a, a’, b, b’ variable which are experimental settings, which is akin to tossing the coin from 3 feet or 4 feet above the surface. And lambda is a hidden variable, which seems to have no direct analogue in an ordinary coin toss. But no matter what goes into H and T, as outcome counters these are indistinguishable from the counts we can make for the toss of a coin.
Jay, as you probably know, I would not use your (14) as a criterion for the elements of reality. So my answers to your questions in Part A and Part B are both in the negative, with more negative for the question in Part B for obvious reasons.
Gill’s “criterion” for the elements of reality as any old functions of B and B’ is also manifestly false, because it admits imaginary numbers like i (for example) as elements of reality. For me, that leaves only the EPR criterion as the only viable criterion for a physically meaningful elements of reality. I am not sure whether this is the answer you were looking for.
OK, Joy, so please articulate with some elaboration: WHY is your answer negative for Part A? And WHY is it even more negative for Part B? And please stay away from imaginary numbers and B and B’. And please let Richard speak for himself. I just want to stay focused on the “reality elements” of a single coin toss: before we toss it, after we toss it and while the coin is still in the air, and after the coin lands and we have a result. Thanks, Jay
An “element of reality” has a name, and it takes on values in some set of possible values. This set might for instance be the set of real numbers, or the set of real vectors of some given dimension. It’s called an element of reality, of a certain physical system, by virtue of it taking a definite value, independently of any measurement or observation of the system. Example: position (in certain units) or momentum (in certain units.
So the physical system might have the property of “momentum = …”
“It’s called an element of reality, of a certain physical system, by virtue of it taking a definite value, independently of any measurement or observation of the system.”
OK, Richard, I like this statement, so let’s start here. But instead of a coin toss let’s make the example a little bit more rich than just a binary result:
I have a six-sided die. The definite values on each side are 1, 2, 3, 4, 5, 6. I plan to throw the die in a few seconds. Or, the die is now flying in the air or in a vacuum, but not yet landed. Or, the die has just landed, and 5 is the result facing up.
A) Ought we define the definite value 5 which results, to be an element of reality? If so, at what temporal event does it become an element of reality? Before I throw? While the die is in the air? If the air is removed so there is a vacuum and no possibility of disturbing the inevitable result once the die is flying? After the landing?
B) Ought we define the values 1, 2, 3, 4, 6 which did not occur, as elements of reality? And what ought to be the temporal sequence, if any, for these to be eliminated as elements of reality, if they are to be eliminated?
C) Before I throw, during, and after, it is definitely known that the probability sum Pr(1)+Pr(2)+Pr(3)+Pr(4)+Pr(5)+Pr(6)=1. Ought we define that number 1 for the complete sum of probabilities, which completeness number 1 is definitely known even though we do not know if it comes from the result 1, 2, 3, 4, 5, or 6, to be an element of reality? If so, again, what is the temporal sequence?
Thanks,
Jay
That is correct and applied to the EPR-Bohm situation, A, B, A’ and B’ cannot be elements of reality until they are measured. The best any theory can do is predict that their averages will be zero after many trials.
The above is the least of the problem with Gill’s “criterion” of reality. According to Gill’s criterion, any old function of A, A’, B, B’, etc. is an element of reality, provided A, A’, B, B’, etc. are elements of reality. In addition to the “imaginary problem” exposed above, to see how absurd Gill’s criterion is, recall that A(a) and B(b) are continuous functions of the 3D vectors like a and b. Thus there are in fact infinitely many A(a)’s and B(b)’s to consider, all which can be added up or subtracted out in infinitely many different ways, the results of which can again be added up and subtracted out in infinitely many different ways. And of course adding and subtracting are not the only mathematical operations we can perform on A’s and B’s. We can indeed construct infinitely many other functions out of them. For example, like { A + A’ + A” + sqrt(B + B’) } / { bunch of other A’s and B’s added up or subtracted out}, and so on. In other words, according to Gill’s criterion of reality there is absolutely no end to the number of different elements of reality we can construct out of the genuine EPR elements of reality like A(a) and B(b), without exhausting any conceivable order of Cantor’s transfinite numbers of infinities. Well, you got the picture. According to Gill’s criterion of reality, absolutely anything at all can be an element of reality, from our universe to infinitely many other universes.
I would like to comment on this train of thoughts, but it would need lengthy writing. However, there is an interesting essay by the philosopher Harry G. Frankfurt. I have forgotten the complete title but part of it reads „On …….“. This essay provides an excellent answer to this and your former comments. Maybe, I will remember the full title again.
Thank you, LJ.
Harry G. Frankfurt’s essay applies perfectly to both Bell’s theorem and Gill’s criterion.
Jay,
the disagreement is between Joy Christian et. al and the overwhelming majority of experimental and theoretical physicists.
Always repeating the same questions by means of a never-ending shift of aspects and definitions brings nothing forward. Richard Gill has explained Bell’s theorem; in case you are unsatisfied with this explanation, you find a lot of alternative explanations in literature.
You should simply think about: “Why has Joy Christian’s paper “Local causality in a Friedmann–Robertson–Walker spacetime” been retracted and why do all his attempts to disprove Bell’s theorem find nearly no attention within the scientific community?”
That are important questions.
Lord Jestocost
Lord Jestocost,
I have recognized repeatedly that Joy holds a view that a large majority of physicists do not agree with. Retraction Watch has brought that to a head, and there are two questions that puts on the table: 1. Why was the paper retracted? 2. Was the retraction justified? Clearly, it was retracted because it is believed to violate Bell’s Theorem, which is regarded in very many circles as inviolate. So now we are discussing the second question, and that necessarily brings us to Bell.
I want to be clear that ***I*** want to know what views people hold about the “elements of reality” associated with such things a coin tosses and dice rolls, because ***I*** have deduced independently of anybody else, ***via a formal logical disproof by contradiction***, that if the CHSH bound=2 is an element of reality so too is the sum 1=Pr(H)+Pr(T) in a coin toss and so too is the probability Pr(1)+Pr(2)+Pr(3)+Pr(4)+Pr(5)+Pr(6)=1 in a die roll. Contrapositively, if 1=Pr(H)+Pr(T) in a coin toss and Pr(1)+Pr(2)+Pr(3)+Pr(4)+Pr(5)+Pr(6)=1 are NOT elements of reality then the CHSH bound=2 is also NOT an element of reality.
These are the possible logical and mathematical and scientific answers to my questions:
1. You can argue is it is irrelevant to Bell’s Theorem whether the 2 in the CHSH sum is an element of reality, and explain why it is irrelevant.
2. You can argue that my line of deduction stated above from CHSH sum=2 to 1=Pr(H)+Pr(T) in a coin toss to Pr(1)+Pr(2)+Pr(3)+Pr(4)+Pr(5)+Pr(6)=1 is incorrect, and explain why it is incorrect.
3. You can argue that 1=Pr(H)+Pr(T) and Pr(1)+Pr(2)+Pr(3)+Pr(4)+Pr(5)+Pr(6)=1 ARE elements of reality thereby allowing the CHSH bound=2 to also be an element of reality, and explain why it is appropriate to include these in how we define relents of reality.
Any other answer is evasion.
You have instead given the political answer, which is a subset of evasion: nobody listens to Joy, go read why nobody listens to him and most people shun him, and stop asking the same old questions in different ways. And: everybody has accepted Bell, so just move on, there is nothing more to see here.
Until 6 weeks ago, I had no interest whatsoever on getting involved in any way in the Bell debates between Joy and most everyone else. I have my hands full enough with other things, and in no way need the grief that has historically been attached to these debates. You will see if you check it out that I had NEVER participated in any prior Bell debates in any other forum nor ever written one word about Bell, until these discussions at RW. I have only learned the details of Bell and all these arguments — and am still learning those from the viewpoint of a student who asks a lot of questions and does not accept anything at face value until I am comfortable with it — thanks to the kindness of the participants here who have indulged my learning curve.
But now that I did agree to “mediate” between Joy and his small circle of followers and Richard and the great mass of opinion, with both Joy and Richard agreeing to my conducting this mediation, I intend to see that through and anybody who knows me knows that once I make a commitment to something I do not quit until I reach a satisfactory conclusion. And I do not accept “everybody says so” unless and until I am comfortable that everybody is right. For scientists above any other group of people, that it a sacred duty.
You imply, or perhaps just assert, that I am “repeating the same questions by means of a never-ending shift of aspects.” However, I have not seen the precise questions I am asking here, asked by Joy or anybody else. Specifically, I not seen the precise, formal, disproof by contradiction I have laid out, posed before by anyone else. Some of the questions Joy has asked here, I have disagreed with his asking. And until 2-3 days ago, ***Joy himself did not understand what I was driving at with my coin toss analogy*** and thought I was on a wrong track. And based on your reply, it seems to me that you do not understand what I am driving at either. So please do not tell me that I am just repeating Joy’s or his supporters’ earlier questions. I am not. These are my own questions, interdependently arrived at.
Show me somewhere these precise questions I am now asking have been asked before ***in the structured logical and mathematical setting of a formal disproof by contradiction*** (which, by the way, is exactly what Bell’s Theorem is in relation to local realism), and where they have been answered in that setting, and I will recant. Otherwise, I will keep asking these questions and pursue this disproof by contradiction until I am comfortable that the answers are scientifically and mathematically and logically sound.
Jay
I can also argue that your coin toss example is not an appropriate analogy of the EPR experiment such that you are on shaky group drawing conclusions from coin tosses about what is true or not true about the EPR experiment. Will that too be evasion?
No, Mike, that would not be evasion, because your argument would be a logical, scientific, mathematical argument if you can make and sustain it, and it would be responsive to my point 2 because it would disconnecting my use of the coin toss from the CHSH sum and / or EPR.
But let me make one other point that you may be missing about my disproof by contradiction, and perhaps that is my fault for not elaborating this clearly enough.
I am making the hypothesis that the bound |2| on the CHSH sum is an “element of reality.” But I am also making the contemporaneous *hypothesis* “Hy” that the CHSH inequality bounded by |2| and the “slip and urn” model with the indexes being the same for all four terms is a true and correct model of EPR-Bell. Unless if have read you incorrectly, you believe that this Hy is a false hypothesis. So the way to disprove a false hypothesis is to assume it is true, Hy=true, derive its consequences C=true, thus deducing:
Hy=true implies C=true.
Then, if you can show that C=false, the logical contrapositive PROVES RIGOROUSLY that:
C=false implies Hy=false.
So I have slavishly started with what all the proponents of Bell (a very large majority of the physics community) regard as true, and taken it on its own terms and at face value to be completely true, but only as as a hypothesis. I believe I can then prove (and did do so here yesterday morning) that if we work from this hypothesis, it can be rigorously proved that the quantity 1=Pr(H)+Pr(T) for a result that absolutely mimics a coin toss with precision, must be an element of reality.
So if this “1” which is the completeness sum for a discrete, two-valued probability distribution is NOT an element of reality, then the starting hypothesis that the bound |2| on the CHSH sum is an element of reality and that slips and urns and all indexes being the same correctly represents Bell and EPR is true — is actually a false hypothesis. Which as far as I can tell, is something you actually agree with.
Long story short: if you think Bell is wrong, assume Bell is completely right, then find some logically-inevitable consequences of that, and show those consequences to be unsustainable and manifestly false, so that everyone will agree with you. Don’t just keep insisting that Bell is wrong by using analogies and examples and making fervent arguments. Look how far that approach has gotten over the past 50 years. That is Einstein’s definition of Insanity: doing the same thing over and over again and expecting different results. Instead, construct a direct, formal, logically-rigorous disproof by contradiction. Or analyze mine and show me if there are holes in it that cannot be fixed.
For all that I pushed back at LJ earlier today, he was right about one thing: Bell is thoroughly ingrained and fortified. You and Joy and others who are not believers will NEVER convince anybody else of this view by any other route than taking what the very vast majority of physicists believe to be true, and rigorously deriving consequences of those beliefs which they will recognize clear as day to be false. Disprove Bell by manifest logical contradiction. Otherwise, forget about it, and go back to insanity.
Jay
I completely agree with everything to just said.
If the Quantum Supremacy project now underway pans out then entanglement will become incontrovertible and Dr Christian’s arguments against it are mooted (disproving entanglement being the point of the whole S^3 exercise). Or do Joy Christian, Fred Diether, Tom Ray and Michel Frodje dispute that it would be a valid empirical test?
Excellent objective account:
https://rjlipton.wordpress.com/2016/04/22/quantum-supremacy-and-complexity/
That is a very big IF. The only unambiguous empirical test of my S^3 model for the EPR-Bohm correlation I am aware of is the following experimental proposal published in IJTP:
http://link.springer.com/article/10.1007%2Fs10773-014-2412-2 .
Ah, had you but a fraction of the opposition’s funding …
https://arxiv.org/pdf/1608.00263v2.pdf
Anyway, my question remains: would Quantum Supremacy, if successfully demonstrated, comprise a valid empirical test, in your opinion?
The answer is, No. Even if it is successful, it would only be further evidence of quantum correlations, not quantum entanglement. No one can ever directly observe quantum entanglement. Hence the significance of the indirect argument by Bell and disproofs of Bell.
That is because more people believe the opposition. If Joy were to manage to convince people to believe him, then the funding would follow the belief. 🙂
Wouldn’t the interpretation of any new experiment as a demonstration of entanglement rather than of something else, still rest upon Bell’s theorem’s exclusion of local realism?
Also I recall discussions on SPF largely involving Joy Christian, Fred Diether and Tom Ray (the latter once with reference to a correspondence from Karl Hess) in which it was predicted that quantum computation was impossible because entanglement is a chimera (my word). They believed D-Wave’s entanglement-free annealing approach was the only way to go to achieve speeded up computation if indeed there were any way to go. But then they also predicted that if a loophole-free Bell test were ever conducted there’d be no measured violation. What to say.
But to Jay’s point. If (for reasons of metaphysical preference or revulsion) you believe we live in an S^3 universe then you can explain (if not prove) anything by that rubric. ‘t Hooft has superdeterminism of course but also a Nobel.
It’s a bit more complicated than that. The problem with Joy’s experiment is that the way he sets up the experiment, the results he predicts are mathematically impossible (completely independent of any imaginable physical theory). So he would not just revolutionize physics, but mathematics and logic as well.
My proposed experiment is not meant to revolutionize physics, let alone mathematics and logic. It is meant to simply verify the well known spinorial properties of the physical space.
The results — namely, E(a, b) = -a.b — I predict are neither mathematically nor physically impossible. If that was the case, then my proposal would not have been published by the International Journal of Theoretical Physics, or have been retracted by now. In fact, my predicted results are already vindicated, both theoretically and by several independently built numerical simulations of the proposed experiment. See, for example, the details here:
http://libertesphilosophica.info/blog/experimental-metaphysics/
Actually, it is much simpler than that. Seventy years ago, Claude Shannon derived his expression for Information Capacity. At the limit of a single bit of information, this Capacity reduces, exactly, to the Heisenberg Uncertainty Principle. Consequently, the reason two variables linked by the uncertainty principle cannot be simultaneously measured, is precisely because they DO NOT EXIST, in the limit at which only a single bit exists, within the entity being measured. Hence, all additional measurements MUST BE CORRELATED with the first, because they are all merely measuring the same bit, just in different ways.
Unfortunately for Bell’s Proof, Bell, along with everyone else, unwittingly assumed that more than one independent bit can be measured in such circumstances. This assumption has been known for decades, and was specifically noted by d’Espagnat in 1979, on the bottom of page 166, of this article in Scientific American (“These conclusions require a subtle but important extension of the meaning assigned to a notation…”):
https://www.scientificamerican.com/media/pdf/197911_0158.pdf
So how do the single-bit correlations actually behave, if one actually goes to the trouble of constructing macroscopic images, that only possess a single bit of information, and then measure their correlations? They behave exactly like the so-called “Quantum Correlations”:
http://vixra.org/pdf/1609.0129v1.pdf
Thus, these correlations have nothing to do with “spooky action at a distance”. They have nothing to do with quantum physics. They have nothing to do with classical physics. It is pure math: the mathematics of Shannon’s Information Capacity, reducing to the Heisenberg Uncertainty Principle, and thus to a single bit of Information, that has no “attributes” or “components”, capable of ever being independently measured, by definition of a single “bit” of information.
Specifically they’re impossible per mathematical logic going back to Boole’s Conditions of Possible Experience (which Bell rediscovered). Classical logic is the infrastructural logic of the macroworld we exist in. We can argue about quantum behavior and meso stuff like photosynthesis at room temp and Zeilinger et aliis’ buckeyballs at 1000 degrees Celsius and Seth Lloyd’s entangled covalent bonds but Dr Christian’s squishy and/or exploding spheres weigh 400 times as much as a mosquito and a thousand times more than a fruit fly and are as macro as the Trump Tower. Their behavior can only be classical, not non-Boolean as Dr Christian demands.
Quanta however can disobey Boolean logic. God knows it’s disconcerting and counterintuitive but there you have it.
Reality has two components. Possible realities (possibilities), and actual realities (actualities). When discussing predictions, we are primarily concerned with possibilities, and when discussing experimental results we are primarily concerned with actualities. Under what circumstances does it make sense to combine actualities and possibilities in the same mathematical expressions? Reasoning about possibilities and actualities is very similar to basic logic. Consider the following statements for a single photon.
1- If you measure it at setting a, you will obtain A
2- if you measure it at setting a’, you will obtain A’
3-You measured it at setting a, and obtained A
4-You measured it at setting a’, and obtained A’
(1) & (2) are possibilities — note the conditionals — which represents things which may yet happen. (3) & (4) are actualities representing things which already happened. As you can see for the above (1) & (2) can be true simultaneously, while (3) & (4) can not be true simultaneously. This is because (1) and (2) are mutually exclusive possibilities and there is no such thing as mutually exclusive actualities. Rather, there is a contradiction between (3)&(4).
It is impossible for two actualities to contradict each other. Therefore, arbitrary combinations of actualities are all actualities, but arbitrary combinations of possibilities are not always possibilities, since the conditionals of the possibilities may be contradictory. If you measure at setting (a) then you can’t at the same time also measure at setting (a’).
This is why notations such as (A + A’) or (B + B’) are so imprecise and problematic, as they give too much freedom to the reader to interpret them as possibilities, or actualities, or sometimes,as both. But as I explained above, some interpretations introduce contradictions.
My main attempt here so far has been to help avoid this problem by demanding more precise notation from the participants.
Bell and followers deduce a set of inequalities which involve combinations mutually exclusive possibilities. Experiments produce terms which are necessarily all actualities. However, mutually exclusive possibilities can’t be tested using actualities.
With that out of the way, here are my answers to Jay’s questions:
Part A:
The *result* from a single coin toss H+T=1 is an actuality. That is because you defined result as “number of heads + number of tails”.
Part B:
With the definition of *result* used above, there is only one possibility. No other possible result exists to be predicted for a single coin toss. There are no mutually exclusive possibilities to consider here if you are being consistent.
If perhaps by *result*, it is meant the mutually exclusive possibilities (H=1) or (T=1)? Then the sum
H+T=1 has to be clarified. It is NOT a sum of mutually exclusive possibilities as may appear at first.
Fair enough. This is in the derivation I presented here yesterday, but a few people emailed me that they got lost in the derivation, which would not be the first time. 🙂 And I don’t like reading or writing math in ASCII, but that is what we have here unless I prepare an attachment which has its own drawbacks in relation to facilitating discussion. So, let me do this is clearly and concisely as I can.
Start with the CHSH sum for a single trial using the slip and urn model. That one slip has A(a, lambda), A(a’, lambda), B(b, lambda), B(b’, lambda) on it. Let’s just call those A, A’, B, B’ to keep compact notation. Each can have the value +1 or -1. Then form the sum:
AB + AB’ + A’B – A’B’ (1)
Each of the four terms is an element of reality. Nobody disagrees. But only one of those four terms can ever be measured in one trial if that trial truly represents EPR. The other three terms may be reality, but we cannot measure them, so they are unknowable reality. The disputes between Joy and others all boil down to what this means for Bell. Joy says it means everything because the other three are counterfactual and unknowable as certainties. Others say it means nothing, because the other three are still elements of reality by postulate, and that this postulate also renders irrelevant that the other three terms are not and cannot be measured.
Forgetting about all of this, I pose only two questions:
First: is the SUM in (1) itself an “element of reality”? But since I do not even want to try to define “element of reality” at this moment, let me borrow from Star Trek and call each term in (1) a “tribble” and ask whether the whole sum in (1) is a “tribble,” knowing that at the end I will set:
“tribble” = “element of reality” (2)
So, is the sum of three tribbles minus the fourth tribble in (1), also a tribble?
Second: What is the maximum possible value for the sum of four tribbles in (1)?
It turns out via the derivation I posted here that by simply renaming some quantities along the way we find that the maximum value of the sum in (1) is:
AB + AB’ + A’B – A’B’ = 2 = 2*(H+T) (3)
where either H=1 and T=0, or H=0 and T=1. Mutually exclusive. Because of this orthogonality, we can also discuss this with two state vectors (1 0) and (0 1). And we all know that this 2 turns into the CHSH bound |2|.
So, now my first question metamorphises to this: is 2*(H+T) a tribble, because that is the sum of three tribbles minus the fourth tribble. If you permit me to divide out the 2, then the question is more simply stated:
Is 1 = H+T a tibble, or is not a tribble?
where (H,T) = (1,0) or (H,T) = (0,1). (4)
If I now liken these to coin tosses then I have proved that:
If the CHSH sum (1) is a tribble then 1 = H+T so-defined is also a tribble. (5)
Contrapositively,
If 1 = H+T so-defined is NOT a tribble, then the CHSH sum (1) is also not a tribble. (6)
And, I can observe that this 1 which may or may not be a tribble, is the completeness sum for a coin toss with either heads not tails, or tails not heads. It is a toss counter. So: can toss counters be good tribbles?
The CHSH sum itself contains no accompanying statement as to whether we are asking what this sum may be in advance of a measurement, or what it actually was after a measurement. That is part of what we all need to fill in.
So I am asking everybody to tell me if 1 = H + T ought to be or ought not to be regarded as a tribble. But if it is not a tribble, then the sum in (1) is not a tribble, even though it combines four individual tribbles.
Jay
My answer: 1 = H+T is NOT a tribble.
Why?
Because (H,T) = (1,0) or (H,T) = (0,1) is not the same as (H,T) = (1,0) and (H,T) = (0,1).
Is “the quantum supremacy project” an “element of reality”? I ask only because my Google search for a Wikipedia page mentioning “quantum supremacy” returned just two hits (neither actually at Wikipedia, but both with “Wikipedia” in the URL), one from 2014 and the other, more recent, a palpable piece of pornospam. Thus my experiment seems to have gotten very close to “the limit of a single bit of information”, leaving me in the dark—indeed, arguably less informed than before performing the experiment.
Leave “project” out of your search terms and you get the Big Rock Candy Mountain:
https://www.google.com/?gws_rd=ssl#q=quantum+supremacy
I do not disagree either. I simply say it is not sufficiently clear to say each of those terms is an element of reality. I say the discussion will go much more smoothly if you identify and everyone agrees what kind of reality they represent *possibilities* or *actualities* there is a risk of losing sight of very important subtleties if that distinction is not made clear at this point.
Rather, I would say all those terms A, A’, B, B’ are *possibilities*. But since the A contradicts A’, and B contradicts B’, their linear combination in the form
AB + AB’ + A’B – A’B’ (1)
is NOT a *possibility* since it amounts to accepting mutually exclusive possibilities as simultaneously true — a contradiction. I think this is Joy’s point, in my own words.
As I hope you now see, these “others” can’t reasonably say that if we were clear from the beginning about distinguishing *possibilities* from *actualities*. Just like it is unreasonable to admit the simultaneously logical truth of two mutually contradictory statements.
The problem does not go away because you use “tribble” it is only made worse, so long as you do not identify that there are two types of “tribbles”. You give wiggle room later, for someone to intentionally or mistakenly switch from one type to another at the expense of clarity. There is a type of “tribble” for which the sum (1) will also be a “tribble” and there is a a different type of “tribble” for which the sum (1) is not a “tribble”. This is the subtlety being missed. If I follow along then if A, A’, B, B’ are “tribbles”, I must interpret tribble later as
“tribbles” = “possibilities”
and at no point in the future is it allowed to use
“tribbles” = “actualities”
With that in mind, I would answer that the sum (1) is not a “tribble” since it involves a logical contradiction, and the maximum value of the sum(1) is 2.
if H, T are mutually exclusive tribbles, then H + T is not a “tribble”. But as I’ve been trying hard to explain, (H, T) in your discussion are not always mutually exclusive. As you say if H, T are head/Tail counters, then they are not mutually exclusive and are therefore NOT “tribbles”! So without the clarity, it may appear reasonable (although it is absolutely not), for somebody to go from your coin toss analogy to conclude that the (1) is also a tribble.
If 0 means down, and 1 means up, then a single coin toss can give you (H=0, T=1) both of which are *actualities*. And because *actualities* are necessarily consistent with each other, H + T = 1 is also an *actuality*. Now if I permit you to be obscure with your “tribble” terminology, then you might think because you called H, T mutually exclusive possibilities. But that is false. For your H/T counters, the mutually exclusive possibilities are not H or T, they are (H=0, T=1) or (H=1, T=0), and H + T is not a linear combination of mutually exclusive possibilities in the same manner as (1) as is suggested in expression (3). And therefore you can’t reasonably draw a conclusion about the “tribbleness” of (1) by using the tribbleness of the expression (H+T=1). You have fallen prey to the trap I’ve been trying to warn you about. H/T counters are *actualities* not *possibilities*!
Here I disagree, it does by implication. *possibilities* vs *actualities* correspond to *prediction* vs *measurement*. The output of predictions are *possibilities* and the outputs of measurements are *actualities*. Again all of this confusion will be cleared-up at once if you would seriously consider my clarifying suggestions to use *possibilities* and *actualities* instead of *elements of reality* or *tribble*.
I would answer that your language has unfortunately not been precise enough to accomplish your noble goal, which I share.
MF, I just made a post moments after your latest came in, to try to address your earlier questions raised about temporal matters. Please take a look at that and advise your thoughts. Jay
I neglected to mention that the 1 = H+T with (H,T) = (1,0) or (H,T) = (0,1) is a direct, rigorous outgrowth of the logic which on page 3 of Richard Gill’s paper at https://arxiv.org/pdf/1207.5103v6.pdf is stated as “In the former case, B-B’=0 and B+B’=+/-2; in the latter case B-B’=+/-2 and B+B’=0.” It is Gill per above, divided by 2, and applied to the CHSH upper bound.
But I wanted to add one other point I have been thinking about all night over dinner with guests (don’t worry, I paid attention to them also 🙂 ). Starting with “tribble” = “element of reality,” before we can really talk about “elements of reality,” there is a precursor concept that we must address: CERTAINTY. Certainty is an atemporal concept that can be defined without reference to the concept of time and without reference to the concept of probability:
A. If I have not yet tossed a coin, but will do so, I can state with certainty that either it will land on heads, or it will land on tails. I can represent this as 1=H+T with 1 representing certainty and (H,T)=(1,0) or (H,T)=(0,1). Not another word.
B. If I have already tossed a coin and now have a known result, I can state with certainty that either it did land on heads, or it did land on tails. I can still represent this as 1=H+T with 1 representing certainty and (H,T)=(1,0) or (H,T)=(0,1). Not another word.
To this degree, A and B above have identical math representations, and certainty so-defined is atemporal and aprobabilistic. But in B, I also have additional knowledge as to whether (H,T)=(1,0) or (H,T)=(0,1), whereas in A the only knowledge I can state about this must include the concept of probability. So probability is temporal in that it is a future-looking concept. But “certainty” as stated in A, prior to uttering a word about probability, has the exact same math representation as “certainty” in B.
One of the confusions IMHO, has been to equate “certainty” to “element of reality.” But in A the only “certainty” is that 1=H+T. I have no information whether the certainty will come about because (H,T)=(1,0) or because (H,T)=(0,1). So can I really say that the certainty that 1=H+T is ALSO ***necessarily*** an element of reality, i.e, that
certainty = element of reality?
Or, is it better to think that:
certainty + definite knowledge of positive result = element of reality?
In B, after the toss, now I do have definite knowledge whether (H,T)=(1,0) or (H,T)=(0,1). So now, 1=H+T, still. But does that now make 1=H+T the “element of reality?” Or, does that make H=1 if I landed heads and T=1 if I landed tails, the element of reality? And, especially, what about T=0 if I landed heads and H=0 if I landed tails. I now have certainty with definite knowledge that these are results that did NOT occur. But ought we define the NEGATIVE results that did not occur — T=0 if I landed heads and H=0 if I landed tails — as elements of reality also? These are negative results, not positive results.
Is President Hillary Clinton = 0 an element of reality? I sure wish it was. Along with JFK was never killed = 0 and two world wars never happened = 0. We can state with certainty that all this did not happen. (Maybe the recount will change the first, but I will not hold my breath.) But does that make them elements of reality? These are negative results. Do elements of reality have to be positive results?
Just moving the football a bit forward, and adding some commentary.
Finally:
So if the momentum = 5 kg m /sec, is a definite certain value, then that is an element of reality. A “value in some set of possible values.” Right?
So what about this same momentum not greater than 6 kg m / sec and not less than 4 kg m / sec? These are both entirely true statements. Both range over the set of “possible values.” But does that make these values from the set, which values did not happen (negative result), elements of reality alongside the one value from the set, that did happen (positive result)?
Maybe a few more yards for the football?
Jay
Let me try a few examples of the difference between “certainty” and “element of reality”:
On an atemporal, aprobabilsitic basis, I can say, before or after:
For a coin toss, it is a certainty = 1 that the result is heads or it is tails. But should that quantity 1 be an element of reality?
For a die toss, it is a certainty = 1 that the result is 1 or 2 or 3 or 4 or 5 or 6. But should that quantity 1 be an element of reality? And should a casino pay me at any odds for betting on this quantity 1? (OK, I snuck in probability to the last statement.)
For any human being it is a certainty = 1 that this person is alive or deceased on January 1, 2017. But should that quantity 1 be an element of reality?
For anybody participating in this forum, it is a certainty = 1 that this person is somewhere on planet earth. But should that quantity 1 be an element of reality?
For any wavefunction psi with many possible states, it is a certainty psi *T psi = 1 as a completeness relation, i.e., it is certain that the wavefunction is in some permitted state. But should that quantity 1 be an element of reality?
For any electron or photon hitting the detector for a double slit experiment, it is a certainty = 1 that this particle passes through either slit 1 or slit 2. But should that quantity 1 be an element of reality?
For any quantum particle which is emitted at point A and is detected at point B, it is a certainty = 1 that this particle exists in the universe between points A and B, and it is pretty well accepted that we describe this using path integrals. But should that quantity 1 be an element of reality?
Those are examples of the reasons why we need to separate the concept of “certainty” from the concept of “elements of reality” and carefully consider each and how they interrelate. And this is before we get to temporal and probability consideration. MF is right that we need to bring those in eventually, what he calls *possibilities* vs *actualities*, but he is in too much of a hurry.
jay
I’m in a hurry to tell you where the landmines are so that you do not get blown up before you have started. For example, you talk of “CERTAINTY”, I ask CERTAINTY of what?
For your die toss, there are 6 mutually exclusive exhaustive *possibilities* (1), (2), (3), (5), (5), (6) before you toss. Therefore you can be certain that one of those *possibilities* will be *actualized* after the toss. But what do you mean by certain =1? Are you suggesting certainty = 0.5 makes any sense?
The mutually exclusive exhaustive *possibilities* for the photon are:
(a) Pass through slit 1, (b) Pass through slit 2, (c) Fail to pass either slit. Therefore the photon will certainly do one of those things. I’m not sure what the point of your certainty = 1 is supposed to mean. By definition, one of the mutually exclusive exhaustive possibilities will be obtained if the toss is made.
If you are clear about what is certain, you will find there is no conflict between certainty and reality.
No, not at all.
Maybe you are being too pedantic or maybe I am or maybe we both are. So let me reset.
Throughout probability theory, all of the possibilities add up to the number one, by definition. Before I toss a die, I know with certainty that I will get one of six results. Their probabilities all add up to one. But I do not know which result. After I toss the die, I know for certain that I actualized one of six results. And, now I also know the particular result. Replicate that logic for any other probability problem.
The question I am asking is whether the quantity 1 which I know with certainty characterizes all of the possible outcomes whether I am looking before or after, ought to be regarded as an element of reality in the sense of EPR?
If it cannot be regarded as an element of reality, then neither are the CHSH bounds |2| elements of reality. And I will leave it to you to read the tea leaves as to what that could mean.
Jay
There are only two possible answers: 1 in “1 = H+T” is an element of reality, or it is not.
But the correct answer depends on what is meant by “+” in “1 = H+T.” If by “+” you mean the “exclusive or”, as is usually the case for a coin toss, then the answer is: Yes, 1 is an element of reality. But if what is meant by “+” in “1 = H+T” is something other than the “exclusive or” (such as the “and” in the CHSH sum), then the answer is: No, 1 is NOT an element of reality.
P(1) + P(2) + P(3) + P(4) + P(5) = 1 represents a *possible* reality because the “+” means OR. That statement simply encapsulates the *reality* that (1), (2), (3), (4), (5), (6) are mutually exclusive exhaustive possibilities for the toss.
However, P(1)*P(2) + P(3)*P(4), or any such combination is not a *possible* reality because the “*” between two probabilities necessarily implies AND, which is a contradiction.
The CHSH contains AND terms (H + T) does not.
What if I told all of you about an ingenious method that allow us to measure A(a) and A(a’) simultaneously? Would that put an end to the discussion about whether (A(a) + A(a’)) is an element of reality?
For space-like separated mutually exclusive 3D vector directions a and a’ ?
If so, then do tell us how that magic trick works.
I will, when the time is right.
HR, that would be a tremendous development and a complete game changer.
Keep in mind, that my 1 = H+T is simply another way of representing the statement from Richard’s paper which I have excepted above.
Were you to present such an ingenious method, Richard’s statement would no longer talk about a “former case” and a “latter case” as if they were mutually exclusive. And so the upper bound on his result very likely would now exceed the number 2 because he could superimpose both possibilities in some way.
And if your ingenious method happened to turn my 1 = H+T into sqrt (2) = H+T, it would also turn Richard’s 2 into 2 sqrt(2).
In that event the tribble which is the sum of terms would also become an element of reality, but if the bounds sufficiently increased, local realism would no longer be excluded by quantum correlations.
So the short answer to your question is, yes, absolutely!
While I only have a very sketchy knowledge of the subject since it has only come to my attention in the last few days due to the replies (thanks to all) of several people here to another question I asked, I am wondering if you are thinking about turning a bit into a qbit?
Jay
One other point also in regard to HR’s presently undisclosed “ingenious trick.” If you were to in fact have a way to simultaneously measure B and B’ then Alice and Bob would no longer have the settings controlled by a coin toss. The coin toss of course is the way we encode locality into Bell’s Theorem. So by removing the coin toss, Bill’s theorem would no longer tell us anything about locality.
Alice and Bob could still select their settings by a coin toss, the only difference is that we now also will have a method to know what the result would be had they picked some other setting.
But that would turn the coin toss into irrelevant window dressing.
Well, the method that allows us to measure both A(a) and A(a’) only gives us increased knowledge about the system; it does not change the physical system itself. I don’t see how increased knowledge about a system could invalidate Bell’s theorem – if anything, increased knowledge should tighten bounds, not widen them.
The “+” sign in 1 = H + T is a direct descendant of the “+” signs in the CHSH sum. So it represents *addition* and is NOT a *logic and*.
However, what also descends from the CHSH sum, is the requirement that either H=1 and T=0, OR H=0 and T=1. This is a “logic or.” This descents from Richard’s paper’s statement that “In the former case, B-B’=0 and B+B’=+/-2; in the latter case B-B’=+/-2 and B+B’=0.” Richard’s semicolon “;” operates as a logic or.
Combining these two facts, 1 = H + T, though containing a “+” sign representing addition, operates as a logic or because of the restrictions on the values of H and T and their mutual exclusivity.
Jay
But what about the “+” in B + B’ and “-” in B – B’ ? I know that as a part of your hypothesis you are assuming B + B’ and B – B’ to be elements of reality, but by not specifying what “+” and “-” in them means (“addition”, “and”, or “or”), you have inconsistency in your hypothesis.
Those are also addition and subtraction signs. But the way in which those signs appear in context leads to a logical exclusive or (xor). As in:
If B=B’ then B+B’=0 and B-B’=2B
or, in the mutually-exclusive alternative (absent a magic trick from HR):
If B=-B’ then B+B’=2B and B-B’=0
This is just another way of stating what is on page 3 of Richard’s paper at https://arxiv.org/pdf/1207.5103v6.pdf.
To summarize:
(B+B’=0 and B-B’=2B) xor (B+B’=2B and B-B’=0)
Alternatively:
(H=0 and T=1) xor (H=1 and T=0).
Jay
PS: I never expected to have to explain what a plus and a minus sign meant. But since I have to, I am. 🙂
Why is that? The purpose of the coin toss is to choose among two elements of reality. Say I have two envelopes with money. You toss a coin to pick one envelope, and get to keep the money (say it contained $10). Now I show you the amount in the other envelope ($100 000). Would you say the coin toss was window dressing?
This is really now becoming a hypothetical discussion with nothing concrete to back it up. So let me get back to my main point.
If the CHSH sum is an element of reality, as an inescapable logical consequence of that, the additive sum H + T = 1 is also an element of reality, with the conditions that H = 1 and T =0, or H = 0 and T = 1, but not both. Logical exclusive or. Again, this is a direct, rigorous outgrowth of the logic which on page 3 of Richard Gill’s paper at https://arxiv.org/pdf/1207.5103v6.pdf is stated as “In the former case, B-B’=0 and B+B’=+/-2; in the latter case B-B’=+/-2 and B+B’=0.” It is Gill per above, divided by 2, and applied to the CHSH upper bound. The semicolon in Richard’s sentence, and his “former case / latter case” dichotomy, operates as his logical exclusive or. I am simply holding up a mirror to that statement in Richard’s paper which is instrumental in getting to the CHSH bound of |2|.
There is NOTHING in the path from Richard’s statement to my H + T = 1 that in any way references or depends upon whether there exists, or not, some “ingenious method that allow us to measure A(a) and A(a’) simultaneously.”
You simply must decide whether H + T = 1, under the mutually-exclusive conditions that H = 1 and T =0, or H = 0 and T = 1, but not both, is an element of reality.
If you want to change my mutually-exclusive conditions in some way, then you must change Richard’s mutually-exclusive conditions in the exact same way. If you can convincingly argue that H + T = 1 under my mutually-exclusive conditions should be an element of reality, then you will equally convincing that the |2| in the CHSH sum is an element of reality. If not, then not. Those two absolutely sink or swim together.
I fully agree that the 1 in H + T = 1 is an element of reality. It is also good to see you state “There is NOTHING in the path from Richard’s statement to my H + T = 1 that in any way references or depends upon whether there exists, or not, some “ingenious method that allow us to measure A(a) and A(a’) simultaneously.”
Because that is correct, and it is also correct that Bell’s theorem doesn’t depend on it either.
But, I thought the issue here was that Bell’s proof fails because (A(a) + A(a’)) can’t be an element of reality, because you can’t measure them both? Isn’t that what Joy and MF claim? So in that case, a method to measure them both should settle the case?
(And the method will be forthcoming, we just have to agree upon the consequences first).
Well, that is the issue which has been discussed into the ground. But believe it or not, I think that Joy and MF were too *modest* in their claim. The real issue is whether the sum:
AB+AB’+A’B-A’B’ == CHSH sum
is a “tribble,” if we assume that all of A, B, A’ and B’ and well as each product in the sum, is a “tribble.” Or, “tribble” = element of reality. This discussion really is about, or should be about, the additive properties of tribbles a.k.a. elements of reality.
Good. If you fully agree that H + T = 1 under the stated mutually exclusive conditions where (H=1 and T=0) xor (H=0 and T=1) should be defined to be an element of reality, then CHSH sum is an element of reality and Bell does not fail. But I am asking a two part question: 1) should this be defined as an element of reality?, and 2) WHY or WHY NOT? So, why?
Joy has answered that this should NOT defined to be an element of reality. But he has also not answered part 2 as to why not. Joy, why not?
And since this all flows from the mutual exclusivity used in connection with Fact 1 in Richard’s paper, I am still waiting for Richard’s views on both these questions.
I personally believe at this time that it is dubious to regard H+T=1 as an element of reality, but I want to keep an open mind and hear from everyone, because that is a logically-permitted answer, and the answer has to come from independent considerations.
The independent though seemingly analogous consideration which has my attention, is whether the 1=psi *T psi for wavefunction completeness, which effectively says that a wavefunction must be in one of its allowed sates, should be defined as an element of reality. And more generally, should the quantity “1” which by definition tells us that in any probability problem all of the possible outcomes add up with certainty to 1, should be defined as an element of reality. It seem to me that individual positive outcomes, and not the sum = 1 of all possible mutually exclusive outcomes, is what ought to be the element of reality.
I have been thinking about all of this for about ten days now, but it is too important a questions for me just to rely on my own musings and I really to want to hear views and reasons from others.
Jay
Well, if each of those terms can be measured, why shouldn’t we be allowed to add them? If we can’t add those, what can we add? And if we in general are not allowed to do additions, we can just stop doing physics (or business, or whatever) altogehter.
Well, if each of those terms can be measured, why shouldn’t we be allowed to add them?
Because the space of all possible elements of reality is not closed under addition, as I have explained many times before:
http://libertesphilosophica.info/blog/wp-content/uploads/2016/11/Fatal.pdf
Because the space of all possible elements of reality is not closed under addition, as I have explained many times before:
http://libertesphilosophica.info/blog/wp-content/uploads/2016/11/Fatal.pdf
******* Sorry, some entanglement seems to have occured within my text editor. I will try it again. *******
Let’s assume that Alice has decided to only perform measurements along two given directions a and a’ and that Bob has decided to only perform measurements along two given directions b and b’.
I) Single experiment:
“Einstein’s demon” – so to speak – prepares four boxes with a lid which he intends to hand over to Alice and Bob: Two for Alice which are labeled with A (spin measurement direction a) and with A’ (spin measurement direction a’) and two for Bob which are labeled with B (spin measurement direction b) and B’ (spin measurement direction b’).
Each box will finally – before handed over to Alice and Bob – contain a card with an inscribed number. Let’s assume that the card stands for the physical property “spin component in direction x”. A physical quantity is a physical property that can be quantified by measurement: Just open the box labeled with X and read out the number written on the card, you have then a value for the physical property “spin component in direction x”. One can also call each card an “element of reality” when following Einstein’s operational measurement procedures to define whether something is an “element of reality” or not. Nothing more has to be defined.
Consider now two entangled spin1/2-particles which are separating. The demon takes a look at both particles and translates his observations into the codings +/- 1. He puts now one single card in each of the boxes, each card inscribed with either +1 or -1 in correspondence with his observations (Alice and Bob would be really surprised in case they would find cards with numbers other than +1 or -1).
When the cards with the inscribed numbers have been put in place, the demon evaluates A*B + A*B’ + A’*B – A’*B’ and enters the result into a one-column list labeled “AB_1 + AB’_1 + A’B_1 – A’B’_1” (A*B + A*B’ + A’*B – A’*B’ can assume only the values +2 or -2). He then closes the lids and couples Alice’s boxes in such a way that a) both lids cannot be opened simultaneously (blocking) and b) one lid is immediately locked when the other is opened (locking). The same procedure for Bob’s boxes. From that moment, nothing can change the boxes’ content, everything is fixed.
The demon now hands over Alice’s coupled boxes to Alice and Bob’s coupled boxes to Bob. The principle of measurement is now that Alice can freely decide to either open box A or A’ and to read out the number on her card and that Bob can freely decide to either open box B or B’ and to read out the number on his card. Alice and Bob have four one-column lists which are labeled with “AB_2-1”, “AB’_2-2”, “A’B_2-3” and “A’B’_2-4”. After opening their selected boxes, they inform each other of their findings, evaluate the product of their numbers and enter this result in the corresponding list. Then all the boxes are handed over back to the Einstein’s demon and another experiment can begin.
II) Series of experiments:
Let’s assume that Alice and Bob decide to “perform” 100 runs (N = 100) of such an experiment whereby they open box A and box B. At the end, the demon has 100 results for A*B + A*B’ + A’*B – A’*B’ in his “AB_1 + AB’_1 + A’B_1 – A’B’_1” list, whereas Alice and Bob have 100 results for A*B in their “AB_2-1” list.
Now Alice and Bob decide to “perform” 100 runs (N = 100) of such an experiment whereby they open box A and box B’. At the end, the demon has added further 100 results to his “AB_1 + AB’_1 + A’B_1 – A’B’_1” list, whereas Alice and Bob have now 100 results for A*B’ in their “AB’_2-2” list.
Now Alice and Bob decide to “perform” 100 runs (N = 100) of such an experiment whereby they open box A’ and box B. At the end, the demon has added further 100 results to his “AB_1 + AB’_1+ A’B_1 – A’B’_1” list, whereas Alice and Bob have now 100 results for A’*B in their “A’B_2-3” list.
Now Alice and Bob decide to “perform” 100 runs (N = 100) of such an experiment whereby they open box A’ and box B’. At the end, the demon has added further 100 results to his “AB_1 + AB’_1+ A’B_1 – A’B’_1” list, whereas Alice and Bob have now 100 results for A’*B’ in their “A’B’_2-4” list.
III) Evaluation:
When averaging over all entries in the respective lists, Alice and Bob get (these results represent knowledge which is acquired by physical measurements):
(averaged over N runs)
(averaged over N runs)
(averaged over N runs)
(averaged over N runs)
When averaging over all entries in his list, Einstein’s demon gets (this result represents knowledge which is acquired by physical reasoning):
(averaged over 4N runs)
When comparing, one will always find:
{ + + – } is rougly (depens on N) equal to
With increasing number of runs N, { + + – } will progessively approach .
As we know that the entries in the demon’s “AB_1 + AB’_1 + A’B_1 – A’B’_1” list can assume only the values +2 or -2, we can thus conclude:
-2 < { + + – } < +2
In case both Alice and Bob want to make measurements along four given directions, Alice along a, a’, a’’ and a’’’ and Bob along b, b’, b’’ and b’’’, the demon has to prepare always four boxes for Alice and Bob – using similar procedures for the codings and for the installation of the blocking and locking mechanisms. In principle, there is – more or less – an infinite number of measurement directions, but this would become a cumbersome job for Einstein’s demon.
Here is an explicit simulation that contradicts your conclusion: http://rpubs.com/jjc/84238
The section “Evaluation” reads:
III) Evaluation:
When averaging over all entries in the respective lists, Alice and Bob get the following averages (these results represent knowledge which is acquired by physical measurements):
av(AB_2-1) (averaged over N runs)
av(AB’_2-2) (averaged over N runs)
av(A’B_2-3) (averaged over N runs)
av(A’B’_2-4) (averaged over N runs)
When averaging over all entries in his list, Einstein’s demon gets the following average (this result represents knowledge which is acquired by physical reasoning):
av(AB_1 + AB’_1 + A’B_1 – A’B’_1) (averaged over 4N runs)
When comparing, one will always find:
av(AB_2-1) + av(AB’_2-2) + av(A’B_2-3) – av(A’B’_2-4) is rougly (depens on N) equal to av(AB_1 + AB’_1 + A’B_1 – A’B’_1)
With increasing number of runs N, av(AB_2-1) + av(AB’_2-2) + av(A’B_2-3) – av(A’B’_2-4) will progessively approach av(AB_1 + AB’_1 + A’B_1 – A’B’_1).
As we know that the entries in the demon’s “AB_1 + AB’_1 + A’B_1 – A’B’_1” list can assume only the values +2 or -2, we can thus conclude:
-2 le av(AB_2-1) + av(AB’_2-2) + av(A’B_2-3) – av(A’B’_2-4) le +2
In case both Alice and Bob want to make measurements along four given directions, Alice along a, a’, a’’ and a’’’ and Bob along b, b’, b’’ and b’’’, the demon has to prepare always four boxes for Alice and Bob – using similar procedures for the codings and for the installation of the blocking and locking mechanisms. In principle, there is – more or less – an infinite number of measurement directions, but this would become a cumbersome job for Einstein’s demon.
Each of those terms represent mutually exclusive possibilities. They each *can* be measured. But only one of them will ever be actualized. Adding them is a contradiction. You can add any numbers mathematically, to your heart’s content, but the result is not necessarily meaningful. you can’t compare your resulting sum with a meaningful sum of 4 actualities.
But I claim that I have a method to measure all those term simultaneously (which I will disclose as soon as the dust has settled here), so they are not mutually exclusive. Would that change your argument?
I will wait to see what magic trick you use to measure the same particle twice. Unless perhaps you don’t understand that what your claim amounts to?
They are mutually exclusive in the CHSH, by definition. If you start claiming they are not, then you aren’t talking about the CHSH.
Why is that? What definition of CHSH makes them mutually exclusive? So if I have a method to measure the same particle twice (which should not be shocking by the way, we measure things twice all the time), would that someow render the CHSH expression meaningless?
You’ll have to go back and follow the derivation of the CHSH to understand that those terms are mutually exclusive. A and B are predicted results from a pair of EPRB particles measured at settings (a) and (b). A’ and B’ are results from the exact same pair of particles measured at two angles (a’) and (b’). Once the EPRB particles are measured at (a), (b), they are destroyed in the process such that it is impossible to measure them again at (a’), (b’). Bell’s assumption is that those outcomes A’ and B’ exist along side A and B as elements of reality even though they can’t be measured along with A and B at the same time. That is what “mutually exclusive” means. I assume you have heard of “counterfactual definiteness”? A’ and B’ would not ever be considered counterfactual if they weren’t mutually exclusive to A and B.
Now you claim to have found mutually exclusive possibilities that are not mutually exclusive, I will wait to see you present he proof that proves itself wrong.
If you have a method to measure the same particle twice, then what you measure are not mutually exclusive possibilities. So why would you think that is relevant to a discussion about mutually exclusive possibilities? Yes, we measure things twice all the time, but not EPRB particles which we can only measure once.
My argument is that a sum of mutually exclusive *possibilities* is meaningless and can’t be compared with a similar looking but totally different sum of mutually compatible *actualities*. Why would you think you could get me to change my mind by presenting a sum of *possibilities* that are not mutually exclusive? Perhaps you have not understood the argument at all.
Where, in the proof of Bell’s theorem, is it a requirement that the particles must be destroyed upon measurement? I have a method of measurement that does not destroy the particles. What does that mean for Bell’s theorem?
Think of the space of all fruits. Is that space closed under addition? Can you add a banana to an apple to create a new fruit? Similarly, you can’t create a new element of reality by adding.
OK HR, I will take the bait. And by doing so, I will prove that your method to measure all of A, A’, B, B’ simultaneously is totally irrelevant to the question at hand, AND that the arguments Joy and HR et al. have been making about counterfactualism are also equally irrelevant to the question at hand.
Your premise, which I love, has enabled me to see that insofar as Bell’s Theorem is concerned, Joy and Richard and MF and Fred and you all might as well have been arguing for years about how many angels can fit on the head of a pin, for all the relevance that discussion actually has. Here is why:
Start with
CHSH = AB + AB’ + A’B – A’B’ (1)
Just for notation, let me rename A’ to C and B’ to D and so rewrite this as:
CHSH = AB + AD + CB – CD = A(B + D) + C(B – D) (2)
I will now grant your premise that A and B and C and D can ALL be measured simultaneously, and although (2) is equivalent to (1) just with a renaming of variables, the reaming is intended to emphasize your premise. So there is no confusion, these are plus and minus signs, meaning ordinary addition and subtraction. We are not yet talking about logical “ands” and “ors” and “xors.” We then make the following supposition / hypothesis: each of A, A’, B, B’ can independently have the mutually-exclusive values of +1 or -1. And, each of these is observable and can be measured and observed simultaneously.
The issue I have been harping on about heads and tails arises from the very structure of the CHSH sum in (2) and this also gets to your question about “why can’t I just add four numbers to get a new number?” And it is because the four terms in (2) weave together two occurrences each of four independent variables in a particular combination with three added and one subtracted, whereby the terms being combined in (2) are NOT FULLY INDEPENDENT of one another. It is like asking why I cannot treat a problem with dependent probabilities the same way I treat a problem with independent probabilities.
Because each of A, A’, B, B’ can independently have the mutually-exclusive values of +1 or -1, each one of these will have same value as or the opposite value from each other one of these. Those are just their correlations or anti-correlations. So similarly to what Richard has done at https://arxiv.org/abs/1207.5103, we can analyze this using two mutually exclusive alternatives:
If B=D then B+D=2B and B-D=0 (3a)
If B=-D then B+D=0 and B-D = 2B (3b)
If I divide through by 2B then I can rewrite the above by:
If B=D then (B+D)/2B=1 and (B-D)/2B = 0 (4a)
If B=-D then (B+D)/2B=0 and (B-D)2B = 1 (4b)
Then, I may again rename the variables by defining H and T such that:
H == (B+D)/2B (5a)
T == (B-D)/2B (5b)
using (5) in (4) now produces the mutually-exclusive possibilities:
If B=D, then H=1 and T=0 (6a)
If B=-D, then H=0 and T =1 (6b)
And in all cases, no matter what:
H + T = 1 (7)
So even if each of A, A’, B, B’ (a.k.a. A, B, C, D) is an element of reality and all of A, A’, B, B’ (a.k.a. A, B, C, D) can be simultaneously observed, the CHSH sum weaves those together into a particular logical combination which produces the mutually-exclusive consequences:
(H=1 and T=0) xor (H=0 and T =1) (8)
with 1=H+T being a completeness sum for those two results in (8).
So we are right back to where I was earlier:
If CHSH in (1) is an element of reality, then 1 = H+T in (7) is also an element of reality.
—BUT, BY CONTRAPOSITIVE LOGIC—
If 1 = H+T in (7) is not an element of reality, then CHSH in (1) is also not an element of reality.
This is logically inescapable. Again, they sink or swim together. Your claim to measure A and A’ and B and B’ simultaneously is irrelevant to this result. Joy’s counterfactual arguments which make reference to what can and cannot be observed simultaneously are equally irrelevant. You have all been barking up a wrong tree. I am throwing cold water on everyone.
The point is that the very structure of the CHSH sum in (2), with its interweaving of four presumably-independent and simultaneously-observable variables, factors out all of these arguments, and STILL leads to the single variable
1 = H+T with (H=1 and T=0) xor (H=0 and T =1) (9)
This variable has an XOR logical property, that is deduced from and arises from the particular correlator structure of the mathematical sums in (1) and (2) and the permitted +/-1 values of each of A, B, C, D.
And, very importantly, this XOR logical property is NOT carried by each of AB, AD, CB and CD on a separate basis. So this also answers your question why you cannot just add the first three then subtract the fourth and not worry about anything. You can add three of these then subtract the fourth, but the result you get has an endemic “heads” and “tails” property of mutual exclusion, which property is not possessed by any single individual term in (1) or (2).
But this does reveal that the number |2| in the CHSH sum has a internal logical property of mutual exclusivity with precisely mirrors that of 1 = H + T for a coin toss with the valuation in (9) that can be represented also by the state vectors
(H, T) = (1, 0) XOR (H, T) = (0, 1) (10)
Because this |2| sets outer bounds on the CHSH sum, one has to decide whether a number with the built-in XOR property 1 = H + T of a coin toss that has the orthogonal states (10), is a suitable candidate for an element of reality used as an outer bound to obtain fundamental, far-reaching conclusions about locality and realism.
Jay
Since 1 = H + T, with your identifications, is completely equivalent to the CHSH sum (1), the coin-toss quantity 1 = H + T is not an element of reality, as I have already noted.
Joy,
The logic structure (for a review of traditional formal logic see https://en.wikipedia.org/wiki/Contraposition) of the disproof-by-contradiction I have proposed, with the state vectors (H, T) = (1, 0) XOR (H, T) = (0, 1), and with the CHSH correlator = AB + AD + CB – CD = A(B + D) + C(B – D) which even accepts HR’s proposition that he can measure all four of A, A’=C, B, B’=D simultaneously, is the following:
Positive logic:
IF CHSH correlator is element of reality THEN 1 = H+T is element of reality. (1)
Contra-positive logic:
IF 1 = H+T is NOT element of reality THEN CHSH correlator is NOT element of reality. (2)
In your reply above, you are NOT using the logic correctly, because you are asserting the positive logic *inverse*, namely:
IF CHSH correlator is NOT element of reality THEN 1 = H+T is NOT element of reality. (3)
That, however, does NOT follow inexorably from (1) as a matter of logic.
The central requirement of the disproof-by-contradiction I have proposed, is that it is necessary to INDEPENDENTLY establish that:
1 = H+T is NOT element of reality = TRUE (4)
In order to prove via (2) that
CHSH correlator is NOT element of reality = TRUE (5) *****which is your goal*****
You have been arguing (5) for a long time implicitly and explicitly. But (4) needs to established INDEPENDENTLY in order to prove (5) via (2). Using the inverse logic (3) is simply using your own prior belief to prove you own prior belief, and logic does not work that way. That is just making or continuing an argument which says (4) is true because (4) is true. That is not applying logic to PROVE (5) on an INDEPENDENT basis via (2), which must be your goal.
If supporters of Bell can be persuaded that (4) is correct on an entirely INDEPENDENT basis, divorced from all the things you have argued previously, then via (2) you establish the correctness of (5) which has been your long term goal, and the supporters of Bell would be forced to concede that (5) is true.
So, I will rephrase and tighten my question:
We all know you agree with (4) as a logic proposition. Please give me / us the reasons WHY everyone should believe (4) as a logic proposition, on and INDEPENDENT footing that does NOT follow the inverse logic path (3) in any way. You are prohibited from using your own conviction that CHSH correlator is not an element of reality to prove that CHSH correlator is not an element of reality. You MUST start with 1 = H+T, or something that is of an identical formal character, on an entirely INDEPENDENT basis from your beliefs about Bell, in order to properly exercise this disproof-by-contradiction.
I have given you some hints, such as my favorite approach below:
Would it be suitable to define 1 = sum (psi^*T psi) (with *T==conjugate transpose) as an element of reality, which states that a wavefunction will be in a permitted state thereof with certainty? In fact, if we define the mutually-exclusive state:
psi_h == (H, T) = (1, 0) XOR psi_t == (H, T) = (0, 1) = TRUE (6)
for the precise case at hand, then another way to write 1 = H + T is (taking out the “conjugate * because the vectors are real) is the following:
1 = sum over h,t (psi^T psi) (7)
If you can convince people that completeness sums “1” formed in the manner of (7) are not and cannot be regarded as elements of reality, then they would have no choice but to concede (5). With that, you can finally declare Checkmate!
Jay
This should have read:
“That is just making or continuing an argument which says (5) is true because (5) is true.”
There is no doubt that EPRB particles can only be measured once? There is no doubt that Bell used the concept of Counterfactual Definiteness in his proof?
Bell’s theorem is a claim based on comparison of a sum of mutually exclusive possibilities, to al sum of actualities. There are 2 sides to every “violation”, the LGS in Bell’s theorem is a sum of mutually incompatible possibilities, the RHS in the case of QM is a set of mutually compatible predictions, and in the case of experiments, the RHS are a set of mutually compatible actualities. There’s no other way of obtaining Bell’s the without making such an invalid comparison.
So whatever you think you have, please make sure you describe the theoretical predictions, the QM predictions and then the experimental result. We shall see if Bell’s theorem survives.
My argument is that a sum of mutually exclusive *possibilities* is a meaningless quantity that can’t be reasonably compared with a sum of mutually compatible *actualities*.
If you think you have found a magic trick that allows you to disprove my argument please go ahead and try. Perhaps you want to present a meaningful sum of mutually compatible possibilities, or perhaps you intend to present a sum of actualities.
OK, Jay. Here is my same old answer, but now respecting your new logic:
Since you have defined H = ( B + B’ )/2B and T = ( B – B’ )/2B ,
1 = H + T is not an element of reality, because H and T are possible outcomes of an impossible coin. There does not exist a coin in Nature that can have H and T as two counterfactually possible outcomes. This includes the procedure (or HR’s “method”) described in my proposed macroscopic experiment: https://arxiv.org/abs/1211.0784 .
Joy,
With the above answer (which is actually not quite your same old answer as I will explain), I believe you can declare checkmate, and that Bell’s Theorem is toppled. But as any student of chess will be aware, the declaring of a checkmate, especially if made five or six moves ahead as we are doing here, requires explanation to the losing party of the moves that will being about the checkmate. So, having developed a checkmate, we enter the phase of the discussion where we are duty bound to explain the checkmate to the supporters of Bell, and develop this explanation fully, so that they will come to see why they have to resign the game.
Let me make just a few preliminary comments about some pieces we have now removed from the chessboard, because they have caused a great deal of confusion and yet have in fact entirely irrelevant to the real issues. As part of this, I need to partially retreat from my earlier statements that your counterfactual arguments were wrong or irrelevant. In fact, your counterfactual argument — which I prefer to discuss in terms of pure logic as mutual exclusivity i.e, the logical XOR — is a correct and relevant argument, but, you have been applying it incorrectly and to the wrong data.
For me the clarity of an impending checkmate comes to a head with HR’s claim to “have a method to measure all those terms simultaneously…so they are not mutually exclusive.” I know you had a problem with that, but to me, it was a tremendously-clarifying move which opened the checkmate of Bell. By these terms HR means A(a), A(a’), B(b) and B(b’) which go into the CHSH correlator in a well-known combination. HR’s move of pushing this piece enables countermoves that allow large amounts of extraneous matter to be removed from the chessboard so that the checkmate can be revealed. And to avoid repetition, I will simply refer to my post here on December 3, 2016 at 5:53 pm for details, in which I took HR’s “bait.”
Let’s first go to the incessant repetition of the argument that B+B’ and B-B’ are not elements of reality because they are counterfactual, i.e., because measuring B is mutually exclusive with measuring B’. This is a right argument, but applied in the wrong way to the wrong data.
As I showed on December 3, 2016 at 5:53 pm, you can concede these chess pieces to HR and Richard and anybody else who wants to grab them, and it still does not hurt you. This is because with
H = ( B + B’ )/2B and T = ( B – B’ )/2B whereby 1 = H + T (1)
and also whereby
[(H=1 and T=0) XOR (H=0 and T=1)] = TRUE (2)
the salient mutual exclusivity is not between B and B’, ***it is between B + B’ and B – B’***. Or, if someone has trouble seeing this, it is between H=TRUE and T=TRUE which faithfully models the single flip of a coin in all respects. Sure, you can observe a and a’ and b and b’ all together. But you still end up with the mutual exclusivity in (1) and (2).
Additionally, you have argued that if HR’s claim that he could “measure all those terms simultaneously” was true, he would be changing the premise of the EPR-Bell experiment, because those premises require that we measure along a xor a’ and along b xor b’. Here too, you can concede those chess pieces without losing the game, still again, because you still end up with (1) and (2) above even after this concession of material.
So now, all of those pieces which caused massive confusion are off the board, and it all comes down to whether the CHSH sum, which we may now write in terms of half the magnitude of that sum as:
.5 |AB+AB’+A’B+A’B’| lt 1 = H+T (3)
is an element of reality. This in turn reduces to the question of whether 1=H+T as this is defined, is a suitable element of reality. Or, whether the number 1 which represents that I flipped coin once, is an element of reality. Or whether, instead, the fact of the coin having landed on heads, or having landed on tails, is what the element of reality needs to be. MF may have some observations here about possibilities versus actualities needed to fine tune this further.
I know that HR has declared that yes, 1=H+T is a suitable element of reality. And while we have not heard recently from him, I am guessing that Richard will also declare that yes, 1=H+T is a suitable element of reality. And / or, he may declare that even if H+T is NOT an element of reality, it can still be used in (3) with no adverse inference that can be drawn against Bell’s Theorem. These are logically-permissible positions. But, as I will try to explain in some forthcoming posts, they are *physically*-dubious positions, on independent grounds.
However, explaining why these are physically dubious positions is not about putting Bell into checkmate. That has already been done in (3). It is tantamount to explaining WHY equation (3) above puts Bell’s Theorem into checkmate.
Jay
Without a definition of “element of reality” it is impossible to agree or disagree.
Without making careful distinction between *name* and *value* of such elements of reality, all we get is confusion.
If B and B’ are simultaneously existing elements of reality (whatever that may mean), taking definite values +/- 1, and if we define H = ( B + B’ )/2B and T = ( B – B’ )/2B , then H takes some definite value, and T takes some definite value. Moreover, one of those values is 1 and the other is 0. Written out in full: the physical system has definite values of H and T, and those values are either 0 and 1 respectively, or 1 and 0 respectively.
If you want to see an analogy with a coin toss, then we have an analogy with a coin which has already been tossed, but whose outcome is still hidden to outside observers.
I do not intend to follow the “debate” here any longer, since all I see is obfuscation and denial.
B and B’ are simultaneously existing mutually exclusive *possibilities*, however (B + B’) is not a *possibility* because it involves a contradiction, it is an *impossibility*. Therefore it is impossible that (B + B’) will ever be an *actuality* in a measurement.
Somebody once, said: “If a theoretician wants a theory tested, then the theory had better predict possible empirically observable data.” I agree.
EPR says:
B and B’ are simultaneously existing *possibilities*, but mutually exclusive *actualities*.
There is no such thing as mutually exclusive *actualities* and EPR do not say such a thing. An actuality represents an event which already happened or an outcome already obtained.
Above, I meant to say:
.5 |AB+AB’+A’B+A’B’| le 1 = H+T (3)
Less than or equal, not less than.
The necessary first requirement for anything to be an element of reality in physics is that it must be a physically meaningful quantity, before we can even entertain the well thought out and universally accepted sufficient criterion for the elements of reality proposed by EPR.
Everyone agrees, or should agree, that the quantities A, A’, B and B’ fulfill both the necessary and sufficient conditions above, and are thus genuine elements of reality, with +/-1 values.
Unfortunately B + B’ and B – B’ do not satisfy even the first requirement of being physically meaningful. Physically they are impossibilities — or absurdities, let alone being elements of reality. If we take such meaningless quantities to define two sides of a coin, then what we end up with is an impossible coin, belonging to a fantasy land. If we insist on defining its two sides by H = ( B + B’ )/2B and T = ( B – B’ )/2B , then H and T can certainly be assigned some definite values, but contrary to Gill’s claim the resulting coin “1 = H + T” most certainly does not represent any meaningful physical system that can take those definite values for H and T.
In nature there simply does not exist a coin that can produce H and T as two counterfactually possible outcomes, because B + B’ and B – B’ can never be actualized in any real experiment.
EPR says:
B and B’ are simultaneously existing *possibilities*, which can never become simultaneously existing *actualities* in a measurement.
Theoretically, you are allowed to conceive terms like B + B’ or B – B’, which you can reflect on.
Quantum mechanics says:
B and B’ are simultaneously existing mutually exclusive *possibilities*, which can never become simultaneously existing *actualities* in a measurement.
Theoretically, you are not allowed to conceive terms like B + B’ or B – B’, you shouldn’t even reflect on.
(1) ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ ≤ 2
(2) ⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ ≤ 2√2
QM is a measurement theory, it is not concerned with the unmearesurable. Bell’s mistake is to claim that (2), an expression from QM, which represents simultaneously compatible *possibilities*, can be compared with (1) which represents mutually exclusive *possibilities*. They are apples and oranges and can’t be compared. Experimenters measure *actualities* and find that they are consistent with QM because all *actualities* are necessarily simultaneously compatible, and prior to measurement were necessarily simultaneously compatible *possibilities*, just like QM predicted.
What they got from QM and experiments does not amount to a violation of (1). They should have been comparing (2) with (3)
(3) ⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ ≤ 4
(3) is the correct inequality involving simultaneously compatible *possibilities* from a local realistic theory, and neither QM not Experiments have ever violated that.
Bell’s theorem is invalid.
EPR says:
A and A’ are simultaneously existing *possibilities*, which can never become simultaneously existing *actualities* in a measurement.
B and B’ are simultaneously existing *possibilities*, which can never become simultaneously existing *actualities* in a measurement.
Theoretically, you are thus allowed to conceive an EPR-“state function” in terms of (A, A’, B, B’) (a complete description according to Einstein) and to reflect on terms like A*B, A*B’, A’*B or A’*B’ or terms like A*B+A*B’+A’*B-A’*B’. This represents knowledge that can be acquired by physical reasoning about an “external local reality independent of our observations”. Use then ensemble statistics to compare the knowledge “acquired by physical reasoning” with the knowledge “acquired by physical measurements”. You will always find with increasing number of experiments:
av_pm(A1*B1) + av_exp(A2*B’2) + av_exp(A’3*B3) – av_exp(A’4*B’4) approaches more and more av_pr(A*B + A*B’ + A’*B – A’*B’)
(Here, “av_pm” denotes an average value obtained by “physical measurements” and “av_pr” an average value obtained by “physical reasoning”)
Quantum mechanics would say:
Wherever it’s needed, you might think of that A and B are existing *possibilities*, which can become existing *actualities* in a measurement.
Theoretically, you only have to conceive a single QM-“state function” in terms of (A, B). Using the QM formalism, you can predict the probability for the occurrence of the *actuality* A*B in measurements. To predict the probabilities for the occurrence of other *actualities*, you merely transform the QM-“state function” by switching to another base (Einstein considered such a merely operational formalism as an incomplete description of reality).
You are referring to the ideas presented in Gill’s paper on the subject, I assume? Here is what is wrong with the paper:
Gill assumes that you a spreadsheet with columns A, A’, B, B’ (your state function). He then assumes that it is possible to randomly extract 4 subsets your spreadsheet (without replacement), in such a manner that the distributions in each sample is the same as the distributions in the population. But this is completely unfounded.
Subsets obtained from a larger set without replacement, are not independent. Therefore, you are ending up with a situation essentially the same as what you started with, in which you do not have 4 independent experiments, like is the case in QM. You have fewer degrees of freedom in this experiment than you have in real EPRB experiments or implied by the QM equations and therefore narrower bounds. Sampling without replacement posses two problems for Gill’s paper:
1) It is contradictory with the claim of having equally distributed sub samples of a population.
2) It is contradictory with the claim of requiring freedom of choice. Once you select your first random sample, the remaining samples are necessarily dependent on the first selection, and therefore not entirely random. It can therefore not be said to have been freely chosen randomly.
In the expression ⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩, each of the 4 terms represents measurements or predictions performed on a freely chosen independent set of particle pairs. But there is an even more serious reason. The terms in ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ involve a cyclical recombination of individual terms A, A’, B, B’ and it has been proven that such recombinations sometimes makes it impossible to recover the joint distributions from independently measured pairs of values such as (A₁B₁), (A’₂B₂), (A₃B’₃), (A’₄B’₄). You might want to check the proof provided in http://www.panix.com/~jays/vorob.pdf
What you (and Gill) have presented is not the same as the expression ⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ from QM and experiments. Your suggested upper bound does not apply to QM and experiments, where there is more degrees freedom between terms.
Nothing wrong with Gill’s paper!
Quoting Joy Christian:
“Unfortunately B + B’ and B – B’ do not satisfy even the first requirement of being physically meaningful. Physically they are impossibilities — or absurdities, let alone being elements of reality.”
But that’s exactly what quantum mechanics says. It is Einstein’s eager argumentation which allows Richard Gill to look at these terms as he does.
Maybe, you might have misunderstood the EPR thought experiment.
If B + B’ is an element of reality, then it shouldn’t be difficult to prove that. I asked Gill for a precise criterion — analogous to that by EPR — which dictates that B is real, and B’ is real; and then, using the same criterion, prove that B + B’ is real. But we have yet to see the proof.
Well, I’ve only been away for a day or so for pressing business, but it’s always nice to feel missed…:-)
So let me now reveal the method to measure all those terms simultaneously. In fact, it’s not my method at all, it’s Joy Christian suggestion for his macroscopic experiment.
Drop the elementary particles, and go to small macroscopic fragments of exploding balls instead (yes, this is his proposed experiment). According to Joy Christian, the correlations will still be the quantum correlations, since the correlations are due to some torsion in space, presumably independent of the size of the particles (in the paper “Macroscopic Observability of Spinorial Sign Changes under 2π Rotations” he suggest they could be around the size of 3 cm, but I have no idea why). Instead of Stern-Gerlach detectors, use high speed cameras that record the motion of the fragments at a prespecified location. From these images, we can now deduce a spin direction s, and then compute A(a) as sign(s.a), and at the same time compute A(a’) as sign(s.a’). Do the same for B(b) and B(b’). This is all verbatim from his paper.
So here we have a clear case where the results of all measurements [A(a), A(a’), B(b), B(b’)] can be computed simultaneously. And this is actually Joy’s own suggestion, which is quite remarkable, since his whole argument against Bell’s theorem hinges on the assumption that such simultaneous measurements are impossible.
Jay, MF and I have already taken into account your misunderstanding of my proposed experiment: http://retractionwatch.com/2016/09/30/physicist-threatens-legal-action-after-journal-mysteriously-removed-study/#comment-1209474 .
Richard,
Thank you for your concise, responsive, and lucid reply. This is exactly what I had hoped for, because it helps me clarify issues that have been challenging for me to think through. Let me provide you with my initial thoughts about all of this:
I could not agree with you more. As you have pointed out, EPR only provided *sufficient* conditions for a definition of reality, not *necessary* conditions. My whole goal these last 10 days where I have repeatedly asked about and given examples of possible elements of reality, has been to develop a consensus about a definition of elements of reality working bottom up from data, rather than top down from a definition to data, which definition would be particularly suited to the values we find in Bell’s Theorem. Let me explain:
The conditions for elements of reality used in EPR were geared toward the particular problem they were studying, namely, simultaneous measurements of momentum and position, back in the historical setting when Heisenberg’s canonical commutation and complementary was still somewhat new and very perplexing. Bell was still several decades away. The “names” EPR particularly had in mind were momentum and position, which have dimension-full values of mass*length/time and length respectively. The “elements of reality” they had in mind were the “operators” which produced definite “eigenvalues” for various “eigenstates,” with like-physical dimensions of momentum and position. The additive properties of these eigenvalues is clear: we can add or subtract the components of momentum along x, y, or z to deduce a total momentum or a momentum difference, at will. These sorts of calculations are used in particle physics all the time, e.g., to calculate s, t, u, interaction channels. And likewise for lengths, we can add or subtract lengths along any given axis, and Pythagoras taught millennia ago how to do this in more than one dimension.
As to Bell, the “name” which seems most appropriate to Bell and CHSH are “correlations,” and these have the values +/-1, which indicate a parallel or anti-parallel correlation between measurements take by Alice along a particular direction, and by Bob along another particular direction. These are *dimensionless* numbers. When I add two correlations to either get +/-2 or 0, e.g., AB+AB’, I effectively have what looks to me to be a “counter” of correlations. +2 means that I counted two correlated measurements between Alice and Bob. 0 means I counted one correlated and one anticorrelated. -2 means I counted two anticorrelated. So the name I would give to AB and the like is “correlation counter.” When the variable we name “correlation” is +1 we have counted one correlation. When it is -1 we have counted one anti-correlation. When we add such numbers, we am effectively counting correlations and anti-correlations. Every time there is a correlation the number ticks up by 1. Every time there is an anti-correlation the number ticks down by 1.
Unlike Joy, I ignore the fact that if B(b) is know to outside observers, then B(b’) is hidden from outside observers. This is for three reasons: 1) By the postulate of realism, we have to regard these as both existing, even if they cannot both be simultaneously measured. 2) It actually does not matter because the important XOR I am seeing is not between B and B’ but between B+B’ and B-B’ which get to me heads and tail analysis. 3) A reason we cannot observe B(b) and B(b’) simultaneously is that what we are producing is sign(b.s) and sign (b’.s), with s being the doublet spin and b.s and b’.s being dot products. HR got me thinking about this, but I would not want to rule out that someone might design a clever experiment to measure the angle in b.s=cos theta directly, in which case one could also know b’.s and thus the signs of each. However: I also point out that the reason we have Alice measure a and a’ and Bob measure b and b’ based on independent, spacelike-separated, fair coin tosses, is to make sure that that Alice and Bob do not conspire to cook their experiments, which is how we encode the locality hypothesis. So an experiment discarding the separate measurements so that Alice and Bob know precisely about one another’s experiments in advance, seems as if it would also discard the locality hypothesis, which we cannot do if we are testing local realism with the disproof-by-contradiction that is Bell’s Theroem. Such an experiment would test realism only.
I agree with every word above and would not change anything were I to write this myself. All I would add are that these “values” are dimensionless integer-valued H and T “counters.” Would you agree to that one amendment, which does advance the definition of this potential element of reality by giving these values a specific name?
I like the way you wrote that as well. Am I correct to amend this to exhibit its temporal independence, by writing the following?
“If you want to see an analogy with a coin toss, then we have an analogy with a coin which has already been tossed, but whose outcome of heads XOR tails is still hidden to outside observers because we measure b XOR b’; or with a coin which has not yet been tossed but which we know will have an outcome of heads XOR tails which is hidden by the veil of a future which has not yet become past.”
Well, Richard, I do hope you reconsider because I find your input and insight extremely insightful.
Jay
Jay,
I always enjoy such “debates” between Bell-adherents and Bell-deniers. Whereas Bell-adherents strictly follow the stage directions given by Einstein to perform “classical” EPR thought experiments, the Bell-deniers or Bell-sceptics, who are commenting here on RW, steadily mash up quantum mechanical and classical thinking.
I am sorry, but there is no need to discuss about real or imaginary coins or dices with faces which cannot be at the same place at the same time. What are currently the elements of reality involved: The tosses? The coins or the dices themselves? The dices‘ faces or the numbers inscribed on the faces? Or another stuff ? Or does it depend on whether you toss the coin or dice in a S^3 or R^3 space? I understand Gill’s comment „I do not intend to follow the “debate” here any longer, since all I see is obfuscation and denial“. The “debate” is getting mor and more boring and predictable.
You can carry out thought experiments for an “external local reality independent of our observations” – that was Einstein’s intention – and you can use the “experimental” results obtained by physical thought experiments to predict outcomes which you can obtain by real physical experiments. Otherwise, there would be no need to speak of an “external local reality independent of our observations” or to invent something like “local hidden variables”.
Typical counterarguments like
“B + B’ and B – B’ do not satisfy even the first requirement of being physically meaningful. Physically they are impossibilities — or absurdities, let alone being elements of reality”
play no role, because non-commuting observables have no meaning in thought experiments for an “external local reality independent of our observations”. Such counterarguments are quantum mechanical diction. The fact that a real physical measurement performed on B “disturbs” – so to speak – B’, plays no role at all for an EPR-thought experiment. Simply follow Einstein’s stage directions. Such counterarguments are nothing else than tricks to keep “discussions” running when you’re a in a blind alley.
Serious consideration should be given to HR’s comment (HR December 5, 2016 at 12:21 pm) regarding the „fragments of exploding balls with their translational and angular momentums”
This would be an EPR-thought experiment. Replace the „high speed cameras“ by – casually speaking – „Einstein’s demon“. Einstein’s demon records the motion of the fragments at a prespecified location. He can now deduce a spin direction s, and then compute A(a) as sign(s.a), and at the same time compute A(a’) as sign(s.a’). The demon does the same for B(b) and B(b’).
Merely our knowledge about that „Einstein’s demon knows the definite set of [A(a), A(a’), B(b), B(b’)] values for every experiment“ allows us to carry out EPR-thought experiments and to derive Bell-type inequalities**. We don’t know the definite sets of values, but we know that each of A(a), A(a’), B(b) and B(b’) can only assume the numbers +/- 1. That’s all. No magic in, simple and straightforward with a pinch of probability theory.
Quantum mechanics, however, states it clearly and harshly: Don’t even think of “thought experiments for an ‘external local reality independent of our observations’”.
** To those who are really interested in Bell’s theorem, I would recommend to read Nick Herbert’s elegant and simple proof of Bell’s theorem: http://quantumtantra.com/bell2.html
Nick Herbert’s unpublished “proof” of Bell’s theorem is seriously flawed. I pointed out the flaws to him several years ago.
The problem with Nick Herbert’s proof is very similar to the problems highlighted in this thread by Joy. He derives an inequality involving probability terms (or relative counts if you like) but if you focus on the RHS and put back the term that was eliminated to create the inequality from the equality, you’ll see immediately that it represents an impossibility.
OK, I have been working toward this for almost two weeks, let me finally give it a try:
PROPOSAL FOR A SUFFICIENT DEFINITION OF REALITY, REVISION 1:
Whether observed or not:
The real occurrence of a physical event, represented by the logic statement “occurrence = TRUE,” is an element of reality.
The non-occurrence of a physical possibility, represented by the logic statement “occurrence = FALSE,” is NOT an element of reality.
The sum of an occurrence with a non-occurrence, and more generally of an element of reality with a non-element of reality, is NOT an element of reality.
Corollary:
It is a *necessary* condition of reality for a physical event, that that physical event occur in reality with certainty, i.e., with the logical attribute “occurrence = TRUE.”
Looking for feedback, and a consensus around a definition.
Jay
Read my comment “Lord Jestocost December 5, 2016 at 5:07 pm”
A physical quantity is a physical property/feature of a physical system that can be quantified by measurements, you get a value.
A “element of reality” is a physical property which can be quantified – following Einstein’s operational measurement procedure – “without in any way disturbing the system”.
Physical events or measurements have nothing to do with the notation “element of reality”.
Measurements just reveal the values of “elements of reality”.
#1) In case you have proved once that the velocity of a moving electron is an “element of reality”, you have proved it for all moving electrons. An electron is an electron! And velocity remains an “element of reality” for all moving electrons, no matter what you are planning to do or what you are really doing with the electron.
#2) In case you have proved once that the position of a moving electron is an “element of reality”, you have proved it for all moving electrons. And position remains an “element of reality” for all moving electrons, no matter what you are planning to do or what you are really doing with the electron.
#3) As an electron is an electron, you can conclude from #1 and #2, that both velocity and position are simultaneous “elements of reality” for all moving electrons. That’s the classical definition. Both are “elements of reality” and measurements or physical events play no role.
A Measurement on one “element of reality” does not “retroact” or does not affect the past in this sense that the “before measurement feature” of being an “element of reality” is erased for the other “element of reality”.
Position and Momentum are pontryagin duals which means there is a basis transformation involved between them with a fourier transform relationship between the basis. As established many years ago (Benedicks’s theorem, Hardy’s uncertainty principle, etc), a nonzero function and its fourier transform cannot both be sharply localized. It therefore makes no sense to talk of position and momentum as being simultaneous elements of reality for the same particle.
In fact, it is well known that it is impossible to make a device that can measure both position and momentum at the same time. In other words, momentum is not well defined at a given position for a given particle.
The physical system can and does include the measurement apparatus.
No According to EPR, if you can predict the value of a physical quantity “without in any way disturbing the system”, then that *physical quantity* CORRESPONDS to an *element of reality*. Nowhere do they say the *physical quantity* itself is an *element of reality*.
This is false. Many physical events and measurements *create* elements of reality. Most often, the physical quality is a result of an interaction between the measurement equipment and the “system”. EPR only say the *prediction* has to be done without disturbing the system, NOT that the physical quantity be obtained without disturbing the system.
According to EPR, the velocity of a moving electron CORRESPONDS to an “element of reality”. But not all elements of reality are independent of the measurement process. For some measured physical quantities, the element of reality to which it CORRESPONDS is in fact created as a result of the interaction with the measurement apparatus. It will be terribly wrong then to argue that unmeasured electrons possess such elements of reality. At best, they can possess the potential to produce these elements of reality on measurement. Your statement above is therefore generally false.
See previous point for a refutation of this.
Therefore your conclusion is completely false. If the act of measure creates an element of reality due to interaction with a measurement device, while at the same time destroying the original “system”, rendering the second mutually exclusive measurement impossible. It will be impossible for both “elements of reality” which were *possibilities* before measurement, to become actualized. It is therefore false to conclude that both can ever be simultaneous *actualities*. Contrary to your claim, measurement on one element of reality does not retroactively falsify the previous *possibility* that wasn’t measured, it simply renders it *counterfactual* and impossible to actualize. No element of reality is erased in the process. Before measurement, you had two *possibilities*. After measurement, it is still true that you had two *possibilities* before you measured. But in addition, you now have one *actuality* corresponding to one of the possibilities, and another counterfactual possibility.
Note the fact that the measurement can’t be done anymore does not amount to erasure of the possibility. For example, the *possibility*
If I measure at (a) I will obtain A.
expresses the truth of relationship between “I measure at (a)”, and “I obtain A”. It absolutely does not express any truth about “I measure at (a)” alone, or “I obtain A” alone. Measuring at (b) does nothing to negate that truth, nor does obtaining B negate that truth unless you also measured at (a). The part that is rendered impossible after measurement is the antecedent (“I measure at (a)”). Thus even though it is no longer possible to measure at (a), it is still true that “had I measured at (a), I would have obtained A”. This is a counterfactual statement.
Jay I think your definition of reality is too strong. I suggest you use my characterization and differentiate between
*possibility*
*actuality*
What you have defined above is *actuality*. According to your proposed definition, there is no room for probability or reasoning from incomplete information. In fact, there is no room for predictions in that view. Theories deal in *possibilities*, experiments deal in *actualities*. Only mutually compatible (or commuting) *possibilities* are simultaneously measurable and can become *actualities*. Non-commuting or mutually incompatible *possibilities* cannot. That means a sum of *possibilities* is only allowed to be compared with a sum of *actualities* if the possibilities commute (are compatible).
This view is more complete, as it provides a framework for theories to make experimentally testable predictions. Your suggested definition of reality is essentially Dewey’s strict instrumentalist view, which basically says, all that exists is what is measured.
Joy has argued that. Read what I have written closely: I have disagreed with Joy and stated that this argument is unnecessary and irrelevant. I am making a different argument that you are trying to tar with the “deniers” pejorative. I am forming my own independent views on this whole subject. The reason I stayed out of the Bell debates for years is because of people hurling accusations around like that.
LJ: That is the point. Bell, as a disproof-by-contradiction, postulates an “external local reality independent of our observations.” Then it makes some deductions, finds a purported contradiction, and thereby disproves local realism. So as Richard has argued correctly, we must have a good definition of realism. And as Richard has also argued correctly, EPR was clear that their definition of reality was *sufficient* not *necessary*. So these definitions are perfectly well on the table for all of us to discuss here.
I have done just that in my December 5, 2016 at 5:17 pm post, and this is something that Richard has consistently and correctly asked for.
Jay
See above for the explanation of what is wrong with it.
Let’s try this; I would be happy for others to edit this if the basic framework is on the right track:
PROPOSAL FOR A SUFFICIENT DEFINITION OF REALITY, REVISION 2:
Whether observed or not:
The real occurrence of a physical event, or the prediction of that event without disturbance and with certainty, represented by the logic statements “occurrence = TRUE,” or “prediction with certainty = TRUE”, corresponds to an element of reality.
The non-occurrence of a physical possibility, represented by the logic statement “occurrence = FALSE,” or the certainty that a physical event will not occur (i.e., that the probability for such an event is zero), does NOT have a corresponding element of reality.
A possibility which may or may not occur, i.e., for which a prediction with certainty cannot be made, but which has some non-zero probability of becoming an occurrence, may or may not correspond to an element of reality, i.e. This is unknown without further information.
The sum of an element of reality correspondence with a not-element of reality correspondence, does NOT have a corresponding element of reality.
Corollary:
It is a *sufficient* condition of reality for a physical event, that that physical event occur in reality with certainty, i.e., with the logical attribute “occurrence = TRUE,” or that it be predicted with certainty, i.e., with logical attribute “undisturbed prediction with certainty = TRUE.” Otherwise, the correspondence of that event with reality is presently unknown, or there is no correspondence.
I will think about these more and see if I can consolidate, and again, I’d like to arrive at a draft of such a definition that gains consensus from Bell and non-Bell adherents alike. That’s what “mediators” try to do.
I now have two questions, based on EPR page 778 second full paragraph:
“If psi is an eigenfunction of the operator A, that is, if
psi’ = A psi= a psi, (1)
where a is a number, then the physical quantity A has with certainty the value a whenever the a particle is in the state given by psi. In accordance with our criterion of reality, for a particle in the state given by psi for which Eq. (1) holds, there is an element of physical reality corresponding to the physical quantity A.”
1) MF, in your reply to LJ you emphasized the word CORRESPONDING several times. Please state precisely what you see as the operational significance of the use of that word.
2) Is not the “element of reality,” or maybe just the “reality,” the logical truth either that a state with (eigen)value a has occurred, or, without disturbance, can be predicted with certainty to occur? That is: observed or not, the occurrence or the inevitability of the state with eigen(value) a?
Jay
Jay, here is an easy answer to your questions. This of a particle in 3D space, and there are 2 observers each with a different reference coordinate system. Both observe the particle at it’s position. Observer 1 in with is own arbitrarily oriented coordinate system writes down the coordinates (x1, y1, z1) for the particle. Observer 2 in his own differently defined coordinate system writes down coordinates (x2, y2, z2) for the particle which will differ from (x1, y1, z1) . Both observations occured and are actualities. Now the question: is (x1, y1, z1) the element of reality, or is it (x2, y2, z2). Or is there an underlying element of reality which both observations CORRESPOND to. Now let us eliminate one of the observers and say there was just a single observer. Does that change anything? What if there is just a single theory by which all observers are observing?
I would say it doesn’t change anything as we couldn’t possibly enumerate all possible theories and all possible physical quantities in those theories. Someone may come along an conceive of a theory which does not have entities such as “position” or “momentum” or other such physical quantities which we now take for granted but instead other physical quantities we haven’t yet imagined. In that case, our definition of reality should be robust enough for account for that. This is the importance of the word “CORRESPONDS” which is often overlooked in the EPR paper. It allows for the possibility that there are elements of reality which we can never directly measure or observe, even though we might perceive their effects in our limited capabilities. Of course you could argue that the perceptions are also real, but what is real is not the perceptions but what the perceptions correspond to as I explained above. If the perceptions were real, it would be impossible for different observations of the same reality to disagree. Two *actualities* can never be in conflict, by definition.
So, MF, is it your view that “corresponds” just subsumes the many-to-one mapping of possible observers to a single actual or inevitable event, i.e, is a statement of observer-independence or invariance?
Also, I like your revision 2 much better!
One other set of questions I have, specifically regarding the correlators such as A(a)B(b), which is intended to come to a common understanding about the best language practices to use when talking about elements of reality in Bell.
Let us talk ONLY about A(a)B(b) for right now. I will call this a “correlator.” I will state that the correlator is permitted to have the value +1, or the value -1. I will state that if the correlator actually has, or can be predicted with certainty and without disturbance to have, the value +1 and not the value -1, for a specific trial event, that A(a)B(b) = +1 = TRUE is the element of reality (or just “the reality?)) for that trial event and A(a)B(b) = -1 = FALSE is NOT an element of reality for that trial event. I will state that if none of the forgoing is known with certainty, that then it is UNKNOWN whether A(a)B(b) = +1 or A(a)B(b) = -1 is the TRUE element of reality, but that it is known with certainty that either A(a)B(b) = +1 or A(a)B(b) = -1 is the element of reality and the other is not an element of reality or that trial event.
I’d like to know what problems there are with those statements, if any, and how you would fix them.
And even more generally, we know that in physics, some things are true, some are false, and some are unknown or unknowable. Underlying all of this, I am really trying to ferret out a universally agree mapping between physically TRUE, FALSE and UNKNOWN events or values, whether “past” or “future,” whether “observed” or “not observed” or “not observable,” and “reality” and / or “elements of reality.” But I would like, if possible, the central hub to be the logic propositions “TRUE,” “FALSE BUT POSSIBLE” “UNKNOWN BUT POSSIBLE,” “UNKNOWABLE BUT POSSIBLE,” and “IMPOSSIBLE THUS FALSE.” Am I missing any?
Jay
How about this? Simple and complete?:
Any value for an event or an event possibility which is TRUE is an element of reality.
Any value for an event or and event possibility which is FALSE is NOT an element of reality.
Any value for an event or an event possibility which is UNKNOWN OR UNKNOWABLE, bears an UNCERTAIN relation to, and may or may not be, and element of reality.
Jay
The sufficient criterion of reality provided by EPR is quite sufficient for my main argument. It tells us that the quantities B + B’ and B – B’ do not satisfy EVEN the sufficient criterion of reality. EPR were very wise to stay well clear of forming a “necessary criterion of reality.”
B + B’ is measurable. If B and B’ are spin components of a particle in two directions, B + B’ can be measured by measuring the spin component in the direction that is half way between those two. In addition, B + B’ is predictable if the entangled particle can be measured.
The spin component B” in the direction that is “half way between” b and b’ would be an altogether different, or third, counterfactually possible component of the spin, say along the direction b”, which would be exclusive to both b and b’. Such intermediate measurements cannot overcome the impossibility of actualizing B and B’ simultaneously, even “if the entangled particle can be measured.” Actualizing B” cannot actualize B, B’, B + B’, or B – B’.
Three quotes from MF’s comment “MF December 5, 2016 at 9:23 pm”:
Quote#1:
“In fact, it is well known that it is impossible to make a device that can measure both position and momentum at the same time. In other words, momentum is not well defined at a given position for a given particle.”
Quote#2:
“This is false. Many physical events and measurements *create* elements of reality. Most often, the physical quality is a result of an interaction between the measurement equipment and the “system”.
Quote#3:
“According to EPR, the velocity of a moving electron CORRESPONDS to an “element of reality”. But not all elements of reality are independent of the measurement process. For some measured physical quantities, the element of reality to which it CORRESPONDS is in fact created as a result of the interaction with the measurement apparatus. It will be terribly wrong then to argue that unmeasured electrons possess such elements of reality.”
Answer: As a said before: Bell-deniers or Bell-sceptics, who are commenting here on RW, steadily mash up quantum mechanical and classical dictions!
One quote from Joy Christian “Joy Christian December 5, 2016 at 6:07 pm”:
Quote#1:
“Nick Herbert’s unpublished “proof” of Bell’s theorem ( http://quantumtantra.com/bell2.html ) is seriously flawed. I pointed out the flaws to him several years ago.”
Answer: It is an “experimental” proof, you can in principle perform such experiments. I think it will be much easier to implement Nick Herbert’s proposed experiment than to implement experiments on the angular momentums of small macroscopic fragments of exploding balls.
One quote from Jay R. Yablon “Jay R. Yablon December 6, 2016 at 12:15 am”:
“How about this? Simple and complete?:
Any value for an event or an event possibility which is TRUE is an element of reality.
Any value for an event or and event possibility which is FALSE is NOT an element of reality.
Any value for an event or an event possibility which is UNKNOWN OR UNKNOWABLE, bears an UNCERTAIN relation to, and may or may not be, and element of reality.”
Answer: If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity. As these “elements of reality” belong to Einstein’s “external local reality independent of our observations”, you can use these “elements of reality” and thus the corresponding “physical quantities” to perform physical thought experiments for an “external local reality independent of our observations”. That are the ingredients which you need to derive Bell-type inequalities by means of thought experiments. Observations or real experiments have nothing to do with such derivations. To compare the outcomes of “real physical experiments” with the outcomes of such “physical thought experiments” you can rely on Gill’s approach.
A thought experiment is not much help if it doesn’t correspond to reality. And the reality is that it is mathematically impossible for anything to “violate” a Bell inequality. Please demonstrate exactly how QM or the experiments actually do this supposed violation. It is pretty easy to show that they use a different inequality with a higher bound than |2|.
You talk of “mashing up dictions”. Specify exactly how. It seems to me you are just handwaving. The whole point of the discussion is that Bell proponents switch meanings of terms on the fly and end up comparing apples and oranges that shouldn’t be compared. When understood under a framework of comon definitions and logic, there is absolutely no conflict between QM and local realistic theories. That is the point.
Gill’s approach is flawed, as I have already explained. To compare two things with each other, you have to be using the same definitions of terms. The terms from Bell’s inequalities including the inequality derived by Gill (in https://arxiv.org/abs/1207.5103) are very different from those predicted by QM and measured in experiments. The comparison is thus invalid. I’ve explained already the reasoning behind the criticism. Once all the definitions are straightened out, as we have done in this thread, it becomes obvious that QM and Experiments do not violate Bell’s inequalities. They can’t. In fact, there is no way in QM or Experiments to produce the kind of terms present in Bell’s inequality.
Interestingly, Gill proved an earlier inequality which confirms this point. The appropriate upper bound for the kinds of terms produced by QM and experiments is 4 not 2.
See equation (11) of https://arxiv.org/pdf/quant-ph/0312035v2.pdf
Joy, You and I disagree here. You can concede B+B’ and B-B’ to be elements of reality as a hypothesis, and still argue against Bell, because H = ( B + B’ )/2B and T = ( B – B’ )/2B with 1=H+T remain mutually-exclusive outcomes as a matter of pure logic, divorced from real versus thought experiments, divorced from measured or not measured and not measurable.
I agree with you that EPR were very wise to stay well clear of forming a “necessary criterion of reality,” and am thinking that I would be wise to do the same. I actually now think it best to focus on a single question about one very particular type of quantity and whether that is an element of reality, because if it is not and element of reality, then all of the other arguments presented here becomes completely and totally irrelevant. I will state that in the separate post to follow momentarily.
Jay
I am no longer going to try to define element of reality in general as either a necessary or sufficient condition. Instead, I would like to cut to the chase with one very simple question, laid out with its predicate, below:
Posit a real experiment or a thought experiment, I care not which, which involves carrying out discrete “trials,” the number of trials to be designated by N. For example, carry out 10 million coin tosses, so that N=10 million. Or 10 million rolls of a die so that N=10 million. Or 10 million EPR trials using the slip and urn model if you wish in which case N=10 million slips are drawn from an urn. Keep in mind, I have said not a word about what the coin toss or the die roll or the slip draw results actually are, e.g., how many heads and tails came up, how many times the die landed on 4, or what + and – signs, hidden or not, were on the slips.
I now have one every simple question for all the rocket scientists here: By ANY reasonable definition you care to use:
Is N=number of trails, an “element of reality”?
Yes or no? And by what reality criteria do you put forward your answer?
Jay
My answer to your question is, No. I put forward my answer using the EPR criterion of reality.
Just so your position is totally clear for the record, please state directly and precisely and specifically how and why the EPR criteria leads you to conclude that N=number of trails is NOT an “element of reality.”
My answer concerns actual experiments. We can imagine that the source emits M pairs of particles, but this number M is usually unobservable in experiments counting microsystems.
In practice the number M of particle pairs emitted during a given time interval is deduced from the counting data for that period, since intervening counters will in practice depolarize, if not totally destroy the systems. But this deduction always depends upon the state-of-the-art theoretical description of the entire phenomenon — the source, the apparatus, and their non-trivial interactions. With that in mind, the only theory-independent way of inferring the actual number N of particle pairs detected is by using the EPR criterion of reality, which says:
“If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.”
Now your question concerns N = number of trial, which is not necessarily the same as M = number of particle pairs emitted by the source. But there is no way an experimenter can deduce the difference M – N. I am aware of the fact that recent “loophole-free” experiments employed some nifty tricks in this regard, so I will be happy to be corrected about this.
Given this background, if we now apply the EPR criterion of reality to the physical quantity representing the number N, then it is clear that one cannot deduce that number without disturbing the entire system. Consequently, there is no way of predicting N a priori. But if N cannot be predicted, then according to EPR we cannot say that it is an element of reality.
But if we now consider an actual experiment with exploding macroscopic balls, would there be any problem with predicting N apriori? Remember Jay told us to consider any experiment whatsoever.
Of course there would be. The only number one can predict with certainty in my proposed macroscopic experiment is the number M, not the number N of trials in the sense of Jay.
I would have no problem considering an experiment in we use M=pairs actually emitted and harvested, because I am not wedded to the part of the EPR definition that makes predictability part of the sufficient condition for reality. I am happy to use either M or N in the sense of Joy, and I do not think it affects the overall result one way or the other.
I would like to further tune what I said above, to ask whether the below is a fair statement that can be agreed to by everybody:
If we do a N = 10 million EPR-Bell trials, we are actually measuring 10 million physical systems. So number of trials = number of systems. Therefore, with N = 10 million, we are measuring the possible or actual properties of 10 million physical systems. The N=10 million experimental or thought experiment trials that one undertakes to measure N=10 million physical systems has nothing to do with the possible or actual properties of those physical systems. Therefore, the number of trials, i.e., the number of systems considered, is meaningless in relation to any elements of reality corresponding to those systems.
Fair statement?
Jay
Yes. However, in an EPRB scenario, N is not even usually known and is estimated taking it even further from ever being an element of reality. But as for each individual system, it has absolutely nothing to do with their reality.
I have to say that I find that so much of the discussion about Bell after six weeks in this muck – much more than I am used to with other physics problems I have worked on – is tied to the written language that people use to talk about these experiments, and people using their own personal languages and talking past each other. So it really is important to try to facilitate agreement about language before it can be fruitful to have a discussion about results. And because Bell finds that “local realism” is contradicted, the definition of realism is all-important. Richard or LJ or HR can define reality in a way that local realism is contradicted. Joy or MF or Fred can define local reality in a way that is not contradicted. And it all hinges on the definitions, and the objective reasonableness of these definitions, which is also difficult to quantify.
In order to obtain greater quantification and less metaphysical argumentation, would there be any benefit to talk in terms of “local hidden variables” rather than “local realism.”? How, exactly, might a framing in terms of hidden variables rather than realism give us more precision?
Thanks,
Jay
If you really would like to “cut to the chase” then ask, “Please demonstrate exactly how QM or the experiments actually do the supposed violation of Bell’s inequality.” It is pretty easy to show that they use a different inequality with a higher bound than |2|. This should hopefully eliminate some “language problems” but we will see. 🙂
I find it ironic that people are so upset about the words “local realism”, when Bell himself never used it, and none of the early authors. In his original paper Bell used the words “determinism” and “separability”, and in the CHSH paper I think they used “determinism” and “locality”. If you go a bit forward in time to Clauser and Horne’s 1974 paper you will see “factorizability”, and in Bell’s 1976 paper “local causality”.
One other question I have been asking myself lately, so let me ask it of everyone:
A singlet “system” splits into two doublets. Alice and Bob are measuring that “interrelated” system for which some would use the conclusory word “entangled.” That system would be the “reality.” Then, Alice can look at it from a or a’ and Bob from b or b’ but they are all observing the same system, simply from different spacelike-separated reference frames. And a versus a’ and b versus b’ are just rotations of the detectors, thus different viewpoints. With some qualifications I will add below.
SO: Let’s talk about A(a)B(b) all by itself. If this is detected to be equal to +1, that means that an element of the reality of this system is that when Alice does a and Bob does b they measure correlated results. We can call this
correlation (a,b) = +1 (1)
Name, and value, that latter being dimensionless. And, an element, namely the (a,b)-view, of the reality that is the doublet.
So far so good?
So, when I now add
A(a)B(b) + A(a)A(b’) (2)
what is the *name* for that sum? We are adding two elements — two views — of a single reality. What do we call that sum, which of course is the first half of CHSH? When I add two momenta, the result is still a total momentum. When I add two correlations, the result is not a total correlation, at least defined as +1 = parallel and -1 = anti-parallel. These binary EPR-Bell systems do have important character differences from continuous dimensional physical quantities.
The other thing that strikes me as I go back over part 2 of the EPR paper again, is that there is an important reason for having both a and a’ measurements, same with b and b’. EPR used wavefunctions, and their point was that measuring Alice’s wavefunction in one eigenstate with one eigenvalue versus in a second eigenstate with a second eigenvalue, actually seemed to impact what BOB would measure even though these systems were no longer in physical communication. They used that to argue the wavefunctions are incomplete, but today most people understand Bell to tell us that, no, this is the leading indicator of entanglement, confirmed by the non-linear quantum correlations with 2 sqrt(2) = 4*sin(pi/4) = 4*cos(pi/4) bounds whereby Alice and Bob’s measurements actually do affect one another.
Am I fairly stating this in a way that both sides can more or less agree with.
And if I cannot get agreement, then can we all at least agree that it is December 2016 in the English calendar on the planet earth, and try to make some progress from there? 🙂
Jay
A(a)B(b) + A(a)B(b’) is pretty “meaningless” without any indexing.
Index = 1 for all both. One EPR-Bell trial.
Jay,
The key is that the additivity of expectation values is non-trivial and cannot be justified for incompatible elements of reality. Ironically, this is the very reason why von Neumann’s own no-go theorem collapsed, at the hands of Bell himself mind you (see http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Bell%20(1966)_Hidden%20variables.pdf).
Now to your question. When you say “system” what do you mean. Are you talking
1) A specific pair of particles in the singlet state?
OR
2) A series of particles in a singlet state?
The distinction is very important because many confuse those two to grave consequences.
When you talk of “expectation”, you have to revisit the previous question in another form, are you talking about
1) The theoretical expectation from measuring A specific pair of particles in a singlet state
OR
2) The experimentally observed “expectation” value after measuring a series of particles in a singlet state.
If you are familiar with the Einstein-Bohr debates, you might recognize that we are going back to these foundational issues. There are significant difference between those answers that impact this discussion profoundly, especially in the case of Bell’s theorem in which you have to compare theoretical predictions with experimental results.
The expression A(a)B(b) + A(a)B(b’) is supposed to represent a theoretical number calculated from a specific pair of particles. That number expresses a true relationship 3 properties of the system. A(a), B(b), B(b’). But then there is a problem, A(a) is a property of a system which includes just one of the particles PLUS a measuring device (a) at Alice’s side, while B(b), is a property of a different system which includes the only OTHER particle, PLUS a measuring device at Bob’s side (b). And also, B(b’) is a property of yet another system, with includes the previous particle, but this time, a different measuring device (b’).
Therefore the number A(a)B(b) + A(a)B(b’) represents 3 different systems, two of which are incompatible because the system of which B(b) is a property, can’t exist simultaneously as the system for which B(b’) is a property. The measuring device can’t have both b and b’ set at the same time. At best, B(b), B(b’) represents 2 mutually exclusive possibilities.
But let us take for granted the suggestion from the other side, that A(a)B(b) + A(a)B(b’) are just a bunch of numbers we want to multiply and contemplate together in an expression. In that case those numbers express a certain logical relationship within those 3 systems involving 2 singlet particles and 3 measuring devices (combined system). It would be unreasonable to expect the same that relationship to hold if we replaced one of the particles with another particle from a different singlet pair, let alone if we replaced one of the measuring devices with a completely different one. It becomes clear why the following expressions make all the difference.
⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ ≤ 2
⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ ≤ 2√2
One analogy: Say a zoologist wants to study the relationship between the head-diameter (H)and the body-length (L) of snakes in the amazon. He writes down an expression relating the two in the form of a ratio.
H/L
This expression represents and internal property *within* a snake, and absolutely nothing *between* snakes. Of course he could do statistics by measuring a large number of snakes, and doing statistics on the numbers. However, there is no way to recover the above relationship in any experiment if it is impossible to actually measure H and L on the same snake. Of course in the case of snakes, we can think of ways of measuring both together, but assume that the two properties were mutually exclusive in the sense that only one could be measured. In that case, while H/L may be an interesting number to write down and contemplate theoretically, it would actually be very foolish to measure H on one set of snakes, L on a different set of snakes and think that law of large numbers will allow you to reconstruct the relationship *within* snakes by using only information measured *between* snakes.
One final point: Perhaps there exists a Grand Wizard of The Amazon (GWTA) who made all these snakes according to specifications he has stored in his lab-book. For each snake he has the exact measurements of H and L he used. Then, H and L are elements of reality, H/L is an element of reality. Perhaps our limited experimenter, has a theory which can predict with certainty for each snake what H will be if measured, and what L will be when measured, even though he can’t measure both. In that case H and L are possible elements of reality, elements of reality, but they can’t both be actualities as observed in a measurement. H/L is meaningful to GWTA but will never correspond to anything that our experimenter will ever observe.
Therefore the issue here is less one of the meaning of element reality, and more about consistency in reasoning from derivation of relationships, to comparison of those relationships with experiments. Bell’s theorem is essentially the derivation of a relationship such as H/L from the perspective of GWTA, and then using results from a measurement performed by the experimenter to claim that snakes do not have head and lengths simultaneously — ie derive a an inequality using a relationship *within* particle pairs, measure outcomes entirely *between* pairs, and use the data to conclude that the relationship *within* pairs is false.
Is that reasonable?
MF, A lot there to digest. But starting with the above highlight, how do you answer the guys on the other side of the net who say that you can use statistics to have ⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ approach ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ in the limit of very large numbers of trials? Jay
Keep reading Jay, you’ll find the answer further down:
In short:
You answer them the same way you answer the person who claims he can recover the H/L relationship *within* snakes by measuring H and L separately *between* snakes. You answer them by pointing out the absurdity of claiming to have falsified the joint existence of A₁B₁A’₁B’₁ *within* the same system, by using data which never measured A₁B₁A’₁B’₁ jointly *within* any system.
MF, one other question, which for is a point of clarification from your side.
From what I can tell, you guys really have two main beefs with Bell. But I want to make sure this is what y’all are saying:
First, even if we never knew a thing about the 2√2 QM limit, it seems to me that you believe there is an inherent logical contradiction embedded in the very formation of the sum ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩, because you are adding “elements of reality” (recognizing the definition disagreements) and regarding the sum with three adds and one subtract to also be an “element of reality” upon which one can then base a finding of contradiction from ANY external data, the 2√2 limits or otherwise, when in fact that sum is not an “element of reality” by any fair and reasonable definition? In other words, you kill Bell in the cradle because what it tests for contradiction is not realism?
Second, you have the disagreement that the other side has “falsified the joint existence of A₁B₁A’₁B’₁ *within* the same system,” and so is using a bogus expression to do their falsification. So even if Bell gets out of the cradle, the falsification argument is based on a false use of statistics?
In other words, are you all saying that the Bell side is a) using a falsified statistical argument that turns ⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ into ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩, and is ALSO b) using a false argument that ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ is an element of reality, to claim that they have disproved local realism by contradiction?
I am not arguing either side, I am trying to make very clear to myself and others the main parameters of the claim of the contra-Bell side.
Jay
PS: I assume you use latex codes, but what meta-tags do you surround that with to get real-looking equations instead of ASCII gobbledygook?
OK, MF, I have reproduced my earlier statistical calculation using the law of large numbers, which Richard Gill has confirmed is in his view correct. Can you specifically pinpoint the flaws that you believe are in this calculation. Thanks, Jay
So Jay, how is your quest for the “low-hanging fruit” going? Recall that you embarked on the quest for the “low-hanging fruit” following the suggestion by Stephen Parrott quoted above. Going back to the very beginning of this thread, please have a look at the length of time and effort you have devoted in understanding Bell’s unjustified claim against Einstein’s local realism compared to the length of time and effort in understanding my 3-sphere model.
The physical question at stake here concerns quantum correlations E(a, b) = -a.b, not some physically meaningless inequalities considered by Bell and his followers. My model proposes directly a physically illuminating understanding of ALL possible quantum correlations, not just the simplest possible singlet state correlations E(a, b) = -a.b considered by Bell.
Even if we concentrate on the singlet state, the physical question is about the actually observed strong correlations E(a, b) = -a.b, not some misguided inequality misapplied to physics. With that in mind, wouldn’t it be a better service to physics to actually study my paper which was withdrawn from Annals of Physics and try to understand my model?
Wouldn’t it be more beneficial to study the reasoning behind the following manifestly local-realistic simulation of my model for the singlet state considered in the withdrawn paper?
http://rpubs.com/jjc/233477
Note that the above is a simplified version of this simulation: http://rpubs.com/jjc/84238.
Regarding JC’s new simulation, it is easy to see the trick. The detection results are not dichotomic! For example, he makes this assignment:
A = +sign(h(a,e)*g(a,e,s)) # Alice’s results A(a, e, s) = +/-1
But in fact it produces 0 outputs as well, which you can know by looking at the definition of the sign() function as well as simply printing out A after this. So the comment on the line is false. Same for B. Then when he calculates the correlation, all pairs of outcomes with either A or B equal 0 are effectively excluded. So it is just the old detection loophole cleverly (or not) disguised.
My simulation has nothing to do with any loopholes, let alone the detection loophole.
There are no zero-outcomes in my simulations, as anyone can verify for themselves:
http://rpubs.com/jjc/84238
See also http://rpubs.com/jjc/105450 and http://rpubs.com/jjc/99993
To understand Graft’s confusion, please read the paper: https://arxiv.org/abs/1405.2355
I’m talking about http://rpubs.com/jjc/233477, which you just pointed us to. As I said, it is easy to show that 0 outcomes are generated. When this error is fixed by mapping the 0 outcomes to (say +1) then your deceitful correlations disappear.
You are making a judgement based on incomplete knowledge. Please read the paper Joy linked to.
As I have stressed, there are no zero-outcomes in my simulations. The latest version, namely
http://rpubs.com/jjc/233477 ,
is just a simplified version of the more detailed version linked above. The full version has been cited at the very beginning of the simplified version. There is no point in repeating the demonstration of the non-existence of zero-outcomes for the simplified version, since it has been done already for the full version: http://rpubs.com/jjc/84238 .
Unless one understands that all my simulations are simulations of the quaternionic 3-sphere model, it is difficult to understand what exactly is being simulated in them. Needless to say, a numerical simulation is not the model itself. It is merely an implementation of the model.
If one adds:
print(A)
after the assignment to A in the 233477 simulation one will see lines like this:
0 -1 0 1 -1 -1 1 -1 1 -1 1 0 0 -1 0 0 0 1 0 1 0 0 1 1 …
Anyone with R installed can do it. It is perfectly obvious that 0 outcomes are generated.
And to top it all off, instead of normalizing by simply taking N as a fixed constant equal to the number of source events the simulation sets the following factor:
N = length((A*B)[A & B])
N should not need to be conditioned by the outcomes, unless one needs to correct for the 0 outcomes that were excluded.
So it’s obvious why we are now being directed away from 233477 and back to earlier simulations where it is not dichotomous results that are correlated. Bell’s theorem applies to dichotomous measurements. One cannot correlate things in an analog space and then pretend that it will apply to dichotomous measurements as well. Basic statistical theory tells us that dichotomization produces a big loss of correlation. Al Kracklauer also made this mistake. He correlated the analog electric fields and obtained -cos(theta). But that does not survive dichotomization.
There are no zero outcomes produced in any of my simulations. It is easy to see that there is one-to-one correspondence between the initial state (e, s) and the outcomes (A, B) observed by Alice and Bob. Thus not a single initial state (e, s) goes undetected, unlike in the detection loophole model. Of course, if one insists on changing my simulation to a totally different one as Graft has done above, then it is not my model that is being simulated in the first place.
Joy,
I think the Yiddish expression that my grandma or my mom would have used is “Oy vey ist mir!” 🙂 If this is low hanging fruit I’d hate to think about the ones at the top of the tree. But I have learned what much of noise and fury about Bell that I assiduously stayed away from all for all those years — before I had a moment of weakness — is all about. 🙂 And I never regret learning something new about science, and the sociology of science.
As I have opined in the past, and will do again here, setting science-qua-science aside, Joy, you have a sociological problem here. The sociology is such that right or wrong, Bell’s theorem has been elevated to gate-keeping position in the minds of many, so that absent a priori proof that Bell in isolation is erroneous, the rule of thumb is to not waste time on any local realism / hidden variables theory. And Stephen Parrott was most certainly channeling that prevailing view.
I think the one place where your S^3 model could perhaps raise interest even to the many who take the same position as Stephen, is if you can use the S^3 model itself as part of the disproof of Bell that you have been advocating for the last nine years. But it would need to be a very direct and clear disproof in order to hold attention.
Recently, I have focused on how it might be that a measurement by Alice could impact what Bob sees even without any supposed communication. That is the puzzle about which EPR first articulated their paradox. Presumably, your S^3 model would have to explain — to couch a phrase that Wheeler might have liked — “entanglement without entanglement.” If you can step back from the weeds of your theory, and show how it leads — as it must — to strong correlations without communication, and illuminate that as directly and clearly and intuitively as possible, that could be a point of interest. Then, you can talk about how that back-washes to Bell-proper. In other words, don’t go directly after CHSH which we have all been beating into the ground. Go after it through entanglement.
In sum: either disprove Bell before you get to talking about S^3, or go from S^3 to a Bell disproof very directly. And at least in terms of what captures my interest and likely that of others, take on strong correlations directly, and really show ***entanglement without entanglement*** in the most direct and simplified way possible. If you can make it simple enough for me, perhaps I can find some ways to make it simple for others who may not be as simple as me. 🙂
I have been tending to my day job this week and picked up a cold which slowed me down, so it may be a few more days before I catch up here. I know I owe MF a reply on his critique of Richards’s use of statistics to collapse the disjoint CHSH indexes into a single range of indexes and thus the EPR-Bell trial readings into the slip and urn metaphor.
Jay
Yes and Yes and for the last point, I’m using unicode characters https://en.wikipedia.org/wiki/Mathematical_operators_and_symbols_in_Unicode
Polling is not a good analogy. Coin tosses are not a good analogy either. Unless you adopt the earlier example of I gave you of the coin reading machine. There is no use for an analogy that does not capture the salient features being discussed. In other words, you’ll simply be proving what you assumed. Now if you can redo your derivations using my coin-reading machine example, then we will be off to a good start and you may even discover in the process what I’m talking about.
Notwithstanding my objections above, let us examine the above. First I need an admission from you. Let Ai, Bi be the outcome of a coin toss where Ai = 1 means Heads Up, and Bi =0 means Tails down.
1) Do you agree that for each specific toss of the coin (i), the relationship Ai + Bi = 1 must hold? (ie, *within a toss*
2) Do you agree that the relationship Ai + Bj = 1 does not necessarily hold if i =/= j ?(ie *between tosses*)
Now the issue is not just one of writing an expression and claiming that the value will converge to a population value due to the law of large numbers. Your expression (1) is not the issue, as it doesn’t say anything about the relationship. For that, you need the RHS of the inequality. Say for example:
⟨AiBi⟩ + ⟨A’iBi⟩ + ⟨AiB’i⟩ – ⟨A’iB’i⟩ ≤ 2
which is derived by reasoning as follows
⟨AiBi⟩ + ⟨A’iBi⟩ + ⟨AiB’i⟩ – ⟨A’iB’i⟩ = ⟨Ai(Bi + B’i) + A’i(Bi – B’i)⟩
wherein, either (Bi + B’i) = 2 when Bi = B’i or (Bi – B’i) = 0 when Bi =/= B’i and therefore
⟨Ai(Bi + B’i) + A’i(Bi – B’i)⟩ ≤ 2
I repeat the derivation here just so that there is no doubt that the reasoning behind the derivation, is based on the relationship *within* the particle pair (i). You can verify Gill’s equations (1) and (2) in his paper to confirm that he uses the same reasoning. Secondly, you have to note that each average in that expression amounts to a cyclical combination of pairs of columns of a 4-column spreadsheet with columns Ai,Bi, A’i, B’i, and the cyclical nature of the recombination in the expression ⟨AiBi⟩ + ⟨A’iBi⟩ + ⟨AiB’i⟩ – ⟨A’iB’i⟩ implies that the data used to calculate ⟨AiBi⟩ is not independent from the data used to calculate ⟨A’iBi⟩,⟨A’iBi⟩ and you the same proceeds to the next term in a cyclical manner. This dependence, is simply an extrapolation of the *within* relationship to a larger number of particle pairs and is the reason why the upper bound continues to apply to the averages. Another way of looking at it is the following: Let us assume that you randomly pick the first set of particle pairs for measuring AiBi. This means you now have one of the columns needed for A’iBi and the other column needed for
AiB’i. In other words, the sets of data AiBi, A’iBi, AiB’i are not independently freely chosen; they have a lower number of degrees of freedom than three randomly chosen sets of data pairs AiBi, A’jBj, AkB’k where i =/= j =/= k EVEN IF those were randomly chosen from the same spreadsheet.
So we have reached our first major problem with the statistical argument: The reason for the upper bound of 2, is precisely due to this reduced number of degrees of freedom. It is made worse (not better) by random sampling of a so-called population, as random sampling necessarily gives you a higher number of degrees of freedom, with less dependencies between the sets of data generated. You can randomly sample as many millions of data points in each set of pairs as you like and it won’t change a thing. The upper bound is determined by the dependence *within* a particle pair, which translates into a dependence between the sets of data used to calculate all the averages. That dependence is lost when you generate the data for each average independently of all the others.
You may ask then, what about Gill’s statistical proof? Let us take the Ai,Bi, A’i, B’i spreadsheet of length 4N, and randomly sample N rows out (without replacement) from which we calculate ⟨AB⟩₁. We may now sample out another N rows (without replacement) from which we calculate ⟨A’B⟩₂. But note something. You may think that the second set of data is independent of the first but it is not! By taking out the first set, you have modified the distribution available for the second one and in the process you have lost some freedom, in a manner not too different from the dependency we described above. In fact, in our earlier expression ⟨AiBi⟩ + ⟨A’iBi⟩ + ⟨AiB’i⟩ – ⟨A’iB’i⟩, once you have all the data for the first three averages ⟨AiBi⟩, ⟨A’iBi⟩, ⟨AiB’i⟩ you automatically have the data for the last average⟨A’iB’i⟩. Similarly, in Gill’s situation, once you randomly extract N rows for the first 3 averages ⟨AB⟩₁, ⟨A’B⟩₂, ⟨AB’⟩₃, you automatically have the data for the remaining average ⟨A’B’⟩₄. You have a similar kind of dependency between the data as you had in the original case. But in the case of an actual experiment, there is absolutely no dependency between the sets of particles used to obtain the data. Each one is completely free. Gill’s statistical argument does absolutely nothing to counter the problem. It merely obscures it.
There is an even stronger argument against Gill’s statistical approach. The point of the Ai, Bi, A’i, B’i spreadsheet is to state that each row represents Bell’s realism assumption equivalent to the claim of a joint probability distribution ρ(A,B, A’, B’). And the main thrust of the statistical argument which Gill attempted to formulate, is to state that according to the law of large numbers, the averages of the averages of random samples of pairs AB, AB’, A’B, A’B’, will be similar to the averages of the pairs from the population. I’ve already explained above why random sampling makes things worse not better. But there is also a rigorous mathematical proof showing that it is not always possible to reconstruct a joint distribution of a set of random variables, by randomly sampling pairs of them, especially when the pairs represent a cyclical recombination of variables. (see http://www.panix.com/~jays/vorob.pdf) or (see http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=181 for an alternative explanation of the problem with random sampling)
Therefore, Bell’s theorem is false for several reasons including:
1) Bell’s claim that “realism” is equivalent to the existence of a joint probability distribution of ρ(A,B, A’, B’) is false, because there is a contradiction between A and A’, and B and B’ which means such a joint probability distribution does not exist for any experimental data, even if a supreme being such as GWTA may know of such a distribution. (see my coin-reading machine for a counter example). In other words, the inability of experimenters to observe such a joint distribution does not mean it does not exist from the perspective of GWTA. It simply means experimenters can’t measure it.
2) Even if we grant that “realism” assumption, Bell’s theorem still fails because such a joint probability distribution (even if it exists) can not always be reconstructed from separately sampled random paired data. Since the cyclical recombination of data in pairs demands a lack of freedom that is contradictory with the concept of random fair-sampling of a population. Trying to use random fair sampling and the law of large numbers to get the “correct” value does not work, you are merely getting the “correct” value of a different term not the one in the original expression. The correct term must have the appropriate dependence with all the other terms that is why I keep returning to the expressions:
⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ ≤ 2
⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ ≤ 2√2
Given the above, I do not see any point in addressing the rest of your proof, or do you think is there?
Simply stated, “It is mathematically impossible for anything to “violate” a Bell inequality.
Rick, I have a quick, simple question. Yes or no answer please: Is my equation 12, with its two valued relation between the wedge product and the cross product, mathematically correct, or is it not? I am not saying that it is, or that it isn’t. I am only asking your opinion. If you want to explain your opinion either way, that is your prerogative. Thanks, Jay
For the record, the only change I made was to add a print statement that shows the 0 outcomes generated by the Christian model.
There are no “0 outcomes” generated in my 3-sphere model, or in any of the simulations of my model. This is because there is one-to-one correspondence between the initial state (e, s) within my model and the dichotomous results (A, B) observed by Alice and Bob. Ideally one should read my actual paper ( https://arxiv.org/abs/1405.2355 ) to understand why there are no “0 outcomes” generated in my model. But to avoid any confusion, I have updated my latest simulation to make this point clear: http://rpubs.com/jjc/233477 .
The “0 outcomes” are not elements of reality, so they don’t exist until someone decides to print them. Right?
In the updated version it is made clear that the simulation does not intend to model the detection loophole. Instead, it simulates the non-local dependence of emission on the detection settings: the number of emitted particles L is computed from the detection directions, proving that for each pair of detector settings a different subset of pre-ensemble is selected.
The model is manifestly local, respecting the geometrical properties of the 3-sphere. To see some kind of non-locality in it is to view it within R^3, missing the very point of the model.
The simulation model is not manifestly local. No location is assigned to any of the formulas. It is easy to observe that the observation history of one observer is affected by the detector setting of the other observer.
That is a false claim. The measurement functions A(a; e, s) = +/-1 and B(b; e, s) = +/-1 are manifestly local functions, where a and b are freely chosen detector directions of Alice and Bob and (e, s) is the randomness shared by Alice and Bob which defines the initial state or the hidden variable of the singlet system. This is how Bell defined local measurement functions in equation (1) of his famous paper of 1964. His lambda = my (e, s). Please provide evidence from Bell’s paper exactly how and where he assigned “location” to any of his formulas for the measurement functions. By the way, the size of the physical ensemble within the 3-sphere is N, not M: http://rpubs.com/jjc/233477 .
The S^3 model naturally incorporates and explains “entanglement without entanglement.” But for the sociological reasons you have mentioned, this message has fallen on deaf ears.
The key ingredient is the twist in the Hopf bundle of S^3, which is a U(1) bundle over S^2.
See especially the last pages of this revised version: https://arxiv.org/abs/1405.2355 .
Unfortunately S^3 — and even its Hopf fibration — is rather counter-intuitive. But the twist in the Hopf bundle of S^3 is analogous to the twist in the Mobius strip. I have a toy model of Alice and Bob living in a two-dimensional Mobius world, reproducing the correlations E(a, b) = -a.b in two dimensions, which may be thought of as a 2D version of the “entanglement without entanglement.” See the appendix of this paper: https://arxiv.org/abs/1201.0775 .
As I have noted before, despite appearances I have no interest in Bell’s theorem per se. My main interest has always been in understanding the physical phenomena of the observed strong or quantum correlations. It turns out that they can be understood as purely local-realistic correlations in terms of the Clifford algebra of the orthogonal directions in physical space.
OK, Joy, As I said, let’s be as simple as we can. Let’s work from the 2-D surface of a mobius strip. I downloaded a picture of such a strip, annotated it slightly, and uploaded it to https://jayryablon.files.wordpress.com/2016/12/alice-and-bob-mobius.pdf.
Let me also quote the EPR paper (http://journals.aps.org/pr/pdf/10.1103/PhysRev.47.777) the middle of the right-hand column on 779:
“We see therefore that, as a consequence of two different measurements performed upon the first system, the second system may be left in states with two different wavefunctions. On the other hand, since at the time of measurement the two systems can no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system. This is, of course, merely a statement of what is meant by the absence of interaction between the two systems.”
So, let’s see if we can use the annotated mobius strip at https://jayryablon.files.wordpress.com/2016/12/alice-and-bob-mobius.pdf to explain what puzzled EPR. Suppose we have a vector which points in a direction, which vector we shall use as a proxy to represent the EPR wavefunction. Suppose Alice (A) and Bob (B) can only live in the “flatlands” of this strip, and are situated on opposite “sides” of the strip as shown in the annotation. So as far as Alice and Bob are each concerned, they are at a spacelike separation from one another and can no longer interact. But suppose that this vector is not constrained by the flatlands reality of Alice and Bob, and has its own reality that bleeds through to both sides of the flat mobius strip. So to Alice and Bob it appears that they are looking at two different vectors. But in reality they are looking at the exact same vector from opposite “sides” of the mobius strip. Alice and Bob see an “entangled doublet” of vectors. But in reality, the vector is a singlet viewable from two “sides” of the strip. Any perceived “doublet” is a property, not of the singlet vector or of the singlet wavefunction, but of the double-sided space.
So if Alice “disturbs” the vector (a.k.a. EPR system) by pointing it in direction 1, then Bob on the opposite side of the strip will also see it pointed in direction 1. And if Alice instead points the vector in direction 2, then Bob will see it also pointed in direction 2. So the conclusion they will reach is that the state caused by Alice’s disturbance of the vector directly affects the state in which Bob sees the vector. Not knowing that they are on opposite sides of a mobius strip, and believing that the only path for interaction to occur between their positions is by travel along the surface of the strip until they meet up with one another, Alice and Bob will have concluded that the two vectors they are observing are “entangled,” and that some non-local superluminal (indeed, instantaneous-at-a-distance) communication has occurred. But of course, it has not. The “entanglement,” such as it is, results from the single vector bleeding through to both sides of the strip, while Alice and Bob cannot get from one side of the strip to the other without traveling a finite path along the flatlands surface of the strip.
In other words, this simplified “entanglement” is the consequence of instantaneous communication between two “sides” of the two-dimensional space, in which the observers are constrained to live on only one side or the other, and can only get from one side to the other by transport along a finite spatial path, and in which the “side” upon which an observer sits is a “hidden variable” which may be called “side A” or “side B,” or, to use numbers, +1 or -1. Then, if we want to talk about the entanglement that seems to be observed in three space dimensions via strong QM correlations, we simply generalize to higher dimensionality in ways that are well-known, and make use of fiber bundles with mobius-type twists and all of the modern topological tools presently at our disposal which were not available to EPR in 1935. And to the question, how, exactly, do I carry out this generalization?, Joy proclaims he already has the answer.
And, from this view, it is far more palatable to understand entanglement as being an instantaneous communication between opposite sides of one space at single two-sided events in that space with the side being a hidden variable, rather than being instantaneous communication between two widely separated events in the same space in which we ignore are or unaware of the two-sidedness of the space.
Joy, simplified down as much as possible, is this what you are contending is the correct explanation for entanglement, and the underlying basis for your theory?
Jay
Joy’s model has two simple physically reasonable postulates. 1. The particle pair is either left or right hand oriented as a system at creation. 2. The topology of the space is S^3 thus giving space unique spinor properties. I think you might do better working from the postulates.
Jay, your toy version is oversimplified. As Fred mentioned, what is missing is the handedness of your “vectors.” Of course, vectors don’t have handedness, nor do complex numbers. So what we really need are quaternions (or spinors). But in the toy model we can replace them with L-shaped objects — like the one we use on cars when we are learning to drive.
Now try to imagine (or even better, make a paper model of L’s on a Mobius strip): What will happen if Alice’s right-handed L goes around the strip to meet up with Bob’s right-handed L? Will Alice’s L still be right-handed and match up with (i.e., be congruent to) Bob’s right-handed L? Let us start with this question. We need at least this much to get started.
Of course it is. But if there is a clear path to start here, then layer in elements of complexity one at a time, and get to your full model without simplification, then that is exactly what should be demonstrated.
I already did cut some mobius strips this morning to do exactly what you suggest, and of course the parity gets flipped. In fact, if you regard the two vectors in my https://jayryablon.files.wordpress.com/2016/12/alice-and-bob-mobius.pdf as an L, this parity flip is already plain from that diagram. It is inherent to the two-sided twisted mobius space that you have opposite parities on each side. And, the component of any vector orthogonal to the direction of passage around the strip, will inherently flip by 180 degrees on the opposite side of the strip. So to an observer not aware of the twist, this will appear as an anti-correlation.
So, yes, it is oversimplified. But I still come back to my question:
Does your model effectively explain “entanglement” as arising from an instantaneous connection between two sides of the same space, in which observers experience reality only on one side of the space? Of course, parity flips and anti-correlations are a natural outgrowth of this. But again, are two-sided spaces in which we only live on one side and the second side is hidden, the root of your entanglement explanation? And conversely, aren’t strong correlations the best evidence we have of such two-sided spaces in which observers live on and directly experience one side only?
If your answer is yes, good, I understand, and I can see the layers of complexity that you would serially build onto that, as well as the features which are already endemic to this space such as parity flips and anti-correlation. So you can call side A and side B “sides,” or “parities,” or “up” and anti-correlated “down,” or perhaps other binary classifications. And those will truly be hidden variables to someone who has no way to directly detect the other side of the space, and only notices them via strong QM correlations.
If your answer is no, then why would you even bother mentioning a mobius strip and Hopf bundles and twists, because all that would do is add unwarranted confusion?
I must declare: You and Fred are overthinking and over-explaining everything, and that may be part of the sociological problem you are running into. KEEP IT SIMPLE. START FROM THE SIMPLEST PLACE POSSIBLE. ADD ONE NEW THING AT A TIME IN CAREFUL PROGRESSION. DON’T OVER-COMPLICATE THINGS. DON’T OVER-EXPLAIN. DON’T LET YEARS OF BEING ON DEFENSE TAINT HOW YOU APPROACH THIS. AND END UP AT YOUR MODEL.
I can see a clear and quite simple path to defending your model against Bell’s Theorem if the answer are “yes” to my questions about using the two sides of the same space, with a twist, to explain entanglement. And I am very familiar with how quaternions turn 360 degrees before a “return to state” into 720 degrees before a return to state and how that would help you formalize this.
I can’t see that path if the answers are no.
Jay
Sadly, Jay, my answer is most definitely No. To begin with, there is no instantaneous communication between Alice and Bob. Neither in the real world, nor in the Mobius world. We can understand the strong correlations completely locally, just as Einstein would have liked.
I am glad that you already made the Mobius strip from a paper and confirmed the parity flip for the 2pi rotation around the strip. But if we go around the strip one more time, i.e., 4pi rotation, then we return back to normal and Alice’s L will be congruent to Bob’s L again.
Now before we continue I urge you to actually read my appendix about the toy model. That will save us all a lot of time. I don’t mean to rush you, but to put us on the right track.
OK, Joy, I have now read Appendix 1 in https://arxiv.org/abs/1201.0775. I get it regarding the mobius strip, and I have a sense for how you go to three dimensions, especially because I am very familiar with the behaviors of quaternions and spinors in three-dimensions.
Query: is there some way to argue that Bell’s Theorem would apply if the space was R^3 but does not apply if it is S^3, and to show with precision how this difference renders Bell inapplicable? This is in contrast to discussing Bell in isolation as we have all done for the last 700 or so posts here following my agreement with Stephen Parrott to confine the discussion to Bell standing alone. I hate to use the word “loophole” because it is already a term of art in this area of study. So let’s call it “loophole type-2.” Can you argue and show how S^3 creates a “loophole type-2” in Bell?
“Loophole” is a terrible term for anything. Originally it meant that the predictions of QM are not correct. I guess in this case, you are saying for type-2, that the prediction of Bell is not correct. Joy wrote a whole book showing how S^3 creates the type-2 for Bell. Bell’s domain never considered anything but R^3.
Absolutely, Jay. Now we are talking! It is all about R^3 versus S^3, and most of my papers are about bringing out the difference between R^3 and S^3 with regard to Bell’s argument. There are number of ways to answer your query. One way is to recognize that Bell did not define the co-domain of the local functions A(a, h) = +/-1 and B(b, h) = +/-1 he used to prove his “theorem.” Mind you, I am not talking about the “image points” (which are just +/-1) or the “range” of these functions, but their co-domain: https://en.wikipedia.org/wiki/Codomain .
Now in my view Bell should have specified the co-domain, and it should have been S^3 = SU(2), not R^3 as he implicitly assumed by not specifying it at all. Ask any real mathematician and they will tell you that one cannot define a function without specifying its co-domain. So in my view Bell blundered mathematically right in his very first step. But I am digressing.
I am not sure which among the many ways I have argued in the past that S^3 is necessary to overcome Bell would appeal to you. Perhaps none will appeal to you. But here is one very clear demonstration that the strong correlations in the real world disappear if we let the torsion (or curvature) of S^3 set to zero: http://rpubs.com/jjc/84238 .
I am afraid the above is a simulation written in the language R, and that is not what you are asking for. But it is quite a delightful demonstration nonetheless. Please scroll down to the part where I set the parameter f = 0 in the simulation, which amounts to setting the curvature to zero, or reducing S^3 to R^3. The detailed theoretical discussion about this is there in the withdrawn paper: https://arxiv.org/abs/1405.2355 .
Another more technical way to see this is in terms of geodesic distance within the manifolds SU(2) and SO(3), which I know will definitely appeal to you. Please check out Figure 4 on page 9 of this published paper to see what I am talking about: https://arxiv.org/abs/1211.0784 .
Jay R. Yablon
Query: is there some way to argue that Bell’s Theorem would apply if the space was R^3 but does not apply if it is S^3, and to show with precision how this difference renders Bell inapplicable?
Yes, it is all about R^3 versus S^3, and most of my papers are about bringing out the difference between R^3 and S^3 with regard to Bell’s argument. There are number of ways to answer your query. One way is to recognize that Bell did not define the co-domain of the local functions A(a, h) = +/-1 and B(b, h) = +/-1 he used to prove his “theorem.” Mind you, I am not talking about the “image points” (which are just +/-1) or the “range” of these functions, but their co-domain: https://en.wikipedia.org/wiki/Codomain .
Now in my view Bell should have specified the co-domain, and it should have been S^3 = SU(2), not R^3 as he implicitly assumed by not specifying it at all. Ask any mathematician and they will tell you that one cannot define a function without specifying its co-domain. So in my view Bell made a mathematical mistake right in his very first step.
I am not sure which among the many ways I have argued in the past that S^3 is necessary to overcome Bell’s restriction would appeal to you. Perhaps none will appeal to you. But here is one very clear demonstration that the strong correlations in the real world disappear if we let the torsion (or curvature) of S^3 set to zero: http://rpubs.com/jjc/84238 .
I am afraid the above is a simulation written in the language R, and that is not what you are asking for. But it is quite a delightful demonstration nonetheless. Please scroll down to the part where I set the parameter f = 0 in the simulation, which amounts to setting the curvature to zero, or reducing S^3 to R^3. The detailed theoretical discussion about this is there in the paper: https://arxiv.org/abs/1405.2355 .
Another way to see this is in terms of geodesic distance within the manifolds SU(2) and SO(3), which I know will appeal to you. Please check out Figure 4 on page 9 of this published paper to see what I am talking about: https://arxiv.org/abs/1211.0784 .
There are no opposite sides on the strip.
But someone not knowing there is twist in the strip would perceive that there are two sides. To wit:
One of my favorite memories from my freshman year at MIT was at a calculus lecture by George B. Thomas (https://en.wikipedia.org/wiki/George_B._Thomas) who had written the calculus book used in my high school calculus AP class. That by itself was impressive to me as a young lad, even before I heard him speak. Thomas was a tremendously-gifted lecturer, and one of a half dozen professors whom I found immensely inspiring in my own development. This also included John Donovan and Pat Winston in Computer Science and the incomparable “Doc” Edgerton and his strobe lights (in whose class I truly graduated into seeing myself as a real scientist and wrote the most challenging paper of my entire time at MIT).
Anyway, in one of his lectures, Thomas held up a mobius strip, talked about how in mathematical and physical thinking it was good for this and good for that, etc. Then, at the end of his monologue, he concluded: “And if it is good for nothing else, you can always hand it to your little brother and tell him to color one side red and the other side blue!”
Jay
OK, Fred,
So with these two statements, are you and Joy admitting that you really cannot get outside the CHSH bounds of |2|? And are you admitting that Bell’s Theorem is true on R^3, but that you can get around it in S^3, to which Joy devoted a whole book? If you are, you might get some of the Bellophiles to take a look at how that “type-2 circumnavigation” gets done. Then the question becomes whether Joy can abstract the “whole book” into a few pages that can be digested without another 700 back and forth posts here.
Jay
That is right. Nothing can get outside the bounds of |2| for Bell-CHSH. Not QM; not the experiments. So what good is it?
What is important is that Joy’s model gives the same predictions as QM in a local-realistic way. Most of the back and forth here was about how Bell was wrong in our view or right in their view. But Joy’s model clearly shows that Bell was wrong about his postulate that no local-realistic theory could give the same predictions as QM. Joy’s model is actually pretty simple physically with the two postulates.
Jay, you still owe me a response but if you understood what I was arguing about earlier then
(1) ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ ≤ 2
(2) ⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ ≤ 2√2
It is clear that (1) is the CHSH, (but 2) is NOT the CHSH.
You will never find a 4xN spreadsheet data with columns (A₁, B₁, A’₁, B’₁) that violates the CHSH. Anyone who claims to have violated (1) should present the spreadsheet of data which accomplishes that feat. The silence on this point is deafening.
The question of whether a theory can produce ⟨AB⟩ in agreement with QM is an entirely different question. Although it was believed after Bell that that was impossible, the collapse of Bell’s theorem implies that it is possible.
On the issue of “loopholes”. Do you know of any other valid theorem that has loopholes? Mind you, a loophole is a scenario which obeys the assumptions of a theorem but yet still violates the conclusion of the theorem. In other words, a loophole is just one more piece of evidence that a theorem is false and in the Bell’s case, every time one has been found, people have rushed to “patch” the theorem with a new inequality. In every such case you find hints of the fundamental problem with Bell’s theorem. For example, Larsson and Gill writes
Exactly! But note that there is no common part between the four different ensembles used to measure the 4 averages from any experiments! So tell me, how can Bell’s inequality still apply to that situation? In fact, Larsson and Gill prove in that paper that the correct inequality for such a case is
⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ ≤ 4
Which has never been violated by QM or any experiment.
But once you understand the fundamental flaw in the theorem, you don’t even have to talk of loopholes any longer. The only question remaining is how to generate the strong quantum correlations in a local-realistic manner that is consistent with established experimental evidence.
On your earlier question about explaining instantaneous connection between two sides, I would say you are assuming a myth that is based on Bell’s theorem, as a premise of your question. The only relevant question is “What is the origin of the strong correlation?”.
Bell’s theorem is a “no-go theorem”! No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.
To get around Bell’s Theorem in S^3, one has merely to devise a probability distribution for the occurrence of {A, B, A’, B’}-combinations in many like S^3-systems which yields a violation of the CHSH bound of Abs(2).
Period.
Bell-CHSH inequalities and CHSH bounds are irrelevant for physics. Bell’s “no-go” claim has nothing whatsoever to do with physics:
http://libertesphilosophica.info/blog/wp-content/uploads/2016/11/Fatal.pdf .
The actual physical question concerns the quantum mechanical predictions of correlations
E(a, b) = -a.b
and 13 different probabilities associated with these correlations. These are all reproduced exactly using purely local and realistic functions A(a, h) and B(b, h) as demanded by Bell:
https://arxiv.org/abs/1405.2355 ; see also http://rpubs.com/jjc/233477 .
1) Bell’s theorem: No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.
You really should look at the word “all”.
2) In case Bell-CHSH inequalities and CHSH bounds are irrelevant for physics, why do you throw a critical eye on Bell in your paper “Local Causality in a Friedmann-Robertson-Walker Spacetime”.
Either you are able to refute Bell-CHSH inequalities by presenting a complete “local hidden variable” model for all probabilities associated with all correlations, or Bell-CHSH inequalities and CHSH bounds remain relevant for physics!
1) Bell, or anyone else, has never proved the comprehensive statement of Bell’s claim you have just quoted. So, to begin with, you are making a false claim with your emphasis on the word “all.” There are claimed “proofs” of Bell type for only very special cases, such as for the singlet correlations. But some of us have argued that such claims of “proofs” are unjustified.
2) I have not thrown “a critical eye” on Bell in my paper “Local Causality in a Friedmann-Robertson-Walker Spacetime”. I have simply reproduced the quantum mechanical correlations for the singlet state in a local realistic manner from a physical perspective, as Einstein would have done. If there is any reference to Bell’s local model, it is for historical reasons only.
3) As I have already noted in this thread, a complete local hidden variable model for all probabilities associated with all correlations already exists: https://arxiv.org/abs/1201.0775 . See the comprehensive theorem on page 12 to 16, and references in the above paper.
Claims are not proofs!
“Local hidden variable” models:
“From a classical standpoint we would imagine that each particle emerges from the singlet state with, in effect, a set of pre-programmed instructions for what spin to exhibit at each possible angle of measurement, or at least what the probability of each result should be. The usual approach to analyzing these phenomena classically is to stipulate that a particle’s pre-programmed instructions for responding to a measurement at a given angle must be definite and unambiguous (rather than probabilistic) because we classically regard the two measurement angles as independent, …..”
Quote from http://www.mathpages.com/home/kmath521/kmath521.htm
Provide a list of the probability distribution for the occurrence of {A(a, h), B(b, h), A(a’, h), B(b’, h)}-combinations in many like systems; produced on base of your S^3-spacetime model for two entangled spin1/2 particles in the singlet state by using the purely local and realistic functions A(a, h) and B(b, h) (the pre-programmed instructions).
LJ,
If you provide a data set (or just a reference to a data set) generated by anything at all — by quantum mechanics, by experiments, by a local hidden variables theory, by a non-local hidden variables theory, or by whatever you prefer — that violates the Bell-CHSH inequality
⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ ≤ 2 ,
then I will immediately provide you the list of probability distributions you are asking for.
Here is something:
https://rpubs.com/heinera/16727
Now we await the list of probability distributions.
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=288&start=60
This is not what I asked for. Let LJ provide what I asked for, and I will keep my promise.
I have had a chance to reflect on the last several days of RW conversations from both sides of the Bell disagreements. And while there has been some recent discussion again about S^3 versus R^3, I would like to go back to Bell-CHSH proper. The reason for this is because I am now pinpointing an apparent logical contradiction in the way these relationships are interpreted, and would like to be corrected if I am seeing this wrongly or confirmed if I am seeing this rightly.
Let me start by simply writing five relations, and giving them numbers, so we can then talk about them. I will use the magnitude of X, i.e., |X| to write these as:
|⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩| ≤ 2 (1)
|⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| ≤ 4 (2)
|⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| ≤ 2√2 (3)
⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ = ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ for large N (4)
|⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩| ≤ 2√2 for large N (5)
I have saved myself the trouble of writing unicode by borrowing and pasting from MF’s posts, so in the above I think of the subscript 1 as a first index i=1…I; of 2 as a second independent disjoint index j=1…J; of 3 as third disjoint index k=1…K; and of 4 as a fourth disjoint index l=1…L, where I=J=K=L are all equal, but the summations are done independently. For example, if each of I=J=K=L=N=10 million, then in (2), for example, we do least 10 million (real or gedanken or simulation) trials (doublet creations and detections) in *each* of the a,b; a’,b; a,b’ and a’,b’ direction pairs, harvest the first 10 million for each pair as a good random sample, and discard the rest. Conversely, the equations such as (1) with only the subscript 1, utilize the “urn and slip” model in which all of the summations are done at once for all four terms, over the same index i=1…K. Now, let’s talk. To avoid personalizing, I will just refer to “pro-Bell” and “contra-Bell.”
Above, (1) is the CHSH inequality. When we started this discussion in October, I thought that contra-Bell were fudging as to whether this could ever exceed the bounds of |2|, while pro-Bell were adamant that it could not. “Show me the set of slips than can get you outside of |2|” was a common refrain. It looks to me like contra-Bell have since admitted that when all four terms are summed at once as in (1), we do have an urn and slip model, and there is not way to choose the slips to break the bound of |2|. So, I believe there is universal agreement about (1) as a mathematical proposition.
On the other hand, I think that pro-Bell fudged for awhile about (2), but they now admit that (2) is a correct mathematical relation. Though they point out that the statistical probability of getting anywhere near the bound of 4 for any substantial number N of trials is nil. Contra-Bell have always agreed with (2). At this juncture I believe there is also universal agreement about (2) being correct, with the pro-Bell caveat about the small statistical likelihood of getting near 4 for large N.
If (2) is true as all are agreed, then (3) is most certainty true, because it fits within (2). And the statistical chance of getting near 2√2 is certainly much higher than that of getting near 4. And we know that we actually do reach 2√2 = 4*sin(pi/4) = 4*cos(pi/4) when the angle between Alice and Bob’s detectors is theta(a,b)=pi/4. This is the QM strong correlation.
It is at equation (4) where the trouble begins. Pro-Bell argue that in the limit of large N, using statistical analysis, the relation (4) is true. So they substitute (4) into (3) to obtain (5). Then they argue that (5) describes the QM correlations. And they argue that because any local realistic / hidden variables (LRHV) model is governed by (1), the existence of (5) offers a contradiction, thereby disallowing any and all LRHV models. Bell’s Theorem.
Contra-Bell argue that (4) is wrong, and that you cannot use (4) to get from (3) to (5) and then use (5) to contradict the LRHV underlying (1). So let’s talk this through logically and dispassionately and civilly.
I do understand the statistical logic underlying (4). One of the pro-Bell principals has confirmed that I do, and I recently re-posted my derivation, which a contra-Bell has disputed, and to which I still owe a reply, which debt I am discharging here, though from a different angle of approach.
The pro-Bell logic of (4) says that once I harvest I=N=ten million trials for ⟨A₁B₁⟩, I know all about what was seen at the a and b orientations, but do not know for sure about the a’ or b’ data which is hidden by the terms of the experiment. But I do know (or at least, I make the ***statistical assumption***, more on that to follow) that when I harvest the L=N=ten million trials for ⟨A’₄B’₄⟩, and now have real data about a’ and b’, the doublet formations during those L= ten million trials should have a statistical profile that is virtually the same as the profile of the I=N=ten million doublet formations from the first set of trials. Ditto for the data with the a,b’ and a’,b harvests. And the order of the harvest within each run of N does not matter, so I can reorder my individual data points any way I choose. So I use the detected data from each run of ten million, to statistically infer the hidden data for the counterpart runs. And because we are postulating local ***realism***, I regard that hidden data as real even if it is hidden, and so am not shy about making that inference. So I collapse the indexes into a single sum by statistical cross-inference, and use (4) to go from (3) to (5). And once I have (5), I contradict the LRHV theories upon which (1) is based, and according to pro-Bell, that’s all she wrote.
Now, instead of trying to argue against (4) as the contra-Bell folks have done, let me assume that (4) is true for large N so that I can now deduce (5) from (3) and assert that (5) describes the QM strong correlations. So the first thing I do is use (5) to bump off the LRHV theories used to get to (1), and then I proclaim that Bell’s Theorem, when considered in view of the strong correlations in (5), has proved that no LRVH theory can be used to explain the strong QM correlations. Of course, the contra-Bell folks then yell loudly that this is wrong, and we are still be arguing about this 800 more more posts into a Retraction Watch discussion.
But suppose the contra-Bell side plays this differently: Suppose they they instead say — as I have detected them starting to say in the last several days — OK, you broke it, you own it. Or, you won the election, now show me that you can govern. Or, your dog chasing the car finally caught the car, now what will you do with it? How might that discussion go? Let have a Socratic dialogue:
Contra-Bell: OK, let me accept for the sake of discussion that your use of (4) to get from (3) to (5) is correct. And let me hold my nose and even accept that (5) via (1) disproves LRHV as a viable approach for explaining QM. You win: nature is entangled, there is spooky action at a distance, and Einstein is rolling over in his grave because he was wrong. But, to realize your victory, you used a relation, (5), which I am now going to throw back at you:
Here it is, again:
|⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩| ≤ 2√2 for large N (5)
This describes QM strong correlations according to pro-Bell. You used this (5) to kill our LRHV theories. These correlations occur, not via any local realism, but because of non-local entanglement. Good! You got what you wanted, you caught the car.
Now that you own (5), let me ask you what you have asked me for the last two months (and likely longer than that before this discussion ever got to RW and Jay ever paid any attention): “Show me the set of slips than can get you outside of |2|.” Take the ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ inside of (5), use your entanglement theory that I have conceded to you, and tell me some slips that get you outside the bound of |2| enforced by (1) that you just used to kill off my LRHV. There are only 16 of them, as you told us. What are they? Give me 16 frequency numbers, all adding to 1, to get a result in (5) that is greater than 2 but less than 2√2. What is might the Pro-Bell replies be?
Pro-Bell 1: Because of (1), there are no slips that can break the bound of |2|. So (5) really is just (1). And because of that, there cannot be any strong QM correlations. Voila! I have just disproved quantum mechanics and strong entanglement. Oh, but wait, that is a non-starter. Let me try again.
Pro-Bell 2: We have assumed all along that A, B, A’ and B’ all have the values of +/-1, with s being the spin direction, based on taking sign(a dot s), sign (a’ dot s), sign(b dot s) and sign (b’ dot s). Maybe the entanglement causes the slips to have different values than +/-1 and I can use that to get outside of |2|.
Contra-Bell: OK, back at you again, “Show me the set of slips than can get you outside of |2|.” Does the fact that we have now conceded entanglement, give you one bit of help with the slips? Or does it not? Show us the slips!
Pro-Bell: Well, no, we really cannot do anything with the slips, and (2) hurts us in the pro-Bell camp now that we have entanglement, just as much as it hurt you in the contra-Bell camp when you were arguing for LRHV. So maybe we need to back off of (5) and go back to (3). Yes, let’s do that. The entanglement which yields the strong QM correlation is actually (3). Then, all with be right with the world, and we can explain the strong QM correlations using entanglement that you have finally conceded to us. So here that is, again:
|⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| ≤ 2√2 (3)
Contra-Bell: Welcome to our world. We now agree with you. But, you just gave away your entanglement, because once you admit that it is really (3) not (5) which governs the strong QM correlations, then you have given up the precise relationship, (5), that you used to beat down our LRHV models. So now, we are at a standoff. We cannot disprove entanglement, but neither can you disprove LRHV. And it is the strong QM correlations which did this to both of us. We are now all in the same boat.
Pro-Bell: Well if that is the case, what went wrong?
Contra-Bell: The only place where this seems to have gone wrong, is in the application of the statistical argument in (4). For some reason, you just cannot do that. One of the Bell-contras gave some reasons a few days ago. If you do not buy those, then let’s dig deeper until we figure that out, because otherwise you not only disprove LRHV with (5), you also disprove QM strong correlations with (5). And we don’t think anybody wants to submit a journal article claiming to disprove Quantum Mechanics.
Now let me speak as myself: While I understand, and explained above, how one makes the statistical argument to justify (4), there is a very subtle assumption which I think (correct me if I am wrong) goes into deriving (4). This it is the assumption that the statistical profiles of the I=N=ten million doublet formations for a, b are the same as the statistical profiles of the L=N=ten million doublet formations for a’, b’, and likewise the profiles for a, b’ and a’, b. But the strong correlations themselves seem to undercut that subtle assumption. How? Back to the Socratic dialogue:
Pro-Bell: Even when you observe A and B, the unobserved A’ and B’ are still there, and by your own realism hypothesis, they are real. So you cannot treat them any differently. And all of your “counterfactual” arguments are just a way to sneak in treating them differently when they are not different. So we can still use the statistical profiles from one measurement set to the next.
Contra-Bell: If the strong, active QM correlations did not exist, and were only classical, passive, linear correlations, then I would agree. But they do exist. That is why (5) in view of (1) messed you up just like it messed us up. And if you go back to EPR from which this all evolved, this is all about the fact that if Alice measures along a, then Bob gets one result; but if Alice measures along a’, then Bob gets a different result. This is what you call “entanglement.” So once I have done all four N=10 million harvests, I know the results for each of a,b; a,b’; a’,b and a’,b’. But as soon as I start to project the a’,b’ harvest onto the a,b harvest by taking about *what would have happened* had I measured a’ not a and / or b’ not b, I am taking about something that ***would not have had a passive effect***. For weak, classical correlations, where Alice does not affect Bob’s observations, yes, you can project the statistical profile from one set of runs to the next. But for strong QM correlations, had I measured a’ not a and / or b’ not b for some of the data in by a,b trials, I would have caused a change to what Bob measured, and so I cannot project the statistical profile of a’,b’ onto a,b precisely because of the *active* effect of the strong correlations whereby Alice’s choices affect Bob’s results and Bob’s choices affect Alice’s results. The generalization of this takes place in (7) and (8) of the EPR paper and the related discussion.
Pro-Bell: But aren’t you using what you are trying to prove, namely, the strong QM correlations, to prove that we cannot use (4), because these QM correlations mess up the mapping of the statistical profiles from one trail set to the next?
Contra-Bell: No more than you are doing. We are using the strong QM correlations to show that (4) cannot be true because if it was there would be no strong QM correlations for entanglement to explain because (5) still goes outside the bounds of (1) and you are as stuck as we are inside those |2| bounds. You are using the strong QM correlations to show that we cannot use LRHV theories to explain strong QM correlations. But if you take this to this logical conclusion, we are proving that NO THEORY, not entanglement, and not LRHV, can explain strong QM correlations, for as long as we regard (4) to be a valid bridge from (3) to (5). Scorched earth for all. But once we use (3) not (5), then there is no problem explaining strong QM correlations, and in fact, we explain the perfectly. But once we are back at (3) and not (5), we have a stalemate, because neither of us can prove that the other is wrong. We cannot disprove entanglement theories, but neither can you disprove LRHV theories. We can TRY to use LRVH to explain strong QM correlations, and you can TRY to use entanglement to explain strong QM correlations. Game on, may the best team win.
Pro-Bell (maybe they won’t say this, but if they won’t, I’d like to see the logic as to why they don’t have to): Well, are you aware of any LRHV theory that can explain strong QM entanglement? We spent all of this time trying to prove that LRHV cannot explain strong QM entanglement, and once you conceded that to us, we realized that as long as we insisted on (5), neither could we use entanglement to explain strong QM correlations. OMG, instead we all proved that NO THEORY can explain strong QM entanglement. We spent all that time and effort trying to prove a negative and we proved the negative for ALL THEORIES. So where are we now? We still need a *positive* theory to explain the strong QM correlations.
Contra-Bell: Funny you should ask that, because we have someone here who tried to do just that, but had his paper retracted by the same journal that published many of Einstein’s seminal papers.
OK, this is me again, and I have given everybody some pretty fat targets. Am I wrong about the logical contradictions I have pointed out, especially if one accepts equation (5) and just asks the same question that the pro-Bell camp have asked all along: Show me the slips that allow (5)? Use any theory you wish, LRHV or entanglement. Show me the slips! And if you cannot, it seems to me that (5) has to fall.
Jay
Your equation (4) is equivalent to the equation 4 = 2. Its LHS is bounded by 4, whereas its RHS is bounded by 2, even for N = infinity.
If you grant me 4 = 2, then I can prove anything you like. Anything at all.
Quoting:
“Let me start by simply writing five relations, and giving them numbers, so we can then talk about them.”
Answer:
Wrong start!
You start with a list of the probability distribution for the occurrence of {A(a, h), B(b, h), A(a’, h), B(b’, h)}-combinations in many like systems; produced, for example, on base of the “local hidden variable” S^3-spacetime model for two entangled spin1/2 particles in the singlet state by using the purely local and realistic functions A(a, h), A(a’,h), B(b, h) and B(b’,h).
In order to translate it:
Both particles emerge from the singlet state with, in effect, a set of pre-programmed instructions for what spin to exhibit at each possible angle of measurement!
Let’s denote A = A(a,h), B = B(b,h), A’ = A(a’,h) and B’ = B(b’,h) with the typical codings +/-1
Then you must get:
|⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩| ≤ 2 (1)
and
|⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| ≤ 2 (2)
LJ,
You seem to have ignored what I asked for. Until you provide what I asked for, there can be no reason to believe the often made claim that quantum mechanics, or experiments, or something else actually violates the Bell-CHSH inequality
|⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩| ≤ 2 .
Recall what I asked for: I asked you to provide a data set (or just a reference to a data set) generated by anything at all — by quantum mechanics, by experiments, by a local hidden variables theory, by a non-local hidden variables theory, or by whatever you prefer that violates the above inequality. This data set must be written in the form of a 4xN spreadsheet, with columns (A, A’, B, B’), just as you have defined these numbers.
My request is not unreasonable. After all, quantum mechanics is much older than Bell’s theorem. It should be quite easy to generate the data set from its predictions. It may exist already in some well known paper, or even as a school project. There are plenty of relevant experiments that have been in the public domain, at least since 1972. Many of them have been published in Nature and Physical Review Letters. Therefore it should be quite easy to simply look up a good paper and produce at least an online reference to such a data set.
So, please, do not ignore my request, and provide at least a reference to the data set I have asked for, for this discussion to move forward, or conclude. Many thanks.
Sorry, wrong.
First: In (2) which I have defined as *disjoint* trial sets (never mind that you tried to change my definition to make (2) the same as (1)) I can harvest one trial from each of ab, ab’, a’b and a’b’ (four disjoint doublets) and can find from that harvest that A₁=B₁=A’₂=B₂=A₃=B’₃=A’₄=+1 and B’₄=-1. That produces ⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩=4. Clear counterexample to your claim.
Second: Suppose you were right anyway, so that we are bound by the constraint:
⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| ≤ 2 (2)
as you contend. In that event:
|⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| ≤ 2√2 (3)
is a false relation, because it falls outside the bounds you contend are true because of (2). Forget about local realism and hidden variables. I grant you entanglement and that no LRHV theories are allowed. Bell is King. So now that I have conceded everything, show how an entanglement theory, if (2) is true, allows (3) to get larger than 2 and up to 2√2. I showed you some slips above; you show me some slips.
Because (3) characterizes the existence of strong QM correlations, if you cannot show me the slips, you have just proved that there are no QM correlations, and that the only correlations allowed in nature are weak, linear, classical correlations. So if you’d like, go ahead and submit a paper to a journal, claiming that you have disproved quantum mechanics, then let us all know how that goes for you. That should be some fun. 🙂
False.
True. However, the source model is not.
In the beginning of “Formulation”.
Correct. M is the size of pre-ensemble, which is a computational device without any indicated physical significance. Both N and L can be called the size of the ensemble as N = L is verified. But N and L depend on measurement directions, which could not happen in a local model.
You are inventing your own personal definitions of what is supposed to be a local model. I am following the definitions of local causality proposed by Einstein and Bell. Accordingly to Bell’s definition, my 3-sphere model is manifestly local and realistic, as anyone can verify for themselves by simply reading my paper: https://arxiv.org/abs/1405.2355 . The simulation is an implementation of the 3-sphere model discussed in the paper.
Your interpretation of Einstein’s causality is erroneous. If the generation of the pair of particles is affected by detector settings then Einstein’s causality is violated, at least in experiments where the detection directions are chosen when the particles are already separated, as they are in most of the experiments that have confirmed the violation of Bell’s inequalities.
But the generation of the pair of particles is NOT affected by detector settings in my model.
Please read my paper I have linked above before making false claims about my model.
It is in your simulation model.
Simulation is not the model. It is an implementation of the model. Without the theoretical model it implements, it has no meaning.
Dear Readers of Retraction Watch,
As an author of the paper which was withdrawn from the journal Annals of Physics soon after its publication, I make the following appeal, involving a request to all parties involved.
To put my appeal in context, let me begin by quoting the editorial statement that provided the only basis for the withdrawal of my paper from Annals of Physics:
“After extensive review, the Editors unanimously concluded that the results are in obvious conflict with a proven scientific fact, i.e., violation of local realism that has been demonstrated not only theoretically but experimentally in recent experiments. On this basis, the Editors decided to withdraw the paper.” (This statement can be found on the publisher’s website: http://www.sciencedirect.com/science/article/pii/S0003491616300975 .)
It is evident from the above editorial statement that the so-called “… violation of local realism that has been demonstrated not only theoretically but experimentally in recent experiments” was the only basis on which the decision to withdraw my paper was based.
Now after two and a half months of discussion and over 820 comments, we seem to have reached consensus here about what this supposed “violation” actually means in a precise mathematical and physical terms. It means a “violation” of the following Bell-CHSH inequality:
| ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ | ≤ 2 ,
where A’s and B’s are equal to +1 or -1. More precisely, this inequality can be written as
| ⟨ A(a, h) B(b, h) ⟩ + ⟨ A(a’, h) B(b, h) ⟩ + ⟨ A(a, h) B(b’, h) ⟩ – ⟨ A(a’, h) B(b’, h) ⟩ | ≤ 2 ,
where h is the hidden variable, or the randomness, shared by both Alice and Bob.
Thus it all boils down to a supposed “violation” of this extremely simple and elementary mathematical inequality which any precocious schoolchild can understand. It involves an array of N x 4 (i.e., N by 4) matrix, with N rows and 4 columns, marked (A, A’, B, B’), where each A, A’, B and B’ is either +1 or -1. The entire matrix can thus be written out on an N x 4 spreadsheet, with N rows and 4 columns, marked (A, A’, B, B’). Nothing can be more simple.
Now my appeal to all parties involved (which is more like a request) is simply this: To justify the withdrawal of my paper and to bring this discussion to a close, all I ask is an explicit data set of N x 4 numbers, +1 or -1, written on a spreadsheet as described above, that “violates” the Bell-CHSH inequality. Which means that the absolute value of the sum of four averages on the LHS of the inequality actually exceeds the bound of 2, at least by a small amount.
Please note that I am not at all interested in where the data written on an N x 4 spreadsheet comes from that in fact “violates” the above inequality. I simply ask for a data set (or just a reference to an existing data set) generated by whatever means — by quantum mechanics, by experiments, by a local hidden variables theory, by a non-local hidden variables theory, or by “magic” — that violates the Bell-CHSH inequality. The data set, however, must be written on an N x 4 spreadsheet, with columns marked (A, A’, B, B’), because that is what the LHS of Bell-CHSH inequality demands. That is what the sum of the four terms on its LHS involves.
My request is not unreasonable. To begin with, the editors of Annals of physics have not provided any evidence for their claim of “violation.” Moreover, since quantum mechanics is much older than Bell’s theorem, it should not be difficult to generate such a data set from its predictions. Furthermore, there are plenty of Bell experiments that have been in the public domain, at least since 1972. Many of them have been published in prominent journals such as Nature and Physical Review Letters. Therefore it should be quite easy to simply look up a good source for the data and provide at least a reference to the N x 4 data set that “violates” the Bell-CHSH inequality as spelled out above. With this simple demonstration the editors of Annals of Physics can justify the withdrawal of my paper and bring this matter to a close.
You have this completely backwards. No one has claimed that quantum mechanics can be represented by a N x 4 spreadsheet. Only LHV models can be represented in that manner. However, I can easily give you a N x 8 spreadsheet that reproduces the QM correlations.
HR, please read my post carefully to understand what I have asked for. The editors of Annals of Physics have claimed that “… violation of local realism … has been demonstrated not only theoretically but experimentally in recent experiments.” Now we all agree that local realism is defined by the following Bell-CHSH inequality
| ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ | ≤ 2 ,
whose violation means “violation of local realism.” Please provide a data set — generated by whatever means — that violates the bounds of |2| on the above Bell-CHSH inequality.
No one agrees that local realism is *defined* by that inequality. That inequality is a theorem, not a definition. But only with local realism is it possible to write down the lhs in the first place. No local realism, no lhs, and thus no inequality.
That is your opinion only, not Gill’s or AOP’s. Use whatever means to generate the N x 4 data set that “violates” the Bell-CHSH inequality as claimed by the editors of Annals of Physics.
Joy, I submit to you: an Nx4 matrix of all +1’s. That violates your inequality. You said it does not matter how I got those numbers. Now what?
It indeed does not matter how you got those numbers. The problem is that Nx4 matrix of all +1’s does not violate the Bell-CHSH inequality. Please try again.
See my coment below. In sum: the novelty of quantum mechanics is that the operation performed by the brackets depends on where the apoatrophes are. So make the double apostrophe version always -1 and the others always +1. (That is not QM, but you said we can get our numbers any way we want in your challenge.)
HR,
IMHO you have always made cogent points here at RW, and over the past two months I have learned a lot of what I now understand about Bell from what you have said here. So I have two questions to start, to facilitate my further understanding of all of this:
1. I think you and I agree that my (1) and (5) above use an Nx4 spreadsheet, (2) and (3) use an Nx8 spreadsheet, and that (4) equates a particular average from the Nx4 and Nx8 spreadsheets in the limit of large numbers, also recognizing that averages become expectation values for large numbers. Am I correct about all of this?
2. If, as I think you do, you believe that (4) involving large numbers is a correct relationship, does (4) apply only to LRHV models? Or is this a general statistical relation which applies to any model?
Thanks,
Jay
PS: If you wish, for extra credit, 🙂 I would be interested if after answering questions 1 and 2 above, you would “annotate” my equation (1) through (5) above by explaining your view of what each equation means, and the scope of its applicability to LRHV and/or non-LRHV theories.
Jay,
1. I will concentrate on your inequalities (1), (2), and (4), because I don’t really see where you have (3) and (5) from. Let’s now discuss what is the operational meaning of the words “local and “realism”. Start with an Nx8 spreadsheet where each row is the predictions for a single run of an Alice-Bob experiment for any possible combinations of setting, so it takes these eight values:
A(a,b), A(a’,b), A(a,b’), A(a’,b’), B(a,b), B(a’,b), B(a,b’), B(a’,b’)
By “realism” we mean that you have a “realist” theory that enables you to fill out all of those eight predictions.
By “local” we mean that this Nx8 spreadsheet can be compressed to an Nx4 spreadsheet because of the added locality restrictions that ensure each column contains duplicates (we have A(a,b) = A(a,b’), etc.), so we can now write the rows as
A(a), A(a’), B(b), B(b’)
So, lets now discuss your inequalities (1), (2) and the equation (4). As I understand your notation, I say they all refer to Nx4 spreadsheets. (1) says that that if you use the CHSH expression on *one* random row, the bound 2 is absolute. (2) says that if you use the CHSH expression on four arbitrary values from 4 arbitrary rows, the only *absolute* bound is 4. But then we have (4), which says that if the settings of Alice and Bob are chosen randomly for each row, then the statistical bounds of (2) will be 2 as well, for a large number of trials N.
Here it is instructive to note that the word “randomly” is key; otherwise it is possible to cherry pick detector settings for each and every row (based on the entries in that row) so that you can get any CHSH value for (2) between -4, 4.
2. The equation (4) only applies to LHV models (aka “local realist”), because only then is it possible to write down the rhs (the rhs requires a Nx4 spreadsheet).
HR, Is the Bell-CHSH inequality | ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ | ≤ 2 “violated” in the EPR-B experiments, or not? Please answer Yes or No without trying to switch to a different inequality with a higher bound of 4. If it is “violated”, then please provide a N x 4 data set.
It is violated in experiments. The operation of the brackets depends on where the apostrophes are. That is the novelty of quantum mechanics. In your challenge, the Nx4 matrix can look like anything if “magic” makes the rightmost double apostrophe bracket always evaluate to -1 and the other 3 evaluate to +1. That is not what QM says, but you said it doesn’t matter how we got our numbers. But it is an extreme example which I hope shows how measurement choice dependence changes the game.
Sorry, I have absolutely no idea what you are talking about. Nothing whatsoever can ever violate the Bell-CHSH inequality, unless someone makes some mathematical mistake.
Typo in my original post: “each column contains duplicates” should be “each row contains duplicates”.
I also don’t really see where the eq.(3) and eq.(5) are from.
Maybe, |⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| ≤ 2√2 and |⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩| ≤ 2√2 are some misleading and arbitrary constructs which were sometime inserted into the “debate” without any physical justification, just to cause some confusion.
OK, Apparently there is some confusion. So let’s straighten it out if we can. The number 2√2 has been used in these RW discussions quite a bit. The first time I recall seeing it was in a post some weeks ago by Stephen Parrott, then others referred to it as well.
So: There is a number 2√2 that has been floating around here in association with the strong QM correlations. To what other mathematical construct, presumably based on an Nx4 or an Nx8 spreadsheet, is that number 2√2 equal?
There is no confusion about the bound 2√2. It is a quite well known and well established physical bound, predicted by quantum mechanics as well as my Clifford-algebraic model. You have very clearly explained all five of your equations and bounds in your original post.
The number 2√2 is the QM prediction for the CHSH expression. But as far as the interpretation of Bell’s theorem goes, the only significance of 2√2 is that it is larger than 2. It could be 3, or 3.5, etc., wouldn’t matter The conclusion of Bell’s theorem is simply that any theory that predicts a value larger than 2 cannot be a LHV theory.
Your claim is false. Here is a published local-realistic model that predicts the bound of 2√2
on CHSH (see the last equation of the last appendix): https://arxiv.org/abs/1211.0784 .
“The number 2√2 is the QM prediction for the CHSH expression.”
That is also false since it is mathematically impossible for anything to exceed the bound of |2| for Bell-CHSH. You have switched to Jay’s eq. (2) or (3) instead of eq. (1) which is the official Bell-CHSH expression.
HR, To be precise, I presume you mean:
“The number 2√2 is the QM prediction for the ***outer bound*** of the CHSH expression.”
So, for which CHSH expression? I want to have these written down plainly as mathematical relations so there is no room for confusion. There are too many words being tossed around in these Bell discussions, and I want to see math. For ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ with an Nx4 spreadsheet (four independent +/-1 numbers)? Or for ⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ with an Nx8 spreadsheet (eight independent +/-1 numbers)? Or of something else?
To rephrase, is the relation into which 2√2 fits the following for which you and LJ did not see the origin:
|⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| ≤ 2√2 (3) ?
Or is it the following for which you and LJ did not see the origin:
|⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩| ≤ 2√2 for large N (5) ?
Or is it something else? This 2√2 is not just some number. It is either equal to something else, or it sets an absolute or a statistically-significant bound on something else. What is that something else?
Jay
2√2 is the upper bound on the sum ⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ with an Nx8 spreadsheet (eight independent +/-1 numbers) + the Clifford-algebraic or geometric constraint imposed by the orthogonal directions in the physical 3D space. If we ignore the geometry and topology of the physical space than the upper bound on the sum ⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ is 4, not 2√2.
You ask, “So, for which CHSH expression?”
The expression is the following:
|⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩|
If you use the angles 0, 45, 22.5, 67.5 in the Alice-Bob experiment, QM predicts a CHSH value of 2√2 for this expression.
If you want to use an Nx4 spreadsheet or an Nx8 spreadsheet to interpret that expression is up to you. If you use an Nx4 spreadheet you can’t statistically violate the bounds of 2 for large N, so you can’t reproduce the value 2√2. If you use an Nx8 spreadsheet one the other hand, it is almost trivial to find a spreadsheet that gives a CHSH value of 2√2 (or a spreadsheet that gives a value of 3, etc).
Hr,
within the context discussed here on RW, I would prefer to rewrite |⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| when talking about quantum mechanical expectation values E. Some commenters immediately associate the brackets and indices with averages calculated on base of data lists etc.
To distinguish, I would prefer to write
|E(AB) + E(A’B) + E(AB’) – E(A’B’)| ≤ 2√2
for the QM prediction for the CHSH correlator, in order to avoid further confusion.
OK, so let’s break this down using three of the five equations I have been referring to the last few days:
|⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩| ≤ 2 (1)
|⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| ≤ 4 (2)
|⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| ≤ 2√2 (3)
Equation (1) tells us the outer bounds on ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ based on a Nx4 spreadsheet in which the spreadsheet entries all have the values +1 or -1. This is a ***spreadsheet entry-imposed constraint on the outer bound***. This bound is calculated, starting with the spreadsheet entries, to be |2|. Correct?
Equation (2) tells us the outer bounds on ⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ based on a Nx8 spreadsheet in which the spreadsheet entries all have the values +1 or -1. This is also a ***spreadsheet entry-imposed constraint on the outer bound***. This bound is calculated, starting with the spreadsheet entries, to be |4|. Correct?
In contrast, equation (3) tells us the outer bounds on ⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ based on theoretical and observational physics, namely, the QM correlations given by:
⟨AB⟩ = – a dot b = – cos (theta) (6)
I should have added (6) above originally, and now do officially add this, to the other equations (1) to (5) I am trying to have everybody sort out. This is a ***QM physics imposed constraint on the outer bound***, which in turn operates as ***QM physics-imposed constraint on the spreadsheet entries***, which entries ***must*** be part of an Nx8 spreadsheet because you cannot break out of (1) with only an Nx4 spreadsheet of +1s and -1s. Here, while each spreadsheet entry still has the values of +1 and -1, we may no longer choose these freely at will independently of one another. Now the physics of equation (3) establishes a set of constraints upon all of the entries on the spreadsheet. Correct?
So whereas in (1) and (2) the spreadsheet entries may be freely chosen and those entries establish the outer bounds, in (3) the outer bounds are imposed by the natural QM world and they establish permissible and impermissible combinations of entries on the Nx8 spreadsheet, which entries are intertwined and mutually-constrained. Correct?
Or, concisely:
In (1) and (2) spreadsheet entries are freely chosen and they establish bounds.
In (3) the bounds are imposed by the natural world, and they establish permissible spreadsheet entries.
Mind you, I am deliberately NOT referencing Joy’s comment about his “Clifford-algebraic or geometric constraint imposed by the orthogonal directions in the physical 3D space.” That is premature. I am only talking about how quantum physics affects the permitted spreadsheet entries in the Nx8 spreadsheet.
Let’s stop here.
Does this clarify, and does anybody disagree with any of this?
Jay
Quoting “HR December 13, 2016 at 10:51 am”
“Start with an Nx8 spreadsheet where each row is the predictions for a single run of an Alice-Bob experiment for any possible combinations of setting, so it takes these eight values:
A(a,b), A(a’,b), A(a,b’), A(a’,b’), B(a,b), B(a’,b), B(a,b’), B(a’,b’)
By “realism” we mean that you have a “realist” theory that enables you to fill out all of those eight predictions.
By “local” we mean that this Nx8 spreadsheet can be compressed to an Nx4 spreadsheet because of the added locality restrictions that ensure each column contains duplicates (we have A(a,b) = A(a,b’), etc.), so we can now write the rows as
A(a), A(a’), B(b), B(b’)”
Does this mean: Any local-realistic hidden variable model can – when decoded into a consistent simulation algorithm – only produce Nx4 “A(a), A(a’), B(b), B(b’)”-spreadsheets?
No it does not. Because one can derive the Bell-CHSH inequality without the assumption of locality or Einstein’s realism. One can derive it purely by assuming a kind of anti-realism:
http://libertesphilosophica.info/blog/wp-content/uploads/2016/11/Fatal.pdf .
Well, now Lord Jestecost and I are one the same page, because I focused in on the exact same part of what HR wrote as LJ. And following that I also went back and dug out Richard’s earlier definitions of hidden variables, locality and realism. All excepted above.
My impression is that “locality” is introduced by having Alice toss a coin and Bob toss a coin for their detector setting each time an experiment is run, at spacelike separation, so that there can be no communication between them about settings. And this is then encoded by giving 25% equal weight (the expected value of HH, HT, TH, TT from large numbers of coin tosses) to each of the four measurement direction pairs ab, ab’, a’b, a’b’. And this is whether we use the Nx4 spreadsheet for ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ or the Nx8 spreadsheet for ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩. In other words, locality is encoded by having N be the same, whether you have 4 rows or 8 rows in the spreadsheet (I like my shorter dimensionality for matrices to run vertically, i.e., to be rows). And early on, as I was learning this material, we heard repeatedly from both Richard and Stephen that you need to keep N the same for all for terms in the CHSH sum.
This now leads me to question the following seemingly-additional locality definitions above:
From Richard:
“Locality: Alice’s outcome depends on Alice’s setting, but not on Bob’s setting.”
From HR:
“By ‘local’ we mean that this Nx8 spreadsheet can be compressed to an Nx4 spreadsheet because of the added locality restrictions that ensure each column contains duplicates (we have A(a,b) = A(a,b’), etc.), so we can now write the rows as A(a), A(a’), B(b), B(b’).”
I am not convinced that these locality definitions are good ones. In fact, they give me grave doubts. I think locality is already built in with the coin toss, i.e, with the same N for each of the four terms in CHSH, and that this by itself is both necessary and sufficient. If you define “locality” by additionally saying that Bob’s outcome cannot depend on Alice’s setting and vice versa, you are not describing locality, you are describing WEAK CLASSICAL CORRELATION!
That is, by defining locality such that there can be NO STRONG CORRELATION between Alice’s settings and Bob’s measurements and vice versa, then your are DEFINING local theories to be impossible, because QM DOES CREATE STRONG CORRELATIONS!
Then, when you say that you have proved it is impossible for LRHV theories to describe strong QM correlations, you are saying that YOU HAVE PROVED YOUR DEFINITION, which is a self-fulfilling tautology! The failure of LRHV theories then BAKED INTO THE CAKE by supplementing the necessary and sufficient “25%-each pair by coin toss / no-communication between Alice and Bob / all terms with the same N” definition of locality, with an unnecessary and indeed tainted definition which says that Bob’s outcomes cannot correlate to Alice’s settings or vice versa. And this, we know is doomed from the start.
Let’s go back to good old physics and put this in a different way: Yes, in an LRHV physical universe Alice and Bob may not communicate when they are at spacelike separations, because in an LRHV world, spacelike signals are prohibited ***by definition***. But God created strong correlations in the natural world and everybody from Einstein on down found them perplexing. But, ***if you DEFINE a local theory as one which cannot permit these correlations***, rather merely than as one in which Alice and Bob cannot communicate superluminally, you have already prejudged that locality cannot explain strong QM correlations. And with that subconscious, understandably-human-experience-based, but tainted assumption and definition of locality, you have ensured that without question LRHV theories are dead on arrival. They never had a chance, by definition. Specifically, you have assumed and asserted right at the start that ⟨AB⟩ = – a dot b = – cos (theta) MUST DEFINE non-local communication (because in our daily physical experience we cannot conceive of it in any other way), when all that is required for non-locality is to prevent communications outside the light come between Alice and Bob about how they will set their detectors. Period. And this, it now seems, is Bell’ Theorem.
The impulsive side of me wants to yell BINGO! But I’d like to hear from everyone. And I can always stand corrected given good evidence. However:
Am I wrong that keeping Alice and Bob out of communication is a necessary and sufficient definition for locality, but that adding to this definition that Alice’s settings cannot affect Bob’s outcomes and v.v. builds in a preconceived, self-fulfilling bias that LRHV theories cannot explain strong QM correlations?
If I am not wrong, and if non-communication between Alice and Bob is a necessary and sufficient condition for locality, and if any prejudgement about the cause and explanation of the strong QM correlations has no proper place in this definition, then does Bell Theorem, in fact, turn out to be merely a self-proving tautology?
It is important to get this right, for everyone. I’d really like to know.
Jay
Sorry, typo in my second paragraph above after the excerpts, it should say “or the Nx8 spreadsheet for ⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩.”
Just an interesting piece of information if it matters. Locality in QFT is given by the cluster decomposition property which excludes correlated systems.
Jay, local causality was defined by Bell as follows:
“Apart from the common cause (or hidden variable) lambda, which originates in the overlap of the backward lightcones of Alice and Bob, the spacetime event A = ±1 depends only on the measurement direction “a” chosen freely by Alice. And likewise, apart from the common cause lambda, the spacetime event B = ±1 depends only on the measurement direction “b” chosen freely by Bob. In particular, the function A (a, lambda) does not depend on either “b” or B, and the function B(b, lambda) does not depend on either “a” or A [my paraphrase].”
This definition by Bell is simply a precise formulation of the original conception of local causality by Einstein (within the context of the EPR argument). I am in full agreement with Bell here. The above is the only definition I agree with.
But as I noted above, local causality need not be assumed to derive Bell-CHSH inequality. It can be derived entirely without it, as I have done in the appendix of this short paper:
http://libertesphilosophica.info/blog/wp-content/uploads/2016/11/Fatal.pdf .
Yes, that’s correct.
No, HR, both you and LJ are wrong. What LJ suspects is mathematically correct if and only if one presupposes a very restricted class of local-realistic hidden variable models in which the co-domains of the measurement functions A(a), A(a’), B(b) and B(b’) are confined to the real line, or equivalently to R^3, only. For topologically more general co-domains such as S^3 it does not hold. In any case locality is not at all required to derive the Bell-CHSH inequality.
Any theorem is a tautology, in the sense that the conclusion follows logically (and inescapably) from the definitions.
But how can you argue that Alice’s outcome can depend on Bob’s setting, and claim that the theory is still local? As I see it, the word “local” then loses all of its meaning. Anyway, of course Bell’s theorem doesn’t apply if you redefine locality to something completely different from what Bell (and everybody else) meant.
Without a proof that the simulation model actually implements the theoretical model the simulation model is irrelevant to the present discussion. But as you mentioned it one should note that the simulation is non-local and proves nothing about any local realistic model.
The proof that the simulation in question correctly implements the theoretical model is explicitly provided in the paper: https://arxiv.org/abs/1405.2355 . Please read the paper.
The article discusses the simulation but does not prove correctness of any detail of the code. The non-locality of the simulation is not discussed, either.
The locality of the simulation is both explicitly discussed and proved in the article. The correctness of the details of the code is self-evident, and discussed in the preamble of the main simulation. The article cites all relevant references necessary for understanding the codes, which have also been independently verified by professional computer programmers.
Earlier I asked a decisive question but so far I have not received an unambiguous answer to my question (apart from one incorrect answer). I asked whether the Bell-CHSH inequality
| ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ | ≤ 2
is “violated” in the EPR-B experiments? There are only two possible answers to this question:
(1) Yes, the Bell-CHSH inequality | ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ | ≤ 2 is “violated” in the EPR-B experiments,
in which case someone should be able to provide a data set in the form of an N x 4 matrix which defines the above inequality, thus demonstrating the claimed “violation”,
or
(2) No, the Bell-CHSH inequality | ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ | ≤ 2 is not “violated” in the EPR-B experiments,
in which case why have there been persistent false claims of “violation” for so many decades?
No single trial of an experiment will give you 4 numbers per row. It will only give you one number per row. If you want to talk about 4 numbers per row, then you are including hypoothetcals. Then you must allow the bracket operation to depend on the measurement choices (where the apostrophes are). Quantum mechanics depends on measurement choices.
That does not answer my question unambiguously. Please answer Yes or No to my question.
Your question cannot be answered yes or no without reinterpretation. Yes, Bell inequalities are violated in experiments. But there is no dataset like the one you describe. A single run of the experiment gives one number, not 4.
It always helps to turn every stone, but I now see that to say the A and B measurements are correlated is different from making the causality statement that they depend upon one another. Then, even if they followed all the other rules, Alice and Bob could use the EPR outcomes themselves to “communicate” with one another superluminally and that is why this need to be part of the locality definition.
So this brings us back to the question of how to explain strong correlations without outcome A depending on setting B and v.v. And it seems the only way to do so would have to utilize the geometric and topological properties of spacetime, which goes back to S^3 versus R^3 and twists and Joy’s 2D mobius simplification.
Let me therefore go back to before I chased after the locality definition, and pick up some unfinished business:
Does the above clarify equation (3)?
Then, let’s go back to HR’s response below to this statistically-based equation:
⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ = ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ for large N (4)
Joy, I assume that you do not agree with HR’s answer, and you believe that if (4) applies to LRHV models it applies to all models. And at the risk of asking you to repeat yourself, can you please state why you believe this, specifically as to equation (4)?
I say this mindful that if (4) is true, then combining this with (3) yields:
|⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩| ≤ 2√2 for large N (5)
And finally, am I correct that (5) above is why you keep asking the question:
And, am I correct that this question arises because you believe that if (4) is true then it, as well as its consequence (5), must apply to LRHV and other models alike?
Jay
It is mathematically impossible for eq. (5) to be true. After all, it is already proven that the bound is |2|.
Actually two numbers A or A’ along with B or B’, allowing you to evaluate one of the 4 brackets. The other 3 brackets are hypothetical in a single trial.
That is, a Bell dataset looks like this:
Alice’s choice, Bob’s choice, Alice’s observation, Bob’ s observation
A’,B,+1,-1
A,B’,-1,-1
…
It does not look like what Joy is requesting:
A,A’,B,B’
+1,+1,-1,+1
-1,-1,+1,+1
…
Because A and A’ cannot occur in the same trial (likewise for the B’s).
Jay, I do not believe that (4) is true, because it leads to (5), which is demonstrably false.
On the other hand, we routinely come across the claim that Bell-CHSH inequality, namely
| ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ | ≤ 2 ,
is “violated” in the EPR-Bohm experiments. If so, then there must exist an experimental data set, generated by whatever means (even allowing for “magic”), in the form of an N x 4 matrix which defines the above sum, that violates the Bell inequality. I ask the followers of Bell and the editors of AOP to produce such a matrix so that we can bring this discussion to a close.
OK, let’s try again to concisely highlight my understanding of the fault line between pro-Bell and contra-Bell, same six equations as before:
|⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩| ≤ 2 (1)
|⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| ≤ 4 (2)
|⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| ≤ 2√2 (3) (statistical bound for large N?)
⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ = ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ for large N (4)
|⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩| ≤ 2√2 for large N (5)
⟨AB⟩ = – a dot b = – cos (theta) (6)
And let me now put on the pro-Bell hat and see if I have that argument straight:
1) Any LRHV theory has to satisfy (1) using an Nx4 spreadsheet. This is because, by locality, A(a,b)=A(a,b’), etc. squeezes any Nx8 spreadsheet such as that underlying (2), down to Nx4. (Thanks to HR for pointing out that relation.)
2) Strong QM correlations are given by (6).
3) The strong QM correlations (6) may be characterized by (3), where having 2√2 rather than 4 is a quantum physics constraint on the associated Nx8 spreadsheet entries. Question: this 2√2 has to be a statistical outer bound for large N, correct?
4) Because of the statistical relation (4) for large N which applies only to locally realistic theories for which A(a,b)=A(a,b’), etc. squeezes Nx8 down to Nx4, we can use (4) to turn (3) into (5) to represent the correlations (6),but only for an LRHV theory, .
5) Because (5) which applies only to LRHV contradicts (1), the LRHV hypothesis underlying (1) is disproved. And that is Bell’s Theorem in a nutshell.
I will wait on trying to talk about the contra-Bell argument until it is confirmed that this is the essence of the pro-Bell argument.
Jay
Dear Yablon,
I’d like to give an answer to your question 3 that I hope will be useful for you.
3) You are right that this is a statistical bound for large N, but I’m afraid you’ve got the rest of the quantum part wrong. A Nx8 spreadsheet is a non-local hidden variable model. You are not going to get the 2√2 bound out of it, only the 4 bound. With quantum mechanics there is no spreadsheet, as the outcomes of Alice and Bob are not deterministic functions of their settings. To get that bound you need to use some linear algebra, as done for example here https://en.wikipedia.org/wiki/Tsirelson%27s_bound
I am afraid the bound of 2√2 has nothing to do with statistics or large N. Note also that despite what is claimed on the wikipedia page, it does not use quantum mechanics to derive the bound 2√2 but uses only the non-commutative Clifford algebra of orthogonal directions in the 3-dimensional physical space, just as originally done by Tsirelson (see also the last page of my very first paper on the subject for a purely local, deterministic, and realistic derivation of this physical bound: https://arxiv.org/abs/quant-ph/0703179 ). Thus it is simply a geometrical constraint, with little or nothing to do with statistics or large N.
Moreover, the Nx8 spreadsheet has nothing to do with non-local hidden variables model. This is easy to see from the following two inequalities:
|⟨A(a, h) B(b, h)⟩ + ⟨A(a, h) B(b’, h)⟩ + ⟨A(a’, h) B(b, h)⟩ – ⟨A(a’, h) B(b’, h)⟩| ≤ 2
|⟨A(a, h₁) B(b, h₁)⟩ + ⟨A(a, h₂) B(b’, h₂)⟩ + ⟨A(a’, h₃) B(b, h₃)⟩ – ⟨A(a’, h₄) B(b’, h₄)⟩| ≤ 4 .
While the first inequality corresponds to an Nx4 spreadsheet, the second one corresponds to an Nx8 spreadsheet. But the outcome variables involved in both inequalities are purely local variables. The only difference between the two is that while each outcome variable in the first inequality depends on the same hidden variable (or the same initial state) h — originating in the same run or trial, each outcome variable in the second inequality depends on a different hidden variable (or a different initial state): h₁, h₂, h₃, or h₄. Thus each of the four terms in the second inequality is independent of the other terms, leading to the bound of 4 and spreadsheet of Nx8.
Nothing can violate a mathematical inequality. Therefore nothing can violate Bell inequalities. Thus — contrary to your claim — Bell inequalities have never been violated in any experiment. Any claim to the contrary — by anyone — is simply false.
What you call “reinterpretation” amounts to switching the Bell-CHSH inequality
|⟨A(a, h) B(b, h)⟩ + ⟨A(a, h) B(b’, h)⟩ + ⟨A(a’, h) B(b, h)⟩ – ⟨A(a’, h) B(b’, h)⟩| ≤ 2 …………. (1)
(where h is a hidden variable) to a completely different inequality, with a different bound,
|⟨A(a, h₁) B(b, h₁)⟩ + ⟨A(a, h₂) B(b’, h₂)⟩ + ⟨A(a’, h₃) B(b, h₃)⟩ – ⟨A(a’, h₄) B(b’, h₄)⟩| ≤ 4 … (2)
Note that the last inequality is NOT the Bell-CHSH inequality, and since it is bounded by 4 and not 2, it is certainly not violated in any experiments (in fact nothing can violate it).
So neither inequality (1), nor inequality (2) has ever been violated in actual experiments.
Here is your Nx4 dataset:
Alice’s choice,Bob’s choice,Alice’s observation,Bob’s observation
A,B,+1,+1
A’,B,-1,-1
A,B’,+1,+1
A’,B’,+1,-1
Take the bracket operation to be +1 if Alice and Bob’s observations are equal and -1 if Alice and Bob’s observations are not equal. Then the left hand side = 4 > 2.
You are not free to choose the “bracket operation.” It is the average or expectation value defined, for example, in equation (2) of this paper:
http://libertesphilosophica.info/blog/wp-content/uploads/2016/11/Fatal.pdf .
Nor are you free to define N x 4 matrix as you have done. The columns should be marked
(A, A’, B, B’ ) ,
as I specified in my initial post, with N representing the number of runs or trials.
Joy, I see your most recent response now. Your inequality (1), with bound 2, cannot be tested in any experiment, as it would require Alice to choose both A and A’ and Bob to choose both B and B’ in each trial. But they can only choose one setting in each trial. (Or maybe you can describe a way to test it that eludes me?) Your inequality (2), with bound 4, is trivially always true, as the bracket opperation yields a number in [-1,+1], and there are 4 brackets. Yet many reseachers are interested when datsets like the one I describe break the bound 2 (with statistical significance).
That is exactly my point.
Please read the short paper I have just linked to recognize that we are in agreement.
But computer simulations of the experiments that give datasets like I described (with rows of the form A,B’,+1,-1) agree with the bound 2, provided they model “local realism”. So whatever mathematical/philosophical objections you throw out, you are going to meet resistamce. I think your best (perhaps only viable) way of convincing your detractors is to code a video game quality version of your macroscopic exploding balls experiment, and show it violates the bound 2. (I must admit that I do not think it will.)
OK, let us assume for the sake of argument that I am wrong and Bell-followers are right.
So let us examine their claim. They claim that the Bell-CHSH inequality is violated in the actual experiments for large N in the following manner, with fair-sampling assumption etc.:
|⟨A(a, h) B(b, h)⟩ + ⟨A(a, h) B(b’, h)⟩ + ⟨A(a’, h) B(b, h)⟩ – ⟨A(a’, h) B(b’, h)⟩| ≤ 2√2 for large N.
Note that “h” in each term is exactly the same, as required by their definition of local realism. This is just Jay’s eq. (5).
Now let us grant the Bell-followers every assumption they want and let them make the above claim. I don’t care how they got the above equation. Whichever assumption they used, or whichever experiment they did, or whichever magic they used to produce the above equation, it involves only Nx4 matrix of data points, with N rows, and the four columns marked as (A = +/-1, A’ = +/-1, B = +/-1, B’ = +/-1). I am willing to forget everything else, and will concede at once that I was wrong, if they produce the Nx4 matrix described above and demonstrate that the Nx4 data points constituting the matrix violate the bounds of +/-2 set by Bell and CHSH. This seems to me to be an entirely fair request.
As I said before, that experiment cannot be performed. It would require Alice and Bob to each select both of their two measurement options in each trial. No real experiment can be done to even write down a dataset like you want:
trial,A,A’,B,B’
1,+1,-1,+1,-1
2,-1,-1,-1,+1
…
Two of the columns are hypothetical in reality. Actual Bell experiments yield datsets that look like this:
trial,Alice’s choice,Bob’s choice,Alice’s observation,Bob’s observation
1,A,B’,+1,-1
2,A’,B’,-1,-1
…
I think your argument above is that for these datasets, the bound 2 does not follow from Bell’s arguments, and so data from actual experiments like this demonstrate nothing. But whatever your assessment of Bell’s arguments, computer simulations of such experiments, modeling local realism, do stay under the bound 2. So when a quantum system gives a dataset in that form that violates the bound 2, it is noteworthy. Thus I think your best way convinving your detractors is to code the simulation of your exploding balls experiment, and show it gives a dataset in this form that also violates the bound 2.
Simulations of my model already exist, several of them. Here is the latest one, for example:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=296&p=7324#p7322 .
I standby my request to the followers of Bell: Produce the Nx4 matrix of +/-1 numbers as I described above, and demonstrate that it violates the bounds of +/-2 on the Bell-CHSH inequality. This is what they have been claiming, not me (see Jay’s posts above for details).
Have you written a paper on your simulation and submitted it to a high quality journal? (Not a journal that doesn’t really check veracity.) If you want to persist on this topic, that seems like where your efforts are best spent.
Hi HR,
I keep coming back to your post on December 13, 2016 at 10:51 am, which for me added a lot of new information that nobody had made explicit before. If I now have 8 entries A(a,b), A(a’,b), A(a,b’), A(a’,b’), B(a,b), B(a’,b), B(a,b’), B(a’,b’) rather than 4 entries A(a), A(a’), B(b), B(b’) in each column because A(a), A(a’), B(b), B(b’), and if each of these is hypothesized to have a real value +1 or -1 whether or not it gets observed, i.e., even if “hidden,” then you are now talking about non-local realism. So let’s pursue what a non-local but still-realistic theory looks like.
We still want to examine correlations between what Alice and Bob detect. But now that we have four A numbers and four B numbers because of the non-locality with realism maintained, there will be 4×4=16 terms to consider. These are:
A(a,b)B(a,b); A(a’,b)B(a,b); A(a,b’)B(a,b); A(a’,b’)B(a,b);
A(a,b)B(a’,b); A(a’,b)B(a’,b); A(a,b’)B(a’,b); A(a’,b’)B(a’,b);
A(a,b)B(a,b’); A(a’,b)B(a,b’); A(a,b’)B(a,b’); A(a’,b’)B(a,b’);
A(a,b)B(a’,b’); A(a’,b)B(a’,b’); A(a,b’)B(a’,b’); A(a’,b’)B(a’,b’);
So there are four CHSH-like sums that cane be produced from these, namely (look at these closely to see how I constructed them):
A(a,b)B(a,b) + A(a,b)B(a,b’) + A(a’,b)B(a,b) – A(a’,b)B(a,b’) (1)
A(a,b’)B(a,b) + A(a,b’)B(a,b’) + A(a’,b’)B(a,b) – A(a’,b’)B(a,b’) (2)
A(a,b)B(a’,b) + A(a,b)B(a’,b’) + A(a’,b)B(a’,b) – A(a’,b)B(a’,b’) (3)
A(a,b’)B(a’,b) + A(a,b’)B(a’,b’) + A(a’,b’)B(a’,b) – A(a’,b’)B(a’,b’) (4)
So how does someone get through the following series of steps?:
a) Get from (1)-(4) above, to the single terms combination:
A(a)₁ B(b)₁ + A(a)₂ B(b’)₂ + A(a’)₃ B(b)₃ + A(a’)₄ B(b’)₄ (5)
b) Convert (5) over to:
⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ (6)
c) Then prove that:
⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ = ⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ for large N (7), my earlier (4).
d) Use the strong correlations:
⟨AB⟩ = – a dot b = – cos (theta) (8), my earlier (6)
to demonstrate that there is some:
⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ GT 2 for large N (9)
Finally, e) therefore get to say that this violates i.e., contradicts the CHSH inequality:
|⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩| ≤ 2 (10), my earlier (1)
I have stayed free of any equations with a 2√2 because you earlier said ” I don’t really see where you have (3) and (5) from.”
There are two points I am making here:
1. It seem to me that (5) above is a representation of 4 disjoint experimental harvests of equal size, with 25% taken from each of the ab, ab’, a’b and a’b’ detector settings, and has nothing to do with (4) which is based on an Nx8 spreadsheet arising from non-local realism. Yes, there are 8xN spreadsheets in each, but in one case you have the eight rows A(a)₁; B(b)₁; A(a)₂; B(b’)₂; A(a’)₃; B(b)₃; A(a’)₄; B(b’)₄ (call them apple rows) while in the other case you have eight rows A(a,b), A(a’,b), A(a,b’), A(a’,b’), B(a,b), B(a’,b), B(a,b’), B(a’,b’) (call them orange rows). And the former gives you a single CHSH sum (6) while the latter gives you the four sums (1) to (4). So it looks to me this these two Nx8 spreadsheets are being improperly conflated, by assuming that 8 apples = 8 oranges. To the extent that this is an ingredient in Bell’s Theorem, I’d like to understand how the apples are converted to oranges and the four CHSH-like sums in (1) to (4) become the single sum in (5).
2. In the above, (10) is the CHSH inequality, obtained based on local realism / hidden variables. Bell’s Theorem proclaims that local realism is violated because of (8) and (9), i.e., because of the QM correlations (8) which produce results (9) greater than 2 which violate (10). But your eight values A(a,b), A(a’,b), A(a,b’), A(a’,b’), B(a,b), B(a’,b), B(a,b’), B(a’,b’) are still regarded as realistic, i.e., as existing even if they are “hidden,” so what you are talking about now is a non-local realistic / hidden variable theory. So you must be able to show that there are actual spreadsheet values of +1 or -1 that can be used in (9), to actually come up with a number greater than 2. In other words, the non-local realistic / hidden variable theory now needs to produce CHSH numbers in (9) which are greater than 2. If someone cannot show how it does that, then what happens to the contradiction used by Bell to disprove local realism and only permit non-local realism? These are my own words, but IMHO that is the question that Joy is now asking over and over.
Now, I do recognize that in addition to abandoning locality, you then have the option to also abandon realism / hidden numbers. But then it makes no sense to even talk about a spreadsheet with the numbers +1 and -1 on it, if some of those numbers are not real unless they are observed. So when you are talking about an Nx4 spreadsheet you are talking about local realism, and when you talk about an Nx8 spreadsheet you are talking about non-local realism. But when the Nx8 spreadsheet is used in (5), that is talking about 4 disjoint equal-sized harvests from each of the ab, ab’, a’b and a’b’ detector settings in a local realistic theory, unless somebody can show that the apples can be turned into oranges. It seems to me that a lot of the confusion in this never-ending discussion is due to a failure to sort this all out, and stems from taking the apples:
A(a)₁; B(b)₁; A(a)₂; B(b’)₂; A(a’)₃; B(b)₃; A(a’)₄; B(b’)₄
and calling them oranges:
A(a,b), A(a’,b), A(a,b’), A(a’,b’), B(a,b), B(a’,b), B(a,b’), B(a’,b’)
Jay
Oops, my (5) above should be (used a + not a – sign):
A(a)₁ B(b)₁ + A(a)₂ B(b’)₂ + A(a’)₃ B(b)₃ – A(a’)₄ B(b’)₄ (5)
Jay,
That is what Vorob showed — oranges can’t be converted to apples. I have a detailed explanation here:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=181#p4912
Jay,
There is a fundamental difference between predictions made by quantum mechanics and predictions made on the base of local hidden variable models.
Consider many like systems of “two spin1/2 particles in the singlet state”.
Denote Alice’s and Bob’s measurement outcomes for the detector settings {a, a’, b, b’} by A(a), A(a’), B(b) and B(b’).
#### Real EPR-experiments ####
“Disjoint” experimental trials as A(a), A(a’), B(b) and B(b’) cannot simultaneously be measured in one run.
Perform N trials (index 1) to measure A(a) and B(b). One observes that the outcomes exhibit some probabilistic features, so you just evaluate an average value for the correlation A(a)*B(b). Let’s denote this average value ⟨A(a)B(b)⟩₁.
Perform “disjoint” experimental trials to get average values for the other correlations. You finally write down:
⟨A(a)B(b)⟩₁ + ⟨A(a)B(b’)⟩₂ + ⟨A(a’)B(b)⟩₃ – ⟨A(a’)B(b’)⟩₄ ……………………………………………..eq(1)
#### Quantum mechanics ####
The quantum mechanical formalism can merely predict expectation values, E: Such an expectation value is nothing else than the answer to the following question: What would be the *expected* average value for the outcomes of real experimental measurements on many like systems. Quantum mechanics can never say anything about a single system. Quantum mechanics must only “know” that all your systems are in the singlet state, nothing else. You get then four answers E (pure numbers) to your questions regarding the interesting correlations. You write down:
E(a, b) + E(a, b’) + E(a’, b) – E(a’, b’) ………………………………………………………………….eq(2)
#### Local hidden variable models ####
Local hidden variable models are “classical” physical models. Such models set up “classical” physical relations (local reality) to predict all outcomes for a single system in case you provide some initial state (e, s) which “implements” that the system is in a singlet state. You can translate these physical relations into a self-consistent local-realistic simulation algorithm which provides *definite and unambiguous* outputs A(a; e, s), A(a’; e, s), B(b; e, s) and B(b’; e, s) for every *definite* input (e, s).
From the fact, that real measurement outcomes exhibit some “randomness”, you conclude a) that there must be a set of different allowed initial states (e, s) and b) that nature randomly “selects” again and again another initial state (e, s) out of this set.
In case you have such a set of allowed initial states, you can now perform virtual EPR-experiments, either *disjoint* trials or *joint* trials.
Randomly select one of the initial states (e, s) from the set of allowed initial states and feed (e, s) into your simulation algorithm to get the outputs A(a; e, s), A(a’; e, s), B(b; e, s) and B(b’; e, s). Read out the values you are interested in, say A(a; e, s), B(b; e, s). After N trials, evaluate the average value for the correlation A(a; e, s)*B(b; e, s). Let’s denote this average value ⟨A(a; h)B(b; h)⟩₁ (here h represents (e, s).
In the same way, you can perform “disjoint” or *joint* virtual experimental trials to get the average values for other correlations you are interested in. You finally write down:
⟨A(a; h)B(b; h)⟩₁ + ⟨A(a; h)B(b’; h)⟩₂ + ⟨A(a’; h)B(b; h)⟩₃ – ⟨A(a’; h)B(b’; h)⟩₄ ………………..eq(3)
for *disjoint* trials, or
⟨A(a; h)B(b; h)⟩₁ + ⟨A(a; h)B(b’; h)⟩₁ + ⟨A(a’; h)B(b; h)⟩₁ – ⟨A(a’; h)B(b’; h)⟩₁ ………………..eq(4)
for *joint* trials.
In case the number N of trials tends to infinity, both terms assume the same value.
#### EPR-experiments find:
|⟨A(a)B(b)⟩₁ + ⟨A(a)B(b’)⟩₂ + ⟨A(a’)B(b)⟩₃ – ⟨A(a’)B(b’)⟩₄| ≤ 2√2 ………………………………….eq(5)
#### Quantum mechanics predicts:
|E(a, b) + E(a, b’) + E(a’, b) – E(a’, b’)| ≤ 2√2 ……………………………………………………..eq(6)
#### Any conceivable local hidden variable model predicts:
|⟨A(a; h)B(b; h)⟩₁ + ⟨A(a; h)B(b’; h)⟩₂ + ⟨A(a’; h)B(b; h)⟩₃ – ⟨A(a’; h)B(b’; h)⟩₄| ≤ 2 …………eq(7)
or
|⟨A(a; h)B(b; h)⟩₁ + ⟨A(a; h)B(b’; h)⟩₁ + ⟨A(a’; h)B(b; h)⟩₁ – ⟨A(a’; h)B(b’; h)⟩₁| ≤ 2 …………eq(8)
eq(8) (or, maybe (7)) is called “CHSH inequality”. Coining the casual phrase “something violates the CHSH inequality” means nothing else than saying “experimental measurements and quantum mechanical predictions violate the bounds given by the CHSH inequality”.
My local-realistic model also predicts |E(a, b) + E(a, b’) + E(a’, b) – E(a’, b’)| ≤ 2√2 , as anyone can verify for themselves: https://arxiv.org/abs/1405.2355 . Moreover, Fred Diether has just produced a new simulation (linked above) which confirms the above prediction.
LJ,
Thank you for your clear exposition, which I appreciate. I will study this closely. I just want to ask for one clarification right now:
Your equation (7) is for disjoint and your (8) is for joint virtual experimental trials. So shouldn’t your equation (7) for disjoint virtual trials read:
|⟨A(a; h)B(b; h)⟩₁ + ⟨A(a; h)B(b’; h)⟩₂ + ⟨A(a’; h)B(b; h)⟩₃ – ⟨A(a’; h)B(b’; h)⟩₄| ≤ 4 (7a)
with the upper bound of 4? (If I have to show how this can be as high as 4 I will do so, but I think everybody is past that.) And if so, then your phrasing “eq(8) (or, maybe (7)) is called ‘CHSH inequality'” is something of a “tell” that you see some uncertain connection between (7) and (8). So, let’s get rid of the “maybe” and be certain about this connection:
How do you (or anyone) justify the connection that gets you from (7a) to (8)? Because if (7a) is really the correct inequality, then this is entirely compatible with your (5) and (6), and nothing violates anything.
Or, to use your phraseology:
“experimental measurements and quantum mechanical predictions do not violate the bounds given by” (7a). So if the connection from (7a) to (8) “maybe” cannot be made, then “maybe” nothing “violates the CHSH inequality” because you cannot compare EPR experiments or Quantum mechanics to the CHSH inequality (8) without going through (7a) and equating that to (8). Which means that you have to mathematically equate:
“something which can be as large as 4” = “something which must be 2 or less.”
It would be helpful if we all can see how that works.
Jay
Jay,
you will find nothing else than I have decribed. Your local-realistic simulation algorithm will always provide “at the same time” *definite and unambiguous* outputs A(a; e, s), A(a’; e, s), B(b; e, s) and B(b’; e, s) for every *definite* input (e, s). There is no escape in case the simulation algorithm operates properly.
Randomly select one of the initial states (e, s) from the set of allowed initial states and feed (e, s) into your simulation algorithm to get the outputs A(a; e, s), A(a’; e, s), B(b; e, s) and B(b’; e, s). Read out the values you are interested in, say A(a; e, s), B(b; e, s) in order to virtually simulate a real EPR-experiment (*disjoint*). After N trials, evaluate the average value for the correlation A(a; e, s)*B(b; e, s). Let’s denote this average value ⟨A(a; h)B(b; h)⟩₁ (here h represents (e, s).
In the same way, you can perform “disjoint” virtual experimental trials to get the average values for other correlations you are interested in. You finally write down:
⟨A(a; h)B(b; h)⟩₁ + ⟨A(a; h)B(b’; h)⟩₂ + ⟨A(a’; h)B(b; h)⟩₃ – ⟨A(a’; h)B(b’; h)⟩₄ term#1
for *disjoint* trials.
However, you have everything at hand when you use the local-realistic simulation algorithm. Randomly select one of the initial states (e, s) from the set of allowed initial states and feed (e, s) into your simulation algorithm to get the outputs A(a; e, s), A(a’; e, s), B(b; e, s) and B(b’; e, s). Now, read out all values A(a; e, s), A(a’; e, s), B(b; e, s) and B(b’; e, s). After N trials, evaluate all average values for the correlations you are interested in. Let’s denote these average values ⟨A(a; h)B(b; h)⟩₁, ⟨A(a; h)B(b’; h)⟩₁, ⟨A(a’; h)B(b; h)⟩₁ and ⟨A(a’; h)B(b’; h)⟩₁ (here h represents (e, s)). Then you can write down:
⟨A(a; h)B(b; h)⟩₁ + ⟨A(a; h)B(b’; h)⟩₁ + ⟨A(a’; h)B(b; h)⟩₁ – ⟨A(a’; h)B(b’; h)⟩₁ term#2
for – so to speak – an “imaginary experiment”.
Now comes the main point!
In case the number N of trials tends to infinity (!!!!!!!!), both terms *must* assume the same value! Otherwise, your local-realistic simulation algorithm is flawed.
Thus, when merely considering the values, it doesn’t matter if you use your local-realistic simulation algorithm to provide equation (7) or (8). Neither term#1 nor term#2 can ever exceed the bounds +/-2.
LJ,
Again, thank you, I appreciate that you have drilled down on this. I have no doubt about the fact that term #2 can never exceed the bonds of +/-2. My doubts are about term #1, which you say will have the exact same +/-2 constraint if I use a correct simulation in the limit where N approaches infinity, i.e., in the limit where averages become expectation values. So I want to do just that, to convince myself one way or the other.
I prefer doing problems analytically to putting then on a computer. The best way I know to analytically look at a problem with something approaching infinity is to use calculus. So let me lay out what I think would be a valid analytical simulation, and before I put in the time to develop this, I want to confirm that this would be a valid analytical simulation.
In this simulation, I would calculate your term #1 in the limit of an infinite number of trials for each of the “disjoint” virtual experimental trials 1, 2, 3, 4. Every A=+1 or -1 and every B=+1 or -1 comes from calculating sign(a.s) or sign(b.s), same with primes. With the three-vectors a, b for the detector directions and s for the doublet spin direction all normalized to 1, a.s=cos (theta(a,s)) and b.s=(theta(b,s)), same for primes, and these theta are angles between the detectors and the spins. So my simulation would be based on five absolute angles, one of which is a spin angle theta(s), and the other four of which are theta(a), theta(a), theta(b), theta(b’). To define theta(s) I would draw an x and a y axis, and theta(s) would then be defined as an angle pointing from 0 degrees to 360 degrees in relation to these axes, representing the spin direction of a doublet produced in one of this infinite number of trials. Then, theta(a), theta(a’), theta(b), theta(b’) would represent the angle of each of the four detector directions a, a’, b, b’ on these same axes. Thus, a.s=cos (theta(a,s)), b.s=(theta(b,s)), a’.s=cos (theta(a’,s)) and b’.s=(theta(b’,s)) would give us relative angles between spins and detections that produce sign(a.s), sign(b.s), sign(a’.s) and sign(b’.s) to get the numbers A and B which are +1 or -1, which plug into your term #1.
For the infinity of trials, we maintain only the four detector directions theta(a), theta(a’), theta(b), theta(b’). But we treat theta(s) which is greater than or equal to 0 degrees and less than 360 degrees as a random variable uniformly distributed from 0 to less than 360 degrees. And as I generate an infinite number n=1…oo of uniformly-distributed angles theta(s)_n from 0 to less than 360 degrees, first with theta(a), theta(b), then with theta(a), theta(b’), then with theta(a’), theta(b), finally with theta(a’), theta(b’), I will be able to calculate each of the expected values in your term #1, and thus, the entire term itself. We may envision this as simply calling out uniformly-distributed random numbers from 0 to less than 360 degrees, time after time, ad infinitum.
Finally, if I use calculus, I can start with say, N=16 sixteen theta(s)_n, n=1…16, given by 0, 22.5, 45, 67.5, 90, 112.5, 135, 157.5, 180, 202.5 235, 257.5, 270 292.5, 315, 337.5 degrees, uniformly distributed, albeit discrete. Then I can segment this to 32, or 64, or 128… until I make this infinitely large in the calculus limit. Or, using calculus, for any N, I can establish a delta theta = 360/N, and utilize the set of angles theta(s)_n = n delta theta. Then I take the limit as delta theta becomes d theta.
If properly executed, would this be a valid analytical simulation?
Thanks,
Jay
In case I understand you right you want to get every A=+1 or -1 and every B=+1 or -1 from calculating sign(a.s) or sign(b.s) where a.s = cos (theta(a,s)) and b.s = cos(theta(b,s)) mean the dot product of two vectors. Here, the first problem appears when relying on classical reasoning. If a and s (or b and s etc.) are orthogonal (theta = +/- 90° etc.), you would get the indefinite values 0.
Yes, I already counted on that. So in the 16-sample seed, I would displace the detectors by 1 degree.
I like Lord Jestocost’s argument that any local realistic simulation cannot exceed the bound 2. Here is another way of looking at it, Joy:
The actual data from a Bell experiment will look like this, when written in your form:
trial,A,A’,B,B’
1,+1,NA,NA,-1
2,NA,-1,+1,NA
…
With NA values for the settings Alice and Bob did NOT choose. Under local realism, there are hidden variables that could be used to determine what to fill in for those NA values, so that every cell of the table is populated with a +1 or a -1. Yet you have already written above that no such table can exceed the bound 2. So you are already certain that local realism cannot exceed the bound 2. Also, on your “Fatal” paper: you object to a step that writes an expectation of a sum as a sum of expectations. But I think Bell thought this step was justified under his assumption of the existence of hidden variables. He actually did make that assumption in the spirit of “proof by contradiction”. He assumed hidden variables, used it to break up the expectation (and in a lot of other steps), derived a bound, noted that quantum mechanics can violate it, which is a contradiction, and thus threw out the hidden variables assumption.
If that were the case, you should be able to just measure singles, and recombine them in pairs. So my question for you is, why is it not OK to just measure A alone in one experiment, A’ alone in a separate experiment, B alone in a third experiment and B’ in alone in a fourth experiment. Then if you measure N entries that way, you end up with your Nx4 spreadsheet.
Why can’t you do that?
MF,
I agree that that is the better way to do these measurements, and I believe that this is what LJ is proposing and what I intend to do myself, analytically with the calculus approach I outlined earlier this afternoon. LJ is claiming that even if you do the calculation this way, that is, even if you measure A alone in one experiment, A’ alone in a separate experiment, B alone in a third experiment and B’ in alone in a fourth experiment, you cannot break the bound of 2 , so long as you do the experiment a very large number N of times. I do not know if this claim is correct or not, but I intend to convince myself one way or the other and will write up the calculation and share it with everyone else.
What I do know a priori is that we can all easily pick values for which:
⟨A(a; h)B(b; h)⟩₁ + ⟨A(a; h)B(b’; h)⟩₂ + ⟨A(a’; h)B(b; h)⟩₃ – ⟨A(a’; h)B(b’; h)⟩₄ = 4 (1)
when we look at one or two or a small number of data points in each trial. Just as how for a coin toss I may assign +1 to heads and -1 to tails, flip 5 times, and there is a 1 in 32 chance I will get the average +1 for all five being heads. But if I flip 5 million times, there is no way in the spacetime of this entire universe that I will ever get the average +1, which has one chance in 2^(5 million). In fact, when we talk expected values — the average as the trials approach infinity in number — the expected value is zero. Which is why LJ and Richard have emphasized expected values not averages, which I understand and agree with. So likewise, I have no doubt that for a large number of trials, even using (1) above which I know you think is the right way to go, the maximum value we can obtain from (1) will drop down from 4. Whether or not it drops below 2 sqrt(2), and actually drops all the way to 2, is something I would like to see for myself, one way or the other.
Jay
Typo: Third paragraph, fifth line, should read: b.s=cos(theta(b,s)
MF: you always measure for exactly one pair of settings. When you check if you violate the Bell inequality, you cannot mix observations from different pairs. That ruins the quantum correlations. Forcing them into an Nx4 format like Joy’s would thus result in an incorrect calculation.
Bingo! So you recognize that there is matching that needs to be maintained between the 4. How do you make sure the matching is maintained if you only measure two at a time?
When the inequality is derived, it uses the matching between all four in the critical step of the derivation as follows
AB – AB’ + A’B + A’B’ = A(B-B’) + A(B+B’)
Do you see that matching between all 4 terms is absolutely essential for the derivation? So that measuring only pairs for each row of the spreadsheet as you suggested earlier (similar to Gill) violates the very requirement that is needed to derive the inequality!
That is all that can be done in actual experiments. As I wrote above, if you think Bell’s arguments break down for such data: computer simulations of local realism with data in that format always stay under the bound. I think I see mathemtically that it wont work, but one can attempt to code a local realism simulation with data in this format that exceeds the bound.
In real experiments you can only measure one pair of settings at a time. You keep the data in my recommended format above. Then the pairing is maintained. A local realist says nothing Alice does affects Bob’s measurments and vice versa. So a local realist contends pairig is irrelevant. Or do I have the wrong idea of what a local realist thinks about pairing?
When the inequality is derived, it uses the matching between all four values for a specific particle pair, in the critical step of the derivation as follows
AB – AB’ + A’B + A’B’ = A(B-B’) + A(B+B’)
The inequality is not valid if this relationship does not hold for each particle pair, whether you measure them or not.
But when you measure only pairs you have disrupted the matching between all 4 terms. It doesn’t matter that you still have matching between the pairs. The full matching is absolutely essential for the derivation? So that measuring only pairs as you suggested earlier (similar to Gill) violates the very requirement that is needed to derive the inequality!
(I am talking about the local realist mixing and matching actual and hidden values along the same row from a single trial.)
A single “theoretical” trial has 4 numbers A₁, A’₁, B₁, B’₁. All four need to by paired in a cyclical manner for the inequality to be derived starting with A₁B₁ – A₁B’₁ + A’₁B₁ + A’₁B’₁ and then to A₁(B₁-B’₁) + A₁(B₁+B’₁) .
A single “experimental” trial has only two of those numbers. The other two can’t be measured in a single trial. Therefore you can’t derive the inequality. If you measure the pairs in different trials you end up with A₁B₁ + A’₂B₂ + A₃B’₃ – A’₄B’₄ which can’t be factorized in the same way. The inequality no longer applies because the matching *between* pairs which was present in the former has been lost in the latter.
Just like you lose the matching between pairs when you measure the singles A₁, A’₂, B₃, B’₄ and recombine them into
A₁B₃ + A’₂B₃ + A₁B’₄ – A’₂B’₄ = B₃(A₁ + A’₂) + B’₄(A₁ – A’₂) <= 2
You certainly have the wrong idea about what this rational local realist thinks.
1. We agree: Nothing Alice does affects Bob’s settings or outcomes; and vice-versa.
2. I trust we agree: The pristine particles are pair-wise correlated by the conservation of angular momentum during their creation.
3. In other words: The outcomes are pairwise correlated, and not otherwise.
4. Thus, under my rational local realism, pairing is essential.
PS: In that Bell variously has a false view re locality or realism or both, Bell’s theorem is false: see draft letter http://vixra.org/abs/1611.0033.
HTH
If you think Bell’s arguments break down for data with one pair per row, then note that computer simulations of local realism in ths data format always end up under this bound. I think I can see mathematically that it won’t work, but one could try to code a local realism simulation with data in this format that exceeds the bound.
MF,
any local hidden variable model must produce a Nx4 master list. Every allowed initial state (h) gets its own row in this Nx4 master list with the entries A(a; h) = +/–1, A(a’; h) = +/–1, B(b; h) = +/–1 and B(b’; h) = +/–1.
To simulate real measurement runs on many like singlet systems, you assume that nature provides the initial states in a random manner.
Proceed therefore in the following way:
## Generation of a *disjoint* trial set for the correlation A(a; h)*B(b; h) (index 1):
(I) Randomly choose a row from the {A(a; h), A(a’; h), B(b; h), B(b’; h} master list
(II) Read out the values of A(a; h) and B(b; h) and evaluate A(a; h)*B(b; h)
(III) Go back to (I)
While repeating, you can keep track of the evolution of the “cumulatively calculated” average value for ⟨A(a; h)B(b; h)⟩₁. In case the “fluctuations” of ⟨A(a; h)B(b; h)⟩₁ begin to remain well below a given threshold, you can stop. You can then rely that you have obtained a meaningful and representative value for ⟨A(a; h)B(b; h)⟩₁.
## Perform loops similar to that described above to obtain meaningful and representative values for the other *disjoint* trial sets: ⟨A(a; h)B(b’; h)⟩₂, ⟨A(a’; h)B(b; h)⟩₃ and ⟨A(a’; h)B(b’; h)⟩₄
This is a simple numerical approach. You will get – without any exception – for all chosen detector settings {a, a’, b, b’}:
|⟨A(a; h)B(b; h)⟩₁ + ⟨A(a; h)B(b’; h)⟩₂ + ⟨A(a’; h)B(b; h)⟩₃ – ⟨A(a’; h)B(b’; h)⟩₄| ≤ 2
This is absolutely wrong. I gave an example earlier of a coin reading machine that is clearly local realistic but can’t produce such a master list. I don’t think you have thought very carefully about the various possible local realistic theories.
Even if such a list can be imagined theoretically for a model, what matters is what can be measured and Vorob proved definitively that it is not always possible to regenerate an Nx4 list from 4 separate Nx2 lists, especially when you have cyclical recombination like in the CHSH.
http://www.panix.com/~jays/vorob.pdf
Nobody has addressed this point because it is devastating to pro-Bell.
The ensembles contributing to each term in QM and experiments is statistically independent of the ensembles contributing to the other terms.
The ensembles contributing to each of the terms in the CHSH are not statistically independent. They are mutually dependent — apples and oranges. If you start from an Nx4 list, and you pick rows randomly without replacement, the disjoint ensembles you generate are not independent. So it is completely false so suggest that picking rows randomly from an Nx4 spreadsheet without replacement is equivalent to the QM/Experiments situation. Why haven’t you addressed this point?
It is devastating to the pro-Bell Statistical argument.
Therefore your “statistical argument” fails for the reasons summarized below:
1) An Nx4 list is contradictory with the experimental fact of impossibility of measuring at 4 angles simultaneously.
2) Since local realism is not to blame for the impossibility to measure, it is false to claim that local realism must produce a 4xN master list.
3) Even if such a list can be imagined, the cyclical nature of the CHSH makes it impossible to reconstruct such a list from pairwise data (see Vorob)
4) 8xN data has higher degrees of freedom that 4xN data, irrespective of whether the data is from local realistic or non-local realistic theories
5) The ensembles in the QM expression and experiments are independent contrary to the ensembles in the CHSH
6) Randomly picking rows without replacement from a 4xN spreadsheet does not generate similar ensembles to QM and experiments
Therefore pro-Bell has not justified their claim that the QM/experiment expression can be compared to the CHSH — apples and oranges.
Jay,
to calculate all correlations you are interested in, I recommend to use the procedure which I have presented in my reply to MF. See “Lord Jestocost December 15, 2016 at 7:11 pm”
LJ:
So long as I can derive and combine the four disjoint terms in
|⟨A(a; h)B(b; h)⟩₁ + ⟨A(a; h)B(b’; h)⟩₂ + ⟨A(a’; h)B(b; h)⟩₃ – ⟨A(a’; h)B(b’; h)⟩₄| (you claim ≤ 2) (1)
which Joy and MF claim can be greater than 2, and you clam cannot be greater than 2, when N becomes infinite so averages becomes expected values, I believe my approach will do the same thing. But I have a Newtonian (the mathematician) belief in using calculus to get exact results, in the limit of small differences (in this case small angle differences) going to zero. I am tired of all the competing computer simulation claims and want to see this done analytically and exactly on paper.
So I will give this a try, and you all can offer critique as to whether it goes off the road.
In fact, let me propose something to everybody: I will try to analytically structure and calculate (1) above though a simulation, which I will drive to N=oo and delta theta = 360/N degrees between spin directions, with calculus, and get a result. I will put it in a PDF so you are looking at real, visual spreadsheet and equation data. I honestly do not know what that result will be. But it will use (1) above, with four disjoint, infinite N, trial sets.
So if after I do the calculation you all agree that I executed it correctly based on (1), then if I get a result no greater than 2 for any detector angle choices, will Joy and MF and Fred agree to concede the point, and if I get a result at least as large as 2*sqrt(2) for at least one detector angle choice, will Richard and LJ and HR agree to concede the point, as to the bounds of (1)?
I will play this right down the middle as an umpire calling balls and strikes with rigorous mathematics, and I will show my work carefully and transparently and will not bully anybody into anything.
Jay
Hi Jay,
Been there and done that.
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=196&start=60#p5486
I already obtained a result greater than 2 with Joy’s S^3 model using the GAViewer computer program. In fact consistent with 2*sqrt(2).
Response to * Jay R. Yablon December 15, 2016 at 7:57 pm *
Jay,
any local hidden variable model *must* produce a unique Nx4 master list in case you consider N different initial states (h). For every initial state (h) which “nature” can provide, you get a unique output {A(a; h), A(a’; h), B(b; h), B(b’; h)} with A(a; h) either +1 or -1, A(a’; h) either +1 or -1, B(b; h) either +1 or -1 and B(b’; h) either +1 or -1.
That was the primordial idea of Einstein. He wanted to explain the strange predictions of quantum mechanics by inserting some probabilistic features into local-realistic physical models: You construct a set of different initial classical states (h) whereby each of them constitutes a possible singlet state of the two-particle system.
When your initial state (h) is given, a local-realistic model “calculates” on base of a consistent set of physical relations what an observer must find, viz. you have a set of pre-programmed instructions to produce a unique set {A(a; h), A(a’; h), B(b; h), B(b’; h)}. This is a coherent set, because the pre-programmed instructions have to consider inherent “classical” correlations.
(see, e. g., http://www.mathpages.com/home/kmath521/kmath521.htm )
In principle, Alice and Bob wouldn’t have to perform real measurements if they know the initial state (h_i) (1 ≤ i ≤ N) of any single system before measurement. In case they can take it that “nature” operates in a local-realistic manner, they can take it that measurements reveal nothing else than pre-scribed or pre-programmed values. Thus, they would merely have to look at the corresponding row (for initial state (h_i)) in the corresponding Nx4 master list (for settings {a, a’, b, b’}) and read out the values.
Alice and Bob don’t know these unique Nx4 master lists. They only take note of some “randomness” when performing measurements. You, however, know what’s going on because you have devised a local-realistic physical model which has produced these unique Nx4 master lists.
Thus, to predict the average measurement outcomes found by Alice and Bob, you simply perform some statistical analyses on base of these unique Nx4 master lists using probabilistic approaches.
Maybe, you prefer to separately produce four Nx2 lists {A(a; h), B(b; h)}, {A(a; h), B(b’; h)}, {A(a’; h), B(b; h)} and {A(a’; h), B(b’; h)}. Then you have to keep attention for consistency (because of the inherent classical correlations) by comparing the entries obtained for some initial state (h_i) in each of the four Nx2 lists. When you find, for example, that A(a; h_i) = 1 in the Nx2 list {A(a; h), B(b; h)}, you *must* also find A(a; h_i) = 1 in the {A(a; h), B(b’; h)} list and so on.
LJ,
If you are correct that your (7) and (8) below:
are equivalent when four disjoint sets of trials with very large N -> oo (7) are conducted, which after taking a close look this past day I am actually inclined to agree is true, then I do not think you need any more “to keep attention for consistency (because of the inherent classical correlations) by comparing the entries obtained for some initial state (h_i) in each of the four Nx2 lists,” because that all comes out in the large N statistical “wash.” At least insofar as the mathematics goes.
But perhaps what you are saying is to get past that math, because physically, an LRVH model must by definition keep track of these values in (8), but a non-LRHV model may discard (8) and use only (7) because they do not have the requirement of having to keep track of what A(a’) would have been in a trial where A(a) was measured. And that (8) only has meaning in an LRHV model but has no meaning in models which are not LRHV.
Is that what you are saying?
Jay
PS: I did a first read through the link you provided to http://www.mathpages.com/home/kmath521/kmath521.htm. Am I correct that your point above is stated by the following text from that link:
“The essence of the so-called “local realistic” premise is that each particle emerges from the singlet state with definte [sic] instructions for the spin it will exhibit for each possible measurement angle…. According to quantum mechanics we would still expect the combined results to satisfy the same correlations. This implies that each particle pair must embody the overall propensities that we expect to find manifested in multiple trials.”
LJ I’ve give a detailed response to your comment to me referred to in your post. I assume you are familiar with the paper by Larsson and Gill (https://arxiv.org/pdf/quant-ph/0312035v2.pdf) in which they show that:
The simulation is obviously non-local, as already noted. Therefore it is not possible to prove its locality in a consistent theory. If a proof exists, it only porves that the theory is inconsistent.
The simulation is manifestly local, as anyone can verify: http://rpubs.com/jjc/233477 .
Jay, it does not really matter which way the result of your calculations turns out to be. Either of your results will be devastating for Bell’s theorem. As we agree, since the CHSH inequality with bound 2 is based on physically impossible “joint” measurement outcomes,
|⟨A(a, h) B(b, h)⟩ + ⟨A(a, h) B(b’, h)⟩ + ⟨A(a’, h) B(b, h)⟩ – ⟨A(a’, h) B(b’, h)⟩| ≤ 2 , ………….. (1)
it cannot be tested in real experiments. Only one of the four terms can be observed in a given experimental run. Therefore all real experiments are conducted on disjoint outcomes,
|⟨A(a, h₁) B(b, h₁)⟩ + ⟨A(a, h₂) B(b’, h₂)⟩ + ⟨A(a’, h₃) B(b, h₃)⟩ – ⟨A(a’, h₄) B(b’, h₄)⟩| ≤ 4 , … (2)
which is bounded by 4, or by 2√2. Now the followers of Bell are arguing that for the large N
⟨A₁B₁⟩ + ⟨A₂B’₂⟩ + ⟨A’₃B₃⟩ – ⟨A’₄B’₄⟩ = ⟨A₁B₁⟩ + ⟨A₁B’₁⟩ + ⟨A’₁B₁⟩ – ⟨A’₁B’₁⟩ . ………………….. (3)
Therefore it is OK, they argue, to perform experiments with disjoint measurement outcome, because the equality (3) permits us to write
|⟨A(a, h₁) B(b, h₁)⟩ + ⟨A(a, h₂) B(b’, h₂)⟩ + ⟨A(a’, h₃) B(b, h₃)⟩ – ⟨A(a’, h₄) B(b’, h₄)⟩| ≤ 2 ….. (4)
for the large N. Now there are only two logical possibilities here. Either the inequality (4) with the bound of 2 turns out to right in the large N limit, or it turns out to be wrong:
Suppose the inequality (4) turns out to be wrong and the correct inequality is actually (2). Then Bell’s theorem fails, because the inequality (2) is NOT the Bell-CHSH inequality, and since it is bounded by 4 (or 2√2) and not 2, it is certainly not violated in any experiments.
Suppose, however, that inequality (4) with the bound of 2 turns out to be right in the large N limit. But in the large N limit we also have the equality (3), so we are back to the Bell-CHSH inequality (1), which is now possible to test in real experiments, assuming fair-sampling. Let us grant this assumption and everything else the Bell-followers want to assume, and let them have their cake and eat it. Let them have a testable Bell-CHSH inequality (1), with equality (3).
But inequality (1) involves only Nx4 matrix of data points, with N rows, and the four columns marked as (A, A’, B, B’). Therefore there must exist (or can be constructed) such a Nx4 matrix of actual experimental data points (allowing whatever statistical fine tuning they want), that demonstrates that the Nx4 data constituting that matrix violates the bounds of +/-2 set by the Bell-CHSH inequality (1). So where is this impossible Nx4 matrix I have been asking for?
There is some mix-up here. The theory and computer simulations of the paper we are discussing give us a way to easily create this “impossible” Nx4 matrix. Simply let the computer simulation sample many values of h, and for each h compute all four of A, A’, B and B’.
My question is about actual experimental data, not what is produced using an uncontrollable hidden variable such as h.
But if what you are saying is true, then you should be able to easily construct an Nx4 matrix written out on a spreadsheet using h and publish it on the Internet. If it breaks the bound of |2| on CHSH, then Bell’s theorem would be proven wrong. If it does not break the bound of |2| on CHSH, then we would still be waiting for experimental data that breaks the bound.
Joy, Richard is saying what a local realist should supply to prove the correctness of a supporting local realism simulation. Real data, no matter what the experimenter’s philosophical beliefs, has only one Alice and Bob setting per row. And technically a local realism simulation need not supply an Nx4 matrix with two Alice and two Bob settings per row. But it can. And it is the easiest way to check the correctness. Otherwise the code must be checked line by line, and endless debate may ensue.
Joy has saved me some work over the next few days, because he offers the logic argument that even if Lord Jestocost is correct that his (7) and (8) above are equal, the logical consequence is still “devastating for Bell’s theorem.” (And as an aside, while I have not yet done the calculus calculation I proposed, my preliminary preparations for that calculation and some “hemispheric” drawings I developed lead me to believe that LJ is correct about his (7) and (8) being equal in the N -> oo limit, but to detail that here would go off point.)
So Joy’s argument brings us back to the following portion of LJ’s clear exposition of the pro-Bell position on December 15, 2016 at 10:04 am:
Jay,
regarding your last comments.
Any local hidden variable model devised for “two entangled spin1/2 particles in the singlet state” *must* be able to produce a unique Nx4 master list, where N is the number of different initial states (h) which can constitute a singlet state. For every initial state (h), the model *must* be able predict a unique and coherent output {A(a; h), A(a’; h), B(b; h), B(b’; h)} with A(a; h) either +1 or -1, A(a’; h) either +1 or -1, B(b; h) either +1 or -1 and B(b’; h) either +1 or -1.
All statistical analyses *must* then be carried out with this Nx4 master list. No other approach is allowed in case you want to perform “local-realistic” analyses! You again and again randomly select a row of this Nx4 master list and read out the values you are interested in. When you are interested in correlations like, e. g., A(a’; h)*B(b; h) you read out both values from row (h). You operate as a mere passive observer (local reality), viz. interventions in the list are no longer allowed after the list has been set up! I have no idea why somebody thinks that by that “you pick rows randomly without replacement“.
All any local-realistic model of the singlet state is required to produce are the correlations
E(a, b) = (1/N) Sum_k { A(a; h_k) B(b; h_k) } = -cosine(a, b)
using the local functions A(a; h_k) and B(b; h_k) defined in eq. (1) of Bell’s 1964 paper.
Nothing else is either required by the known experimental facts or demanded by realism.
LJ:
I just want to be clear so I know we are understanding the same thing. I like to stay close to what the physics actually is, so the way I look at this is. Please tell me if you agree:
Alice and Bob each set their detectors to one of two angles. That never changes. They must pick one or the other, based on spacelike-separated, local coin tosses. Then, we fire off doublets, each with a definite associated spin angle randomized in a direction anywhere over a 2pi circle. Each doublet registers +1 or -1 values readings as against the detectors for two of the four numbers in A(a; h), A(a’; h), B(b; h), B(b’; h) — the ones which were selected by spacelike coin tosses. The other two numbers are hidden, but they are still real. Those other two numbers also get put on the “readout,” because realism requires this. That gives you one four-element column of the Nx4 spreadsheet, what has been called the “slip.” Repeat over and over. And that is how we produce slips and the master list. Correct?
If so, then there are two things I have found confusing in this discussion, which may bear a fair share of the responsibility for the gap in reasoning between pro and con Bell:
First, I believe pro-Bell views these “hidden” slip values as “hidden variables.” I believe that Joy (contra-Bell) does NOT regard those as hidden variables because to him the HV are parity numbers of S^3. This point should be clarified from both sides. So you would say that in a non-LRHV there is no longer a need to specify the two “slip” values that did not get read out. Joy, I think would say that you dp still need to read out all four values from each slip, even if you do not believe in LRHV. He would let you off the hook only if you expressly abandoned realism which I thinks he sees as a separate issue from hidden variables. I’d like to hear clarity on this from both sides.
Second, related to this, if you are pro-Bell, then what actually gets killed off by the CHSH inequality when it is compared to the implications of QM correlations? Is it locality? Is it realism? Is it hidden variables? Is it one or all? Can we go down different routes in terms of which of these we give up prior to creating a theory of strong correlations? And I ask about realism and hidden variables separately because of what I wrote in the preceding paragraph, which makes me think that Joy would say you give up hidden variables (as he sees them) but not realism (because the reality of the four datum on the slip is not the hidden variable).
Thanks,
Jay
Just to nitpick a bit; the original polarization angle will be over a 2-sphere not just a 2pi circle. The a, a’, band b’ angles will most likely be somewhere over a 2pi circle orthogonal to the flight path of the particles. Also, don’t forget the polarizers that are before the detectors. They actually align the original spins to their angle, a or b, and then you detect either up +1 or down -1. If you are going to do the hardcore math of simulating EPR-Bohm, you need to account properly for the actual physics.
Fred, you have been at this way longer than me. So, based on your experience, in the simulations you all have done, does using a 4 pi sphere rather a 2 pi circle help you break out of the outer CHSH bound of 2, up to, say, 2√2? For example, with a sphere, I can place four vectors (corresponding to a, a’, b, b’ direction) to start at the origin, point each at an equilateral tetrahedron vertex circumscribed on the sphere surface, and give myself a set of ab, ab’, a’b, a’b’ angles that are unavailable if I restrict movement to a circle. (For example, the dihedral angle has tan(theta) = 2√2, the tetrahedral angle has 2*tan(theta/2)=2√2, and the face-vertex-edge angle has 2*tan(theta)=2√2. All rather convenient numbers which maybe change the CHSH correlation bounds? (See https://en.wikipedia.org/wiki/Tetrahedron#Formulas_for_a_regular_tetrahedron)
I was cognizant of using a sphere not a circle, but I made the simplifying assumption (which could be wrong) that staying on a circle would not change the substantive outcome, at least long enough to get me to the point where I would set up the calculation and discover that I could not make that assumption. Thanks, Jay
Jay,
Adding to Fred’s comments above. It is even more complex than that. The original vectors on the 2-sphere will not be stationary but rotating and precessing in anti-unison with the same frequency. In which case the hidden variables will correspond to all the parameters that govern the common dynamics of the particles, including the time. Then to make things even more complicated, you also have hidden variables in the polarizers that include measurement times, and local dynamics of the components that directly interact with the incoming particles.
Given all of that, I really will like an answer to my question I asked LJ yesterday:
Jay, no one thinks of “slip values” as “hidden variables.” You shouldn’t be using the word “hidden” for them at all. The four possible numbers A(a; h), A(a’; h), B(b; h) and B(b’; h) from the infinitely many points of a unit 2-sphere are counterfactually possible measurement outcomes. This terminology is not specific to my S^3 model. It is Bell’s original terminology, and it has been universally accepted. Hidden variable is the h in A(a, h), not the number A = +/-1 itself. Realism within Bell’s framework simply means the “counterfactual definiteness” of the infinitely many numbers A(a, h) — which all exist counterfactually, whether or not Alice and Bob go anywhere near their respective labs. The hidden variable h, on the other hand, simply means the initial state of the singlet system, originating independently of A, a, B and b in the overlap of the backward lightcones of Alice and Bob. Therefore it is also the “common cause” of the results of Alice and Bob, or the “randomness” they both share.
Secondly, for me what gets killed off by the CHSH argument is neither locality nor realism of the numbers A(a; h) and B(b; h), but the impossibility of the number A(a, h) coexisting with both B(b, h) and B(b’, h) simultaneously, as in the quantity A(a, h) { B(b, h) + B(b’, h) }. I have shown this explicitly by deriving the CHSH inequality without assuming locality (one cannot kill off hidden variables h by Bell’s argument, because Bohm’s non-local realistic theory already exists — it has existed since at least 1952, long before the Bell’s argument of 1964):
http://libertesphilosophica.info/blog/wp-content/uploads/2016/11/Fatal.pdf .
Actually, let’s switch this around. Instead of putting the detector angles on a tetrahedron, what we do is use the N vertices of a regular polyhedron (call it an N-hedron) to point at the regularly-distributed random directions of the single-to-doublet orientations. In two dimensions a circle has four quadrants separated by 90° but in three dimensions if you define four non-orthogonal axes projecting from an origin (and think SU(4) for this if you know group theory), this 90° maps into 2*tan(theta/2)=2√2 which is 109.4712°. So you can actually build a geometry for the singlets around this key number 2√2. This would use OA, OB, OC and OD in https://en.wikipedia.org/wiki/File:%D0%92%D0%BF%D0%B8%D1%81%D0%B0%D0%BD%D0%BD%D1%8B%D0%B9_%D1%82%D0%B5%D1%82%D1%80%D0%B0%D1%8D%D0%B4%D1%80.svg as your four axes, obviously with a redundant degree of freedom.
Now, when Fred tells me that the experiments use a polarizer, that makes me nervous, because unless I misinterpret the polarizer function, I interpret that to mean that you are shaving off one dimension and turning the sphere into a circle, as well as “selecting” your data by processing out data that does not pass through the polarization screen. I assume people have thought about this before, and that (what I am interpreting as) the reduction down from sphere to circle does not substantively impact the detected results?
No, it doesn’t. For the “Bell test angles” 0°, 45°, 90° and 135° on a circle you can get 2√2.
The measurement settings are on a circle and the results of the measurement will be a result of some projection of the sphere (of the particle vectors) onto the circle determined by the settings.
That is just what I envisioned originally. But just to be sure, are you confirming that the figure at https://en.wikipedia.org/wiki/Bell's_theorem#/media/File:Bell_test_for_spin-half_particles_(entangled_qubits).svg shows how we can lay this out conceptually, on a flat page, with three circles? Two of these on the left and right represent measurement directions. And the one in the middle represents the projections of the particle vectors. Correct?
As the putative “mediator” of these discussions, and to provide a common point of reference for our discussions, I am going to make the suggestion at all that we focus our discussions around the Wiki article about Bell at https://en.wikipedia.org/wiki/Bell's_theorem. Wiki articles are never perfect, but the Bell article was put together by many of the people here, and at least it gives us a way to know that we are all talking about the same things at any given point in time. Perhaps in this way we can perhaps filter out “miscommunication” and “misunderstanding” and have left only true “disagreement.”
I would like to start at https://en.wikipedia.org/wiki/Bell's_theorem#Bell_inequalities_are_violated_by_quantum_mechanical_predictions. Joy has been repeatedly asking the question, which I will paraphrase, “show me the A and B correlations that permit us to obtain:
|⟨A(a)B(b)⟩₁ + ⟨A(a)B(b’)⟩₁ + ⟨A(a’)B(b)⟩₁ – ⟨A(a’)B(b’)⟩₁| gt 2
and in fact allow us to have this reach as large as 2√2.”
Joy is expecting that A(a), A(a’), B(b), B(b’) will all have the scalar number values +1 or -1. And of course, with this constraint, there is no way to exceed 2.
But in the Wiki section at https://en.wikipedia.org/wiki/Bell's_theorem#Bell_inequalities_are_violated_by_quantum_mechanical_predictions, we find that A(a), A(a’), B(b), B(b’) are no longer scalar numbers. Rather, they are treated as 4×4 matrices obtained through the tensor products shown in the fourth set of equations in this Wiki section. Obviously, if you permit yourself this change in the structural properties of A(a), A(a’), B(b), B(b’), then you can go outside the bounds, as is shown in the final result of that section.
So, to contra-Bell and Joy in particular:
This is how pro-Bell breaks the CHSH bounds. Is this legitimate? Why or why not?
And to pro-Bell, since you get outside the CHSH bounds in this way by changing A(a), A(a’), B(b), B(b’) from scalar numbers +1 and -1 based on sign(a.s) etc., to matrices with eigenvalues +1 and -1, on what basis do you justify this? And how do we talk about what Alice and Bob “record” on their spreadsheets from their individual data point measurements? Are they now writing down matrices instead of numbers?
Jay
Jay, your paraphrasing of my question leaves escape routes for the Bell-followers. Therefore I prefer my original carefully preambled question, which was spelled out at the following post:
http://retractionwatch.com/2016/09/30/physicist-threatens-legal-action-after-journal-mysteriously-removed-study/#comment-1224819
It is of course legitimate to use operators, or matrices, or Clifford algebra to derive the quantum mechanical prediction of the correlation and claim that they exceed the bounds of |2|. But what is not legitimate in my eyes is the often made claim by the Bell-followers that the bound of |2| is also violated in the actual experiments, which by construction only observe and record +/-1 clicks of the detectors. If such actual detector results — which for the Bell-CHSH correlator necessarily constitute an Nx4 matrix of numbers with N rows and four columns marked (A, A’, B, B’) — do “violate” the Bell-CHSH inequality as often claimed, then I would very much like to see an explicit demonstration of such a violation.
Joy,
Let me see if I can paraphrase without leaving an escape route. But it is important to do so with equations, and with a paucity of words since those can be misinterpreted:
Experiments show a correlation:
⟨AB⟩ = – a dot b = – cos (theta) (1)
For a locally-realistic hidden variable theory, the CHSH sum with h = (lambda, s) is constrained according to the joint trial expression:
|⟨A₁(a; h)B₁(b; h)⟩ + ⟨A₁(a; h)B₁(b’; h)⟩ + ⟨A₁(a’; h)B₁(b; h)⟩ – ⟨A₁(a’; h)B₁(b’; h)⟩| ≤ 2 (2)
where the measurement functions are defined according to A₁(a; h)==sign(a.s)=+/-1; A₁(a’; h)==sign(a’.s)=+/-1; B₁(b; h)==sign(b.s)=+/-1; B₁(b’; h)==sign(b’.s)=+/-1, and as such, are all scalar numbers +1 or -1. (To be clear: 1=i is an index that is summed, and is summed jointly for all four terms. I am borrowing the subscripted ₁ from earlier posts to avoid having to play with Unicode.)
Your bottom line position is that contrary to popular belief, the experimental correlations (1) do NOT contradict the CHSH sum (2) and therefore do not contradict local realism.
Is that a paraphrase you can accept as leaving no escape route?
Jay
Yes, that is what I believe and that is my position. But I am quite happy to be agnostic about the actual correlations [your eq. (1)]. They need not enter my argument, nor my request of an Nx4 matrix of actual experimental data that violates the bounds of +/-2 set in your eq. (2).
However, if you want to argue that LRHV theories are not contradicted by experimental correlation data, then the experimental data you are referring to and arguing from is that which is represented in:
⟨A(a)B(b)⟩ = – a dot b = – cos (theta) (1)
Correct?
Let me put this slightly differently: Can the actual experimental data be written as an Nx4 matrix of N rows and four columns (A, A’, B, B’), or must it be written only as an Nx8 matrix?
If the data can be written as an Nx4 matrix, then please show me explicitly how the data violates the bounds of |2| on the CHSH sum.
If the data can only be written as Nx8 matrix, then we are done here, because then the bounds on the CHSH sum are not |2| but either |4| or |2√2|, and those are not violated in any experiments that I am aware of.
A relevant related paper just came out https://arxiv.org/abs/1612.03606
Referring again to the fourth set of equations in https://en.wikipedia.org/wiki/Bell's_theorem#Bell_inequalities_are_violated_by_quantum_mechanical_predictions in which A(a), A(a’), B(b), B(b’) are defined as operators, not as the scalar numbers +1 and -1 which are the operator eigenvalues and are the actual direct datum extracted from experiments, the answer to your request for explicit data is that the CHSH sum can be written so as to exceed |2| using data entries on an Nx4 spreadsheet, but in that situation, each data entry must itself be a 4×4 matrix operator, not the scalar eigenvalues of these operators. If you restrict to scalar numbers which are operator eigenvalues and which are actually extracted from experiments, then you must use an Nx8 matrix.
Getting to the root of this, I can write an eigenvalue equation as:
A psi = a psi (1)
for an operator A operating on a state vector psi and producing a scalar eigenvalue a in a given eigenstate of psi. But I cannot strip off the psi and write:
A = a. (2)
The latter, (2), is an invalid equation.
I think what you are actually saying is that the attempt to invalidate LRHV theories using the CHSH inequality effectively includes turning a valid relation (1) into the invalid relation (2), i.e, equating an operator to a scalar number, and then using (2) to consummate the disproof of LRHV.
Might this be a fair restatement of your position?
Jay
Needs to be considered in conjunction with an oldie but goodie from Howard Wiseman:
https://arxiv.org/abs/quant-ph/0509061
Jay, I am not sure why you want me to rephrase my very clearly spelled out request above:
http://retractionwatch.com/2016/09/30/physicist-threatens-legal-action-after-journal-mysteriously-removed-study/#comment-1224819
There is no need to bring in quantum mechanics, or operators, or local realism, or anything else to understand my simple request. I am only concerned in my above request about the actual experimental data. What is observed are nothing but yes-no “clicks” of the detectors, represented by +1 or -1 numbers. Therefore I am only concerned about +1 or -1 numbers.
The question then is: Can the actual experimental data collected from recording only such +1 or -1 numbers be written as an Nx4 matrix of N rows and four columns (A, A’, B, B’), where each A, A’, B and B’ is equal to either +1 or -1? Or must the data be written only as an Nx8 matrix? These questions arise because there are claims that at least in the large N limit we have the following equality (the claims I am not necessarily questioning here):
⟨A₁B₁⟩ + ⟨A₂B’₂⟩ + ⟨A’₃B₃⟩ – ⟨A’₄B’₄⟩ = ⟨A₁B₁⟩ + ⟨A₁B’₁⟩ + ⟨A’₁B₁⟩ – ⟨A’₁B’₁⟩ .
Again, these are all averages of the actually observed +1 or -1 numbers, nothing else. But it is an elementary mathematical fact — recognized by George Boole some 100 years before John Bell — that the RHS of the above equality cannot possibly exceed the bounds of |2|.
Now, if the experimental data can be written as an Nx4 matrix, then someone please show me explicitly how that N x (A, A’, B, B’) data violates the bounds of |2| on the CHSH sum
⟨A₁B₁⟩ + ⟨A₁B’₁⟩ + ⟨A’₁B₁⟩ – ⟨A’₁B’₁⟩ ,
which is the sum of pure number (cf. CHSH paper; their sum only involves +/-1 numbers).
If, however, the data can only be written as an Nx8 matrix, then (as I have noted before) we are done here, because then the bounds on the above CHSH-like sum are no longer |2| but either |4| or |2√2|, and those bounds are not violated in any experiments that I am aware of.
Joy, you and the Bell-ists agree the rhs cannot exceed 2. Bell equated the lhs and rhs for a “proof by contradiction”. He said if local realism were true that equality would hold, and thus the lhs could not exceed 2. But in quantum experiments the lhs exceeds 2. Hence he discatded the hypothesis of local realism.
This is a false claim. I am asking all followers of Bell to prove it. No one has proved it so far.
Joy,
What you stated above does not have to be rephrased at all. It is perfectly-stated as is. However, the reply by Earl of Snowden is exactly what I would expect to be the reply from what he calls the Bell-ists, and illustrates perfectly why I keep rephrasing you anyway, and talking about operators. This is because you see your point plainly, but the Bell-ists do not see the relevance of the question you keep asking, for the very reason EOS stated.
Let me put it this way. Take the operators A(a), A(a’), B(b), B(b’) defined in the fourth equation set at https://en.wikipedia.org/wiki/Bell's_theorem#Bell_inequalities_are_violated_by_quantum_mechanical_predictions, and write them as A(a)-op, A(a’)-op, B(b)-op, B(b’)-op. Extract from these, the usual +/-1 numbers for the A(a), A(a’), B(b), B(b’) observed in EPR-Bell experiments using state vectors psi and the eigenvalue equations:
A(a)-op psi = A(a) psi (1a)
A(a’)-op psi = A(a’) psi (1b)
B(b)-op psi = B(b) psi (1c)
B(b’)-op psi = B(b’) psi (1d)
The CHSH eigenvalue inequality based on local realism is (again, 1=i is a summed index):
⟨A₁B₁⟩ + ⟨A₁B’₁⟩ + ⟨A’₁B₁⟩ – ⟨A’₁B’₁⟩ ≤ 2 (2)
However, the CHSH operator inequality based on QM experiments is:
⟨A-op B-op⟩ + ⟨A-op B’-op⟩ + ⟨A’-op B-op⟩ – ⟨A’-op B’-op⟩ ≤ 2√2 (3)
The question you have been repeating, which has fallen in deaf Bell-ist ears, arises because they are using the operator CHSH inequality (3) as a contradiction to the eigenvalue CHSH inequality (2). And your question to them is based on the view that using (3) to contradict (2) is invalid, and is equivalent to removing the psi from (1) and writing the invalid equations:
A(a)-op = A(a) (4a)
A(a’)-op = A(a’) (4b)
B(b)-op = B(b) (4c)
B(b’)-op = B(b’) (4d)
EOS said “in quantum experiments the lhs exceeds 2.” I think you would state that this is false, and rather, that “in quantum experiments the CHSH sum (lhs or rhs) does NOT exceed 2. But if one replaces A(a), A(a’), B(b), B(b’) which are experimental outcomes with QM operators A(a)-op, A(a’)-op, B(b)-op, B(b’)-op for which those are eigenavlues, then a CHSH sum formed out of these operators can go as high as 2√2. But, you cannot use the operator CHSH (3) as a contradiction to the eigenvalue CHSH (2) because operators do not equal eigenvalues.”
Jay
Yes, I agree. What you spell out is indeed the underlying rational for my repeated request. What I am questioning is what I see as comparing apples with oranges to justify Bell’s claim.
But my question can be phrased quite simply in terms of Nx4 and Nx8 matrices we had been discussing lately, without having to involve anything more elaborate. If the sum of quantum correlations -a.b exceed the bounds of |2| on an equality | CHSH | ≤ 2√2, then why should anyone be surprised? If apples do not turn out to be like oranges, then why should we care?
Yes, Joy, it can be phrased that way also. But you have been saying that forever and it is not hitting home outside the contra-Bell group. So it is time to apply Einstein’s definition of insanity and take a different tack.
In this vein, perhaps the thing to do is ask the pro-Bell folks to admit that whenever they derive a CHSH sum outside the bound of |2|, they MUST be using operator matrices rather than the simple eigenvalues +1 and -1 of those matrices which are extracted from sign(a.s) etc. in real experiments for A and B.
I think the conclusion you are driving at is that when a CHSH sum using operators is compared against a CHSH sum that uses the ordinary scalar numbers, and that when that comparison is used to claim a contradiction, the claim of contradiction is invalid, because one is comparing apples (operators) to oranges (the observable numbers associated with eigenstates of those operators).
In this light, another way to then phrase Joy’s question is this: Show how to get a CHSH sum outside the bounds of |2| a) using only the numbers +1 and -1 which are extracted from sign(a.s) etc. for A and B, and b) without using any operators for A and B, and c) using only Nx4 spreadsheets of As and Bs. The answer: it is impossible to do so. You can only use Nx8 spreadsheets. If you demand an Nx4 spreadsheet you must use operators not numbers.
Slightly restated: With only the numbers +1 and -1 you MUST use an Nx8 spreadsheet to get outside of |2|. If you want to get outside |2| with an Nx4 spreadsheet, you MUST use operators in each spreadsheet entry, and CANNOT use only the numbers +1 and -1. Pro-Bell team: Am I correct? I am the student asking my Bell’s Theorem teachers a question.
If I am correct, then the question becomes: can you compare a CHSH sum that uses operators for A and B, against a CHSH sum that is restricted to only the scalar numbers +1 and -1 for A and B, to assert a contradiction from which to draw any fundamental conclusions?
Jay
And I should also add to my last post, that when N->oo and averages approach expectation values, so that:
⟨A₁B₁⟩ + ⟨A₂B’₂⟩ + ⟨A’₃B₃⟩ – ⟨A’₄B’₄⟩ = ⟨A₁B₁⟩ + ⟨A₁B’₁⟩ + ⟨A’₁B₁⟩ – ⟨A’₁B’₁⟩
(which I have studied enough to agree with pro-Bell that this is indeed a correct relation), this means that even with an Nx8 spreadsheet, you can no longer get outside the bounds of |2| without using operators in place of scalar number eigenvalues.
In the final CHSH-like string where they show that it is 2*sqrt(2), there is no indexing. When and where do those four expectation terms happen? Certainly for QM those four terms can’t happen all at the same time. It is just plain bad math. Someone should fix that on Wikipedia.
Actually, Fred, you are a bit ahead of me, because I was going to suggest that this august group of RW participants agree on some modifications to clean up the Wiki Bell article in some places, including this section. But after we clean up our own arguments. Where the article says “citation needed” in advance of the four expectation terms right before the 2√2 result, more than a citation is needed. The expectation value calculation should be shown. I have been playing with that exact calculation today because of its absence in the article. Keep in mind that A and B in this section are operators, so that you get to expectation values using https://en.wikipedia.org/wiki/Expectation_value_(quantum_mechanics)#Formalism_in_quantum_mechanics. But I agree that the calculation needs to be very explicit, and the summation needs to be shown, or explained. This is a clear weakness in the Wiki article. Maybe somebody wants to do that now? Everybody who cares about Bell’s theorem is here!
Also, the operators laid out in that article are specific to the case of the 45 and 135 degree angles that one uses to get the maximum strong correlation, with
⟨AB⟩ = – a.b = -cos(pi/4) = 1/√2. (1)
I would like to see those operators generalized so that one can obtain
⟨AB⟩ = – a.b = -cos(theta a,b) (2)
in the fully general case. That is the other thing I have been playing with today. The place to start for this project, is with the boxed equation in https://en.wikipedia.org/wiki/Pauli_matrices#Exponential_of_a_Pauli_vector, and also the angle difference identity (alpha and beta below are now angles)
cos(alpha-beta) = cos alpha cos beta + sin alpha sin beta (3)
The above (3) appears necessary, because the A operator needs to be independent of the b direction and the B operator independent of the a direction, yet when A and B are multiplied you have to get cos(theta a,b) =cos(alpha-beta).
More generally, multiplying the exponential (unitary) form of the Pauli matrices is the best way there is to add and subtract angles. After all, that (and generalizing imaginary numbers) is what Hamilton was doing when he first conceived quaternions, which as far as I have been able to find (and I have looked), was the first time that non-commuting objects were ever used in math, much less physics. Nothing non-commuting from Gauss or any obvious predecessors. And today you cannot take even a step in physics, without having to ask whether two objects commute or do not.
Jay
Jay, here is the full derivation of your eq. (2).
https://drive.google.com/file/d/0B67qmvk4E9ZzODFmNjE1MWEtNGY4ZC00Y2EyLTllN2UtNTU4MGEwYWViZjBi/view
Here is a proof by contradiction that sqrt(2) is irrational. Suppose it were rational. Then sqrt(2)=m/n where m and n are integers with gcd(m,n)=1. Then m^2=2n^2. Thus m^2 is even. Which implies m is even. So 4 divdies 2n^2. So 2 divides n^2. So n^2 is even. Thus n is even. So gcd(m,n)>=2. This contradicts gcd(m,n)=1. Hence sqrt(2) is irrational. A like objection here to your objecion to Bell would be: “I won’t accept your proof until you supply integers m and n with gcd(m,n)=1. and m^2=2n^2.”
My objection to Bell is nothing like what you are suggesting, because to begin with only +1 and -1 numbers are observed in the experiments; and even if one can demonstrate that the sum ⟨A₁B₁⟩ + ⟨A₂B’₂⟩ + ⟨A’₃B₃⟩ – ⟨A’₄B’₄⟩ exceeds the bounds of |2|, so what? The sum ⟨A₁B₁⟩ + ⟨A₂B’₂⟩ + ⟨A’₃B₃⟩ – ⟨A’₄B’₄⟩ is not bounded by |2|. It is bounded by |4| or |2√2|.
But “operators” are not what is observed in the actual EPR-Bell experiments. What is observed are just yes-no “clicks” of the detectors, traditionally represented by +1 and -1 numbers.
And yet, we frequently come across claims such as “experimental violation of Bell inequality” in both theoretical and experimental literature published in prominent journals. But no such violation has ever been demonstrated by anyone, either theoretically or experimentally.
Now it is quite legitimate to use operators, or matrices, or Clifford algebra, or wavefunctions to derive the quantum mechanical prediction of the singlet correlations theoretically, and claim that their CHSH-type sum exceeds the bounds of |2|. I have no problem with that.
But it is illegitimate to claim that the bounds of |2| are exceeded in the actual experiments, when by construction experiments only observe and record + or – clicks of the detectors. If there are genuine violations of the Bell-CHSH inequality in Nature, then experimentally they must be demonstrated by plugging-in the + or – numbers directly into the CHSH sum
⟨A₁B₁⟩ + ⟨A₂B’₂⟩ + ⟨A’₃B₃⟩ – ⟨A’₄B’₄⟩ = ⟨A₁B₁⟩ + ⟨A₁B’₁⟩ + ⟨A’₁B₁⟩ – ⟨A’₁B’₁⟩
in the large N limit. If the actual detector results do “violate” the Bell-CHSH inequality as claimed, then I would very much like to see an explicit demonstration of such a violation.
Joy, data from quantum experiments can be plugged in to the lhs and exceed 2. The equality is only thought to hold if local realism is true. In real experiments the rhs could only be evaluated if local realism is true, and someone could figure out a way to access the hidden variables and supplement the data. Computer simulations of local realism can easily output data to evaluate the rhs, and it is the “low debate” way of checking them. You and the Bell-ists agree the rhs cannot exceed 2. This contradiction with QM experiment is why Bell-ists reject local realism. If you want to see real data on this, maybe the authors of QM Bell violation papers would share it with you if you asked?
You have been missing my point consistently. If the lhs = rhs (or Nx8 = Nx4) does not hold for the actual experiments, and if the experiments only apply to the lhs, then the bounds on the lhs sum is NOT |2|. Then lhs is bounded by either |4| or |2√2|. Why should then anyone be surprised if the sum on lhs — with the bounds of |4| or |2√2| — exceeds the bounds of |2|? On the other hand, no experimental data can violate the bounds of |2| on the rhs.
I would be surprised because if I took an Nx4 matrix with every cell filled by a +1 or a -1, which we agree cannot exceed the bound 2, and tried to turn it in to a matrix like one would see in an actual experiment, by randomly replacing one of the two Alice observations with an NA and one of the two Bob observations with an NA, then for a LARGE number of rows, the lhs calculation would have almost no chance of exceeding the bound 2 by a statistically significant amount. Try it in R. (The choice of where to put the NA’s must be random and independent of what the rows look like.)
That argument which was first proposed by Gill fails for several reasons:
1) The 4 Nx2 spreadsheets you generate in this way are not statistically independent, but the 4 Nx2 spreadsheets in ⟨A₁B₁⟩ + ⟨A₂B’₂⟩ + ⟨A’₃B₃⟩ – ⟨A’₄B’₄⟩ are.
2) Even if they were independent (which they aren’t cf (1)), it is not true that Local Realistic theories MUST produce an Nx4 spreadsheet of outcomes. The EPR elements of reality do not have to be the outcomes themselves. Just because the outcomes may not be simultaneously available does not mean the hidden elements of physical reality are not simultaneously present.
3) It is a fact that the outcomes in the EPRB experiment are not simultaneously available so a 4xN spreadsheet does not exist for that experiment, irrespective of the kind of theory you want to use to model it.
4) Even if a 4xN spreadsheet of outcomes existed (which it does not cf (3)), and even if the 4 disjoint Nx2 spreadsheets were independent (which they are not cf(1)), the cyclical recombination of rows in the paired Nx2 spreadsheets means that it is not always possible to faithfully sample the Nx4 spreadsheet using data from 4 separate disjoint Nx2 spreadsheets (see http://www.panix.com/~jays/vorob.pdf)
5) Anything you try in R will be one specific model (your imagination) of what a hidden variable theory is. Just because your imagined model does not exceed the bound says absolutely nothing about the possibility of any other model violating it. In other words, the probability of find a model which violates it may be 0.00001. So long as it is not 0, the argument fails. It does not help pro-Bell one iota to suggest that the probability of finding a model which exceeds 2 is low. You need to show that it is P=0.
6) See https://arxiv.org/pdf/0907.0767.pdf, https://arxiv.org/pdf/1108.3583.pdf, https://arxiv.org/pdf/1605.04887.pdf
(The procedure I just described creates data modelling local realism.)
It has been rather quiet here for the past several days. 🙂
I would like to break the silence with a nine-page document I prepared and posted at https://jayryablon.files.wordpress.com/2016/12/unitary-operators-and-quantum-correlations.pdf.
As I have stated before, one of the thing I find incredibly frustrating about the EPR-Bell discussion is that there are too many words and too many different uses of language and definitions, all of which causes a great deal of confusion and miscommunication and makes it even more difficult to resolve issues on which there is true disagreement.
I prefer sticking as closely as possible to the mathematics, and that is what I have done in this document. Basically: the EPR-Bell experiments and the quantum correlations all center around five (normalized to magnitude 1 unit) vectors which point in various directions in three-dimensional space: the four a, a’, b, b’, detector vectors for Alice and Bob which are given are fixed orientation in a single plane, and a fifth set of vectors s_n in which a succession of n=1…N randomly-distributed “arrows” or “pointers” or “spins” are thrown toward Alice and Bob to be detected.
It seems that we should at least engage in the basic geometric exercise of carefully mapping out these vectors in physical space using the unitary matrices of SU(2) which conveniently allow us to represent angle and angle differences in space whether we are looking at a classical problem or at a quantum mechanics problem. And it does not hurt as I shown in (1.10) of this document, that these same matrices are organically-embedded in the correlations predicted by quantum mechanics. So that is what I have done here. I do not think that anything in this document should generate any disagreement from any Bell faction, and would like to know whether I am correct in this belief.
Happy holidays and new year to everyone, whether that means Chag Sameach for Hanukkah or Merry Christmas or anything else!
Jay
So long as you understand that your simplification assumes Alice and Bob measure at exactly the same time. Otherwise, it is too simplistic.
The spin does not have to be stationary, but is most likely following some local dynamics, changing direction with time such that measuring at t1 will give a different result from measuring at t2. Thus both measurements need to be measured at the same time, or synchronized times to have consistent results.
Secondly, when you say s is randomly distributed between 0 and 2pi, that is incomplete. Even if the vector was uniformly distributed on a sphere, which it doesn’t have to be, the projection on the measurement plane won’t be. In other words, you need to be explicit about your chosen distribution of (s), and justify it or assume it. It is not enough to just say “randomly”.
Other than the hidden implicit assumptions, your description is good.
Jay’s description is good as far as it goes, but I prefer brevity. QM predictions can be more elegantly derived in just three lines, as done here: https://arxiv.org/abs/quant-ph/0703179
Although Jay’s goal seems to be different, my main concern with his description is that spin should not be represented by a vector at all. Vectors commute, just like scalars do; whereas rotations in the physical space do not commute. Therefore spin must be represented by a non-commuting bivector, as I have done in my paper linked above. The rest of Jay’s analysis can then be put in a much more compact and elegant form.
Joy, your (3) to (6) in https://arxiv.org/abs/quant-ph/0703179 and my (1.10) and (2.10) in https://jayryablon.files.wordpress.com/2016/12/unitary-operators-and-quantum-correlations.pdfsay exactly the same thing. But I felt it important to unpack those relations and especially show how they embed the unitary matrices of SU(2). Not only do those unitary matrices put the geometric layout of EPR-Bell on clear display, but they have the “incomplete information” properties I lay out on the final page which upon further development appear to go to the heart of the EPR question whether, and under what circumstances, the “quantum mechanical description of physical reality” can be “considered complete.”
Also, I wanted to start from a broader range of possibilities which includes uniformly-randomly oriented macroscopic “arrow” pairs being tossed toward Alice and Bob, in which case the “arrow” vectors are ordinary vectors. Narrowing these pointer vectors from “arrows” to “spins” with SU(2) rotational properties then becomes a discrete step, which allows the differences between “classical” and “quantum” EPR-Bell experiments to be made clear.
Yes, of course.
I do not know where the verbiage went, but when I was drafting this I had written also about the random distribution being “uniform” on the surface of the sphere over large numbers of trials, projecting uniformly onto the plane over 0 to 2pi. Agreed.
Also, when you project unit vectors from a sphere to a plane, you do not get unit vectors and when orthogonal, the projection is a point. I would think the magnitude of the projection will be significant as far as the physics is concerned.
Yes, but that is all accounted for in calculations like (1.10) of https://jayryablon.files.wordpress.com/2016/12/unitary-operators-and-quantum-correlations.pdf which work in three space dimensions. Then steps like those used to go from there to (1.14) tell you what happens when you restrict to a plane, measurement of spins that can be directed anywhere in three dimensions. I say calculations “like these” because spin is not explicit in those (see (1.5) where they cancel out) but when we are using spins in measurement functions the calculations will be similar, and will use the unitary operators in all three dimensions until we decide to look at only a plane.
As to orthogonal vectors projecting onto a point, if I am to answer for those, then anybody who has ever used the measurement function sgn(cos(a)) = +1/-1 has to answer for that, because by leaving out sgn(cos(a)) = 0 which forms an entire equator on the sphere one is always assuming that when you get out to enough significant digits, the angle will not be exactly orthogonal, but will have at least a tiny bias in one direction or another.
That is right Jay, I’m just putting it out there in case anyone was trying to actually simulate this by generating the vectors.
But keeping that aside, there is an even more intuitive way to look at -a.b. “a.b” is the magnitutde of the projection of “a” on “b” (or “b” or “a”) which is a measure of the “same-direction-ness” of both measurement vectors. (-1) is the internal anti-correlation between two spin-half particles, ie from within their joined reference frame. (-1)*(a.b) is just the internal perfect anti-correlation scaled according to the “same-direction-ness” of the measurement directions.
I don’t think that matters so much as the polarizers before the detectors align the spin “s” with its angle if it can as I have been saying for EPRB. At that point it is just up or down detection. Of course you have to have some kind of timing to match pairs. Not sure if it has to be exactly at the same time.
Fred, the “aligning” does not necessarily eliminate the problem. If the polarizer has two ways to align a spin, then it must have a mapping from what it sees to the two directions it aligns to.
Let us assume that the particle is like a spinning top and the polarizer aligns all clockwise to UP and all counter-clockwise to “DOWN”. Now assume that the two particles are always spinning in opposite directions. Time will not matter if there is no additional dynamics of the spins. But if the spinning tops we tumbling in space with the same frequency then although the spint will always be opposite when observed at the same time in the same direction, if you delay one of the observations, you might find that the top which should have been anti-clockwise has rotated until the spin now appears clockwise. Thus, the polarizer only aligns based on what it sees and if there is dynamics involved, what it sees will depends on the time it saw it. You could even detect at the same time but if the polarizers were positioned with different path-lenghts you end up with the same problem. I don’t think the existence of the polarizer makes any difference. You can combine it with the detector and just all it the “detector”. The effective time is the time at which the whole process started. That is the time that needs to be the same. Unless of course you have a different idea about how the aligning is supposed to work but the point I’m raising is that it is important to explicitly state all assumptions so that the implications of any conclusions are not broadened beyond the scope of the starting assumptions.
Hmm… that puts a different “twist” on polarizer action. Thanks. We were able to mimic polarizer action in the complete states simulations a couple different ways. Joy in R by
sign(cos(a – e))
And I in Mathematica by,
If[Abs[a – e] < pi, then polarizer = +1 else polarizer = -1]
Both for A station and similar for B.
As things settle down for everybody during this final week of the year, I now have a few questions to ask based on what I posted the other day at https://jayryablon.files.wordpress.com/2016/12/unitary-operators-and-quantum-correlations.pdf. And as background, I work from the viewpoint I have seen expressed here that there is a “tripod” of three -cos(theta) correlations C which need to be explained and reconciled: the C_qm predicted by quantum mechanics, the C_lrhv which (depending on one’s viewpoint) can or cannot be explained by a local realistic hidden variables theory, and the C_ex which are observed in physical experiments.
1. Does everybody agree that these are the three correlations which need to be explained and reconciled?
2. Does everybody agree that (1.10) which includes (1.11) in the file linked above correctly describes the correlations predicted by quantum mechanics, and that the equality of the expected value to –a.b using expected values in the final line is properly obtained via (2.9) leading to (2.10)?
3. Referring to (1.6) and (1.7), given that sgna.s=+/-1 and sgnb.s=+/-1 are widely regarded to be “elements of reality,” is there any reason that a.s and b.s themselves, which contain even more information than their sgn functions, ought not also be regarded as elements of reality, in accordance with accepted “realism” definitions?
4. Therefore, would there any any problem with regarding the operator functions A and B in (1.6) and (1.7) to be actual elements of reality, with the only “hidden” variables being sgn(axs) and sgn(bxs)? And do not these A in (1.6) and B in (1.7) actually contain more information than would sgn((1.6)) and sgn((1.7)) by themselves? That is, sgn((1.6)) and sgn((1.7)) only know a sign, while (1.6) and (1.7) know everything except for a sign.
5. Because Alice’s operator A in (1.6) is not at all dependent upon Bob’s detector direction choice b nor is Bob’s operator B in (1.7) dependent upon Alice’s detector direction choice a, is there any reason to not regard A and B as local functions in accordance with accepted “locality” definitions?
6. Further to question 4, because cos(alpha-sigma) in (1.6) and cos(sigma-beta) in (1.7) can be predicted with certainty irrespective of the +1 or -1 values of the “hidden” variables sgn(axs) and sgn(bxs), is there ant reason to not regard sgn(axs) and sgn(bxs) as, in fact, “hidden variables” in accordance with accepted definitions of these?
7. Therefore, is there any reason not to claim that (1.10) and (1.11) in view of (1.6) and (1.7) and (2.9) and (2.10) prove by direct mathematical deduction that the C_qm predicted by quantum mechanics are in fact explained by a local realistic hidden variables theory whereby C_qm=C_lrhv?
8. Of course, the unitary measurement functions in (1.6) and (1.7) are operators, not the numbers +/-1. So would it also be correct to enter the admission that while all of this does show C_qm to be locally realistic and the same as C_lrhv, all of this does not yet explain the the third leg of the tripod, which are the experimental correlations C_ex based on sums over many trials of discrete binary correlations and anti-correlations?
That should do it for now.
Jay
Only the experimentally observed C_ex needs be explained. As the quantum mechanical correlation C_qm is the same, C_qm = C_ex, it is explained by the quantum mechanics. A hidden variable theory is not needed but if one is desired anyway, it should predict the same correlation. In addition, it should agree with other quantities observed in the same experiments.
See point ( 7), or better, read Jay’s PDF.
Hi Mikko,
Your reply contains a mix of correct statement and confused statements, the latter of which represent understandable and very common confusions. So let me walk this through carefully to see if I can help clarify.
Yes and no. Yes, I agree, insofar as
C_qm = C_ex = C_lrhv = –a.b (1)
must coincide for all of quantum mechanics as a theory, the correlations observed in experiments, and the correlations predicted — if possible — by an LRHV theory. No, insofar as (1) represents three different ways of concurrently obtaining a.b, and these three different ways are not well-reconciled. Let me explain:
This is correct, as we see in (1) above…
This is confused, and is a common source of confusion, because C_qm = –a.b is obtained in one way, while C_ex = –a.b is obtained in a different way, and the differences are very significant and very important. I will refer primarily to (1.10) in my PDF at https://jayryablon.files.wordpress.com/2016/12/unitary-operators-and-quantum-correlations.pdf to explain:
In quantum mechanics, if sigma denotes the three Pauli spin matrices, and if we define the 2×2 measurement functions:
A == sigma.a (2)
B == –sigma.b
for Alice and Bob respectively, then as I show in my (1.10) with E denoting expected value, that (*T = conjugate transpose), one may calculate:
C_qm = E(AB) = E(-sigma.asigma.b) = -E(a.bI+isigma.(axb)) = -E(U(alpha).U^*T(beta)) = –a.b (3)
In experiments, we use the scalar numbers +1 and -1 to represent correlations and anti-correlations for how Alice measures the spin, and for how Bob measures the spin, such that:
A == sgn (a.s) = +1 or -1 (4)
B == sgn (s.b) = +1 or -1
are binary scalar number measurement functions for Alice and Bob. Then, when we conduct experiments with a large number N of trials such that averages approach expected values, we find that:
C_ex = E(AB) = E(sum_N (+1 or -1)*(+1 or -1) ) = –a.b (5)
So as shown in (1), the result in (5) is equal to the result in (3), both –a.b, but the measurement functions (2) which lead to (3) are not the same as the measurement functions (4) which lead to (5).
So, quantum mechanics explains how C_qm = –a.b based on the 2×2 operator measurement functions (2), but quantum mechanics does not explain how to obtain the experimental C_ex = –a.b based on the binary scalar measurement functions (4). In other words, quantum mechanics does not explain the experimentally observed correlations. It predicts the same –a.b which are observed in experiments, but it does not explain how these come about via (5) from the binary-valued measurement functions (4). It only explains how they come about via (3) from the operator measurement functions in (2). These operator functions (2) do not have the binary values +1 or -1, they do not even have the eigenvalues +1 or -1, and they do not even have expected values of +1 or -1.
The moral is that it is an often-repeated false conception that “quantum mechanics explains the experimental correlations.” Let me use my words carefully: It is correct that “quantum mechanics predicts the experimental correlations.” It predicts C_qm = –a.b. However, it is incorrect to state that “quantum mechanics explains the experimental correlations.” In order to explain the experimental correlations, QM would need to explain how (5) comes about from the binary-valued scalar (4), which it does not. It only explains how (3) comes about from the operators (2).
And, by the way, the same holds true for the CHSH inequality. QM cannot explain how to get outside the CHSH bounds if the measurement functions are restricted to the binary-valued scalars in (4). It only explains how to get outside these bounds if the measurement functions are the operators in (2). But these operators are not reflective of the binary-valued experimental setup and to make believe they are reflective of this is comparing apples to oranges.
This is confused as to “a hidden variable theory is not needed but if one is desired anyway,” but correct as to ” it should predict the same correlation.” To explain:
One of the relations in (1.10) which is also in (3) above is the following:
C_qm = E(AB) = -E(U(alpha).U^*T(beta)) = –a.b (6)
Now let’s focus on the 2×2 unitary matrices U(alpha) and U(beta) above, which are the three unitary matrices of the SU(2) unitary subgroup. As is seen clearly in (3), these unitary matrices:
–U(alpha).U^*T(beta) = –sigma.asigma.b. (7)
This is to say, the matrix product U(alpha).U^*T(beta) is exactly equal to the operator product sigma.asigma.b used to obtain the QM correlations. There is no choice or question about this. I cannot “desire” (7) to be true. It is true. I can “desire” to talk about the correlations using U(alpha).U^*T(beta) and I can “desire” to talk about them using sigma.asigma.b. What (7) tells me is that I am permitted to talk about them in either way if I desire. So let me now talk about them using the unitary matrices.
Using these SU(2) unitary matrices, I may define a third set of measurement functions
A’ == U(alpha) (8)
B’ == -U^*T(beta)
to go alongside of (2) and (4). (Here, I use the primes to denote that these are different measurement functions from (4), but that they still get to the same result via (6). The primes are NOT the A’ and B’ of a different detector orientation, so please do not get confused by the primes in this notation.)
So if I “desire” to adopt the measurement functions (8) in lieu of (4), then I may rewrite (6) as:
C_qm = C_lrhv = E(A’B’) = -E(U(alpha).U^*T(beta)) = –a.b (9)
and this is just an equivalent restatement of (3). But now, I have shown that the QM explanation of the –a.b correlation is entirely equivalent to an LRHV explanation of the –a.b correlation, in which the LRHV theory uses the operators defined in (8). How so?
These operators (8), a.k.a. the SU(2) unitary matrices: a) are local because Alice’s measurement function is independent of Bob’s detector setting and Bob’s measurement function is independent of Alice’s detector setting; b) are elements of reality because, for example, U_y contains the realistic observable -(a.b)_y in its diagonal elements, and has a realistic observable expected value E(U_y)=-(a.b)_y; and c) contain hidden variable because, for example, U_y contains -(axb)_y in its off-diagonal elements which cannot be known from E(U_y)=-(a.b)_y and yet which leave the expected value E(U_y)=-(a.b)_y invariant with respect to whether the hidden variable sgn((axb)_y)=+1 or sgn((axb)_y)=-1.
So: the quantum mechanical explanation of the correlation C_qm = –a.b in (3) is entirely equivalent to a local realistic hidden variable theory explanation of the correlation C_lrhv = –a.b in (9), for a particular LRHV which uses the measurement functions in (8).
However, just as earlier for quantum mechanics, it is correct that “the LRHV measurement functions (8) predict the experimental correlations.” But, it would be overreaching and incorrect to state that the “the LRHV measurement functions (8) explain the experimental correlations.” They do not explain this any more than quantum mechanics does. In order to explain the experimental correlations, this LRHV would need to explain how (5) comes about from the binary-valued scalar (4), which it does not, any more than does QM.
So all of this still leaves open the theoretical problem of explaining the experimental correlations in (5), and in particular, of explaining these C_ex = –a.b based on measurement functions which, for each trial, have only the binary values +1 or -1. QM does not explain this. The equivalent LRHV I laid out in (8) and (9) does not explain this either.
However, Joy Christian claims to explain this. That is, he claims to explain how –a.b can be theoretically predicted from E(sum_N (+1 or -1)*(+1 or -1) ) in (5) when the measurement functions are only the scalar numbers +1 and -1 in (4), and when the measurement functions are NOT the operators in (2) or in (8), which operators have neither eigenvalues of +1 and -1 nor expected values of +1 or -1.
I agree with Mikko’s final statement, but would expand this. I would say:
A hidden variable theory should agree with other quantities observed in the same experiments, and it should agree with and properly represent the experiments being performed wherein Alice and Bob can only measure a +1 to denote a detected correlation, or a -1 to denote a detected anti-correlation. So too, should any quantum mechanical theory agree with and properly represent the experiments being performed. Unfortunately, neither QM nor the LRHV theory I pointed out is embedded in QM, are able to explain how A == sgn (a.s) = +1 or -1 and B == sgn (s.b) = +1 or -1 in (4) leads to C_ex = –a.b in (5).
For now, it is important for people to stop claiming that “quantum mechanics explains the experimental correlations C_ex = –a.b. It does not. It only predicts them to be –a.b, but it does not explain them in relation to measurement functions which can only be the scalar numbers +1 or -1. Rather, it explains them using operator measurement functions which are not the scalar numbers +1 or -1, and which do not even have +1 and – as eigenvalues or as expected values.
It is therefore desirable and important to explain the –a.b which QM can predict but not explain, using measurement functions which can only be the binary-valued scalar numbers +1 or -1 representing correlations and anti-correlations. This is what Joy Christian claims to be able to do, and claims to have done in his paper which was retracted by Annals of Physics.
Jay
Jay, this is one of the most lucid and insightful posts you have made on RW to date! Thanks.
One issue I have with it is that the operator-valued LRHV you have described may not qualify as “realistic” in the sense of Bell, although it may just pass as “realistic” in the sense of EPR.
Thank you Joy.
As to your substantive point, let’s talk a bit about lexicon in a way that might clarify.
If the experimentally-measured binary correlation quantities of my (4) above, namely:
A == sgn (a.s) = +1 or -1 (4)
B == sgn (s.b) = +1 or -1
are “realistic,” then the quantities a.s and b.s themselves are certainly “realistic,” because these latter quantities contain even more information than (4), because (4) truncates the magnitudes of a.s and b.s and only keeps their signs. Put differently, (4) are realistic as to a sign, while a.s and b.s are realistic but for a sign, namely, the signs from sgn (axs) and sgn (sxb).
So it would be misleading and probably just wrong to change whatever necessary or sufficient definitions we might have for “elements of reality” in order to deal with the binary-signed nature of (4). Further, I am not aware of anything that would prevent a clever lab partner of Alice and Bob, from helping them design an experiment that measures the full a.s and b.s which are projections onto the detectors, and not merely the correlations in (4) which are truncations of a.s and b.s that discard the projection magnitudes and only keep their signs.
I propose that in order to have our language clarify rather than obscure, we refer to (4) as measurement functions which are “binary-correlative-valued elements of reality,” or just “binary elements of reality” for short.
Then, I would say that because of the pertinent part of my earlier (3) which says that:
C_qm = E(AB) = E(-sigma.a sigma.b) = -E(U(alpha).U^*T(beta)) = –a.b, (3a)
quantum mechanics is a local realistic hidden variables theory which predicts but does not explain the strong EPR-Bell correlations insofar as one may choose to use the operators U(alpha) and U(beta) to define one’s “elements of reality.” However, I would also say that the experimental correlations
C_ex = E(AB) = E(sum_N (+1 or -1)*(+1 or -1) ) = –a.b (5)
which cannot be explained either by quantum mechanics or its embedded LRHV theory in (3a), MIGHT be explainable with a “binary correlative-valued LRHV theory,” which I will denote by the acronym BLRHV and call “binary LRHV theory” for short.
I would then say that to date, there appears to be no accepted BLRHV theory which successfully explains the experimental correlations C_ex = –a.b, nor is there any theory at all which does so, because as I have shown, quantum mechanics does not do so either on a BLRHV basis. It only does so on an ordinary LRHV basis. That one letter “B” in the acronym makes all the difference.
So Joy, what you are doing, in this lexicon, is proposing a BLRHV to try to explain the experimental correlations C_ex = –a.b. This has not been done by any ordinary LRHV theory to date. Nor has this been done to date by any other theory including quantum mechanics.
And again, to state that QM explains the experimental correlations is just plain wrong. It is like saying the following: draw two vectors a and b at some angle theta thus some dot product a.b=cos theta. Then rotate (or reflect, cosines don’t care about parity) these vectors by 180 degrees to obtain –a.b. Then proclaim that I can explain the experimental EPR-Bell correlations simply by rotating or reflecting two vectors through 180 degrees. That is just plain silly. How you obtain a.b is just as important as the fact of your having obtained a.b.
Jay
One correction, I misspoke, and would have my third-from-last paragraph say:
“I would then say that to date, there appears to be no accepted BLRHV theory which successfully explains the experimental correlations C_ex = –a.b, nor is there any theory at all which does so, because as I have shown, quantum mechanics does not do so either on a BLRHV basis. It only predicts these without explanation, on an ordinary LRHV basis. That one letter “B” in the acronym makes all the difference.”
Great! This is basically the same argument that I put forth to Richard Gill and others a long time ago though you have presented it in complete mathematical detail. Thanks. Concerning Joy’s GA model, why should it be expected that it has to use +/- 1 outcomes to produce -a.b when QM can’t even do it? However, the complete states version of the S^3 model does explain -a.b using +/-1 outcomes. Thus the paper should not have been retracted by Annals of Physics.
Great post Jay,
What you have arrived at here is the realization that LRHV and QM are fully consistent with each other, which implies that there is something wrong with Bell’s theorem — “No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.”
But to the specific point you make about explaining C_ex, I would say there be specific. You will find that there are many experiments, each with different observations and most of them have LRHV explanations. Therefore, I suggest the question of explaining C_ex by LRHV should be more specific. What specific experiment are you talking about. You will find that the situation with experiments is more murky than the theoretical discussion we’ve been having.
Jay,
You are still early in your Bell endeavors so perhaps you are not aware of the full literature on the subject. But some of your claims above are not quite accurate.
1) Do you know of any experiment which produced C_ex = –a.b, exactly.
2) Do you know of any BLRHV theory which reproduced –a.b exactly but was not acceptable?3) Do you personally agree that the model identified in (2) is not an acceptable explanation of the experiment identified in (1).
It is however true that there has not been a Binary version of QM reproducing the –a.b
Well, MF, speaking as a newbie learning EPR-Bell through this discussion and the papers that people point me to as recommended reading and what I dig out myself, here are my answers:
1) The distinct impression I have gathered is that experiments do reproduce C_ex = –a.b exactly, up to explainable statistical errors / deviations.
2) “Acceptable” or “accepted”? From what I see here Joy’s theory is a BLRHV which claims to reproduce –a.b exactly, but is not widely “accepted.” Whether it is “acceptable” is a different matter. But because the rule of thumb imposed by Bell’s Theorem (and my original agreement with Stephen Parrott here) prohibits us from even discussing LRHV theories without first showing an error in Bell, I have not yet started to discuss there whether Joy’s theory is an acceptable explanation of the experiment identified in (1), because the Bell rule prohibit that unless we show a problem with the Bell rules.
Since I have now shown with mathematical rigor that quantum mechanics as it is used to predict C_qm = –a.b is in fact an ordinary (not binary) LRHV, Bell’s Theorem, ironically, would seem to now bar us from ever talking abut quantum mechanics again. That by itself should demonstrate that something preposterous has been going on.
3) If the model identified by 2) is Joy’s model, then I would say it has not been “accepted” based on the sociological evidence I have witnessed here and elsewhere. The question to which I shall soon turn is whether it is “acceptable” and should be “accepted,” once we clear away the detritus of the idea LRHV theories cannot explain strong correlations when, as I have rigorously shown above, QM is itself and its prediction of the strong correlations is an LRHV theory.
Jay
Jay,
Fair enough. As you look more closely at experiments, you will find that the experiments usually proceed as follows:
Alice records a series of time-stamped data points
Bob records a series of time-stamped data points
At the end of the experiment, the data is brought together but they have no way of knowing which record at Alice corresponds to Bob. So they run an algorithm which tries to match records at Alice and Bob under the assumption that the measurements would happen at the same time. Every thing that does not match is discarded as “noise”. What is left is used to Calculate C_ex(AB) and it is this number that is found to be similar to QM.
There are a few variations to the above but we can get to those as they arise. The discussions are usually simply too simplistic when compared to what is actually done in the experiments. There are a lot of assumptions involved.
With that in mind, you may be interested in one of my simulations of a BLRHV theory reproducing C_ex (https://github.com/minkwe/epr-clocked).
PS: That is my understanding also. Which makes for a rather ironic doubles standard.
That is not correct. It is trivial to produce a hidden variable binary model that reproduces the QM correlations exactly. But it has to be non-local.
That is not correct on at least two counts: (1) MF is talking about a “Binary version of QM”, not a hidden variable theory. And (2), a hidden variable binary model that reproduces the QM correlations –a.b exactly need not be non-local. A fine counterexample of a purely local and realistic model is presented in eqs. (54) to (80) here: https://arxiv.org/abs/1405.2355 .
Whether you call it a “Binary version of QM” is simply a matter of interpretation. The point is that you can construct a model with binary outcomes that reproduces the QM correlations.
I would be interested in seeing such a model, rigorously formulated, as part of my education about EPR-Bell. Also, Joy’s model claims to do the same thing with locality, so I’d like to again revisit why that model does not also qualify as acceptable if a) it can predict the strong correlation E(AB)=-a.b and b) if it can reach the |2 sqrt(2)| CHSH bound and c) if it can do this on the basis of binary A=+1 or -1 and B=+1 or -1 measurement functions averaged over a large number of trials whereby averages approach expected values.
To use acronyms and words, I am asking to see a BRHV model (no “L”) that reproduces the strong correlation and reaches the CHSH outer bound, and I am asking why Joy’s BLRHV model would not be acceptable if in fact it turns out to also reproduce the strong correlation and reach the CHSH outer bound.
Jay
This model reproduces the QM predictions exactly:
https://rpubs.com/heinera/16727
It can also easily be turned into an urn model, where the slips have eight values (as opposed to four).
For an explicitly quantum model see the following paper and the simulation cited in reference 5.
https://arxiv.org/pdf/1607.01808v2.pdf
Hi Donald,
Thank you for providing your paper as a reference in reply to my query. Before getting into the details, I cannot help but note your statement at the end of section 2 about the EPR paradox:
“Notwithstanding the apparent horror, the current consensus is that the paradox is re-
solved by accepting the existence of ’quantum nonlocality’, whereby the correlations are explained. Concerns about the conflict with special relativity are set aside by right of the
no-signaling theorem. I show that such expedients are superfluous and that the paradox can be dissolved without appealing to nonlocality. I also argue that the concerns about a conflict with special relativity cannot be set aside.”
So am I to conclude that you, like Joy Christian, are advocating for a local model of the strong correlations E(AB)=-a.b which I see is in your equation (10)?
Jay
The key is to understand what is meant physically by the density matrix (1) and the tensor product (4). For example why is it a 4×4 density matrix to begin with when we have 2 particles and 2 possible outcomes each, and why are there negative numbers in a density matrix.
I will be interesting to see the A(.) and B(.) function from those models which produce (+1, -1) output values.
In the model I linked to, the A and B functions are both the same, and is called “obs” in the code. Defined around line 12 or so.
Of course not. As I clearly stated, the sampling involves both meaurement angles, so it is clearly nonlocal.
I am trying to reconcile what appears to be a conflict the in language you are using. On the one hand you say “the paradox can be dissolved without appealing to nonlocality” and then on the other you say “the sampling involves both measurement angles, so it is clearly nonlocal.” That confuses me. Am I the only one who is confused by that? Please help us become un-confused.
The quantum joint solution and the quantum separated solution using Luders’ rule both require nonlocality. The quantum separated solution with Von Neumann projection (or null projection) do not require nonlocality, but they do not give -cos(theta). This is my point of departure. I claim that Luders’ projection is wrong for EPR and that a validly designed and analyzed experiment cannot show -cos(theta). I hope it is clear. It’s all in the paper, please read it carefully. By the way it has now passed peer review and been accepted for publication.
Feel free to debate my claims if you like but the cited simulation for the quantum joint solution with dichotomic measurements clearly gives -cos(theta), which is what we were discussing, and what has been denied by MF, etc. Download the code and run it, you will see. This has been shown many times, it is not controversial at all.
Write out the function mathematically.
It is written out mathematically in the code. I can hardly think of anything more mathematical than R code. If you want a different mathematical notation, it should be obviuos how to transform it from the code (it’s just if-else statements). If you’re not versed in R, I can point you to some introductionary resources.
Sorry I’m not interested. If you want me to invest any time in your model, you will have to write out the A and B functions as mathematical expressions. Then we can evaluate if it is indeed a binary model of Quantum mechanics or just an impostor. Perhaps others will be interested in your model.
To all those unfamiliar with the formalism of QM expressed in density matrix form, don’t worry about all that. Go straight to my expressions for the eigenvalue probabilities and see that they are just those given by orthodox quantum mechanics. Dichotomic samplings of these probabilties yield the result -cos(theta) for largish N.
You can also look at the following paper where I show the same thing using spinning disks that directly implement the quantum eigenvalue probabilities.
https://arxiv.org/abs/1309.1153
No serious quantum theoretician doubts this. Do you really want to claim that dichotomic sampling of the quantum probabilities cannot yield -cos(theta) for largish N?
I was not aware of any such calculation prior to seeing your paper which you linked to above. The quote above simply states the fact that there is mathematics, then there is physical interpretation of the mathematics. Your paper does not provide a physical explanation for the density matrix in equation (1).
It’s a density matrix for a singlet state.
But why do you avoid my question to you:
“Do you really want to claim that dichotomic sampling of the quantum probabilities cannot yield -cos(theta) for largish N?”
Let’s get a clear answer from you on record. There are so many ways to show it that it seems almost inconceivable that you would deny it. I gave you two methods, HR gave you one, and I can give you a few more if you continue to deny it.
I don’t think you understood my question. What is the physical meaning of a density matrix for a singlet state, with negative numbers.
I’m not avoiding your question. The premise of your question implies you think I’m claiming “dichotomic sampling of the quantum probabilities cannot yield -cos(theta) for large N”. But where did I make that claim? I previously stated that to my knowledge there was no binary QM based simulation reproducing the -cos(theta) correlation. That was before you mentioned your paper which I wasn’t aware of. Is that equivalent to avoiding your question?
You said this: “It is however true that there has not been a Binary version of QM reproducing the –a.b”. But that is not correct. I had published two demonstrations some time ago and many others have shown it. OK, you weren’t aware of it. That’s why HR and I corrected you.
So it’s good to know that you accept that dichotomic measurements of QM can produce the cosine and thus violate CHSH. This is the thing that FD denies and JY has been wondering about. We should spend our time on real issues and not things that are well-known.
Of course a density matrix can have negative off-diagonal entries. It is only the trace that must equal 1. The density matrix represents the state. Then you multiply the density matrix by the matrix for the operator and the trace of the resulting matrix is the expectation value. But as I said, this is not important. It’s clear that QM can produce dichotomic statistics giving -cos(theta).
BTW, just in case you still think I’m avoiding your question, there are tons of models showing dichotomous sampling reproducing the -cos(theta) correlation. I’ve produced quite a few myself. HR’s model is definitely not QM based as far as I can tell since he did not provide the expressions.
I admit that your paper which you posted after that claim, appears to disprove the claim by providing a monte-carlo simulation producing binary outcomes by using “QM probabilities”. I do not admit that HR’s model is a “QM calculation”.
Absolutely not. If you think this is what FD and JY are denying, you have not understood their arguments, nor mine in this discussion. We have repeatedly argued that the inequality
⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ ≤ 2√2
is absolutely correct, and QM does reproduce it. How else will QM obey that inequality unless it did in fact produce the correlations -cos(theta)? Nobody has argued that it is impossible to produce binary data maximizing that inequality. Why would anybody argue such a thing, when there is experimental binary data achieving it? Surely you understand the difference between “there hasn’t been …” and “it is impossible to …”.
The CHSH inequality:
⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩ ≤ 2
on the other hand, is also absolutely correct, and nothing can violate it. Not even QM. I claim that it is impossible? Do you have a QM binary simulation which violates this inequality? This is the issue. It is invalid to compare the RHS of the former expression against the LHS of the latter. That is the whole point.
That is not a physical interpretation of the density matrix. The question is very important. What do the rows represent, what do the columns represent, what does each number represent physically, for example -1/2 off-diagonal entries. What do they represent. Once you start thinking about this, you’ll find that most of the “spooky stuff” is already introduced in equation (1).
BTW, what you call separated sampling in your paper, does not really classify as separate sampling. Separate sampling must be independent. Yours is not — just saying.
Larsson and Gill, stated in their paper (https://arxiv.org/pdf/quant-ph/0312035v2.pdf, page 4):
If you think the result of your QM simulation can be compared with the CHSH, then let me ask you. What is the common part of the ensemble you used to calculate each of the terms from your simulation. Isn’t each term is calculated from an ensemble generated from a distinct pair of probability distiributions? If your answer is that there is no common part, then it makes absolutely no sense to even talk of the CHSH in relation to this simumlation. Similarly, the CHSH does not apply to experiments or to LRHV models in which there is no common ensemble. For those cases, you have to use a different inequality
⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ ≤ 4
Which has never been violated by anything and never will.
Considered system: Two spin1/2 particles in a singlet state
In case „nature“ works in a local-realistic way, one would experimentally get
⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ ≤ 2
I have previously debunked this claim, where is the mayhematical proof of it?
Manifold proofs can be found in modern textbooks or on diverse internet blogs.
Please provide a link to one such proof. BTW, you still have e not answered the question
Larsson and Gill, stated in their paper (https://arxiv.org/pdf/quant-ph/0312035v2.pdf, page 4):
What is the common part of the four disjoint ensembles used to calculate ⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ from experiments?
See, for example, John R. Boccio’s textbook “In Search of Quantum Reality”.
You did not answer the question:
What is the common part of the four disjoint ensembles used to calculate ⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ from experiments?
If you are referring to the argument in chapter 15 of Boccio’s book, I already debunked it. It is exactly the same as Gill’s sampling without replacement from an Nx4 spreadsheet. If you are ready to respond to my counter-argument, I can repeat it for you. Or you can scroll up and just read it.
Casually speaking, “nature” randomly “selects” one row from the Nx4 spreadsheet and “provides” it to you to perform an experiment. “Selecting” doesn’t mean “deleting”. It’s simple probability theory.
The above inequality — if it holds at all — does so only in the large N or infinite limit. But in the latest state-of-the-art “loophole-free” experiments only 256 events were claimed to have been observed. That is a far cry from the large N or infinite limit. And for such a small number of events as 256 the only inequality that can possibly hold is the following:
⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩ ≤ 4 ,
which has never been “violated” in any experiment (because of course nothing can violate it).
That is a prevalent view, provided in the large N limit the following equality is meaningful:
|⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| = |⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩| ≤ 2.
In that case, let alone any local-realistic model, nothing can violate the bound of 2 on the above mathematical inequality, which was derived by Bell and CHSH by assuming locality and realism. “Violations” of the Bell-CHSH inequality are therefore usually taken to imply “violations” of either locality or realism, or both. But in the appendix of the short paper linked below I have been able to derived the Bell-CHSH inequality by assuming only that Bob can measure along b and b’ simultaneously while Alice measures along either a or a’, and likewise Alice can measure along a and a’ simultaneously while Bob measures along either b or b’, without assuming locality. “Violations” of the Bell-CHSH inequality therefore only means impossibility of measuring along b and b’ (or along a and a’) simultaneously:
http://philsci-archive.pitt.edu/12655/
The inequalities
|⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| = |⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩| ≤ 2
say nothing else than:
No flawless, local-realistic physical model can ever reproduce the experimental findings!
If that is the claim, then that claim has been explicitly proven wrong in this published paper:
http://link.springer.com/article/10.1007%2Fs10773-014-2412-2
Further extensive evidence can be found in the bibliography of the paper I linked previously.
http://link.springer.com/article/10.1007/s10773-015-2657-4
That paper by Gill has been criticized also on PubPeer:
https://pubpeer.com/publications/4C65BF0EDD5500B6D460273C68E70E
As the new year approaches and many people take stock of many things, I’d like to take stock of the overall flow of this Retraction Watch discussion about the retraction of Joy’s paper by Annals of Physics.
Back in late October we agreed to focus on Bell’s theorem before jumping into the “geometric algebra” (GA) which Joy uses as the foundation for the theory he presented in the paper retracted by Annals of Physics. This GA is a part of well-established mathematical physics, though there are many physics problems one can do without knowing or making use of GA. In late October, I was very new to Bell’s theorem. Today, two months later, while I may not yet be on top of the some of nuances which many of the other participants here have in mind, I have learned enough to have concluded that the low-hanging fruit restriction effectively required Joy to fight with one hand tied behind his back.
Specifically, I have now taken the time to study geometric algebra for myself (whether or not Dr. Parrott or anybody else thought it would be a wise investment of their own time). And I have also taken the time to try to understand how Joy uses GA in his model and in relation to EPR-Bell-CHSH.
What now seems plain to me based on my studies — without prejudging whether or not Joy’s development is “flawless” per LJ’s comment this morning — is that Joy’s purported disproof of Bell’s Theorem is effectively in the form of a specific counterexample by which he shows, using GA, that a BLRVH (binary-correlative-valued LRHV) theory can in fact be used to obtain the experimental correlations
C_ex = E(AB) = E(sum_N (+1 or -1)*(+1 or -1) ) = –a.b (1)
and can likewise break the CHSH bound of ||2|| using binary measurement functions. But in order to make this showing, Joy uses, and indeed must use, geometric algebra.
So based on what I have come to understand following my own study of GA and how Joy seeks to make use of GA, it would seem manifestly unfair to say to Joy that before anybody takes a look at how he uses GA to disprove Bell’s Theorem by counterexample, he must first disprove Bell’s Theorem without using the mathematical physics tools that he uses to disprove Bell’s Theorem. In other words, “I’ll look at your disproof, but only if you first do a disproof which allows me to ignore the tools you use to do your disproof.”
I will not prejudge whether Joy carries out his disproof flawlessly, as he must. But if we purport to be fair-minded, then we must now start to directly entertain the question, here at RW, a) what his actual disproof is, and b) whether or not he does execute it flawlessly. And if he uses GA to do his disproof, we cannot ask demand that he disarm and first do the proof some other way without the tools essential to his proof.
The first step that I will take in this direction, will be to facilitate a tutorial of sorts about geometric algebra here at RW, to bring everyone up to a common baseline of understanding of this rather nifty set of mathematical physics tools. If you are familiar at all with differential forms (DFs) which are used to represent Maxwell’s equations in integral form with general covariance, you will be aware that the differential volume element is defined by the wedge (^) product:
dx ^ dy ^ dz == dx (dy dz – dz dy) + dy (dz dx – dx dz) + dz (dx dy – dy dx) (2)
It should be clear that this is indeed a three-dimensional volume element, and that it has an intrinsic parity to it, whereby dx ^ dy ^ dz = – dy ^ dx ^ dz, for example. I happen to be deeply familiar with differential forms which I have used on my own work on electrodynamics and Abelain and non-Abelian magnetic monopole theory, so for me, DFs are my starting point for understanding GA.
Then, for example, in DFs you have the two-dimensional area elements:
dx ^ dy = dx dy – dy dx
dy ^ dz = dy dz – dz dy (3)
dz ^ dx = dz dx – dx dz
These also have a parity / direction associated with them, dx ^ dy = – dy ^ dx, because currents and fields can flow through a surface in a left-handed, or a right-handed direction.
Geometric algebra, which I have only studied myself in the last ten days and which has many fine points I am still learning, has a very similar structure. But instead of the infinitesimal volume elements dx, dy, dz which are suitable for calculus (dt is also there in spacetime), we use the unit vectors e_x, e_y, e_z (also e_t) which have a scalar length of 1, and a vectorial direction pointing in the x, y, z (and t) directions.
So if we stay in 3D space, just imagine that you draw the mutually-orthogonal x, y, z coordinate axes, and then you draw three such vectors (also called “basis vectors”) each with a length of 1, and each pointing along one of the three axes. These can even be thought of as e_x, e_y, e_z = 1_x, 1_y, 1_z, i.e., as a 1 pointing to z, a 1 pointing to y, and a 1 pointing to z. Then, any vector v at all can be formed by writing:
v = ( v_x, v_y, v_z) = ( v_x e_x, v_y e_y, v_z e_z) (4)
It should be clear that the product e_x e_y e_z will be a three-dimensional cube with a volume equal to 1 = 1_x 1_y 1_z. So in one sense, this is the number 1, but it is really 1^3=1, and it is a good idea to not forget about this dimensionality. Especially, as with (2) above for DFs, the starting point for GA is the unit volume element:
I == e_x ^ e_y ^ e_z
= e_x (e_y e_z – e_z e_y) + e_y (e_z e_x – e_x e_z) + e_z (e_x e_y – e_y e_x) (5)
This also has a magnitude of 1, but not only is it a three dimensional cube, it now also has a direction i.e., parity.
As with (3), you will also appreciate that there are some two dimensional unit squares also with areas of 1^2 = 1 which can be formed, namely:
e_x ^ e_y = e_x e_y – e_y e_x
e_y ^ e_z = e_y e_z – e_z e_y (6)
e_z ^ e_x = e_z e_x – e_x e_z
These too, all have an associated direction / parity, because for example, e_x ^ e_y = – e_y ^ e_x.
So what GA is doing, in effect, is creating the scalar number 1, but giving it certain attributes that scalar numbers alone do not have. For example, (5) is a three-dimensional number 1 with + or – parity, and it is called a “tri-vector.” And, for example, (6) contains three numbers 1 each of which is two-dimensional, and these also each have a parity of +1 or -1. These are each “bi-vectors.”
So what is going on, is that GA is taking the number 1, giving it an intrinsic dimensionality (scalar, vector, bi-vector, tri-vector, quad-vector), and giving it an intrinsic + or – parity. All “1s” are therefore not created equally. If I then take one of these 1s and multiply it by a scalar number, say, 2, then I have now created a number 2 which also has a dimensionality and a parity.
Now, when you look at (4), you may say “why complicate my life and write a vector as v = ( v_x e_x, v_y e_y, v_z e_z), when I can just write it as v = ( v_x, v_y, v_z)?” Which, by the way, even though I was aware of GA, is the reason that I never looked at it closely until now: I wasn’t working on any physics problems that required me to use it. But now, Joy says that EPR-Bell-CSHS is a physics problem where were are required to use GA to gain a proper understanding. I think Joy is owed the courtesy of people seeing if that is actually the case. But we cannot extend that courtesy unless people invest some time to understand what GA can do, that cannot be done without GA.
So, even without any heavy lifting, you should see right away that GA opens up some possible manipulations of numbers (such as the 2 in the CHSH bounds) which might not be possible for ordinary scalar numbers, because scalar numbers in GA actually house intrinsic dimensional and handedness attributes that we do not normally think about with scalar numbers, starting with multiple forms taken on by the scalar number 1. And you should be able to see that because the scalar numbers 1 have a + or – parity, the GA will naturally introduce various forms of the numbers +/-1 which could be helpful for a problem where we need to explain binary-valued correlations such as in EPR-Bell. And, you should be able to see that a volume element of 1=1^3, which also has an intrinsic parity, might well be a good source for a variable which is “hidden.”
So in the very broadest sense, what Joy is doing is essentially making use of these unusual (but entirely physically legitimate) numbers, to explain the strong correlations on a BLRHV basis, and to get outside the CHSH bounds in ways that would be impossible if we restricted ourselves only to the ordinary scalar number 1 (and other scalars numbers multiplied by 1). This becomes possible, according to Joy, only when we use scalar numbers with dimensionality and parity such as are provided by GA.
That is why we really cannot discuss whether Joy has or has not succeeded in disproving Bell, unless we are first willing to study these numbers 1 which are more than 1, how they are used in GA, and how Joy uses them in his theory. I am not saying that Joy’s execution is flawless. But I am saying that he is owed the courtesy of vetting this out.
Wishing everybody a very happy new year in 2017!
Jay
Happy new year!
OK, let’s discuss Joy Christian’s model. There are a few preliminary points that can be answered without any GA.
1) Does his model contain functions A(a, lambda) and B(b, lambda) that predict the measurement results for Alice and Bob? Do these functions return a value of either -1 or 1? (These are the values the experimenters will record).
2) It seams that his hidden variable lambda can take on two values (I will follow his own notation and call these two values -1 and 1 respectively). Then, what are the 8 values
A(0, -1), A(90, -1), A(0, 1), A(90, 1), B(45, -1), B(135, -1), B(45, 1), B(135, 1)?
Those eight values predicted by my model are exactly the same as those predicted by quantum mechanics and observed in Aspect-type photon-based experiments. You will find those eight predictions explicitly derived in this paper: https://arxiv.org/abs/1106.0748 .
Didn’t find them there. Since it’s just a vector of eight numbers, I guess it would not be to much of an inconvenience for you to just post it here?
They are there just before eq. (40) in the paper I have linked, used for predicting eq. (41).
But since they are exactly the same as those predicted by quantum mechanics, you can also find them in any good textbook on quantum mechanics. Alternatively, you can find them in this highly cited experimental paper:
http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.81.5039
See (16) and (17) of https://arxiv.org/abs/1106.0748
We find A(a, lambda) = lambda and B(b, lambda) = -lambda, whatever the settings a and b may be, where lambda = +/-1
Equation (16) and (17) of https://arxiv.org/abs/1106.0748
does not say A(a, lambda) = lambda and B(b, lambda) = -lambda.
A and B are outcomes of measurements, occurring at the space-like separated detectors of Alice and Bob, whereas lambda is the hidden variable, or an initial state of the singlet system, originating at the source, in the overlap of the backward light-cones of Alice and Bob. Therefore the functions A and B, or there values +/-1, should not be confused with lambda.
„Einstein rejected the indeterminate, quantum nature of the Universe. This one is still controversial, likely primarily due to Einstein’s stubbornness on the subject. In classical physics, like Newtonian gravity, Maxwell’s electromagnetism and even General Relativity, the theories really are deterministic. If you tell me the initial positions and momenta of all the particles in the Universe, I can — with enough computational power — tell you how every one of them will evolve, move, and where they will be located at any point in time. But in quantum mechanics, there are not only quantities that can’t be known in advance, there is a fundamental indeterminism inherent to the theory.“
http://www.forbes.com/sites/startswithabang/2016/12/29/the-four-biggest-mistakes-of-einsteins-scientific-life/#16c28878889e
Let’s now consider the system: Two spin1/2 particles in a singlet state
From a classical local-realistic standpoint one would imagine that both particles emerge from the “singlet state” with, in effect, individual sets of *pre-programmed instructions* for what spin to exhibit at each possible angle of measurement.
That means: Once some initial state – which “realizes” the singlet state – is given, the *pre-programmed instructions* are firmly tied up and, from that moment on, both particles have – so to speak – not anymore “freedom” to change anything; no spooky interactions etc.
Select four measurement angles: {a, a’} by Alice, {b, b’} by Bob
Any local-realistic physical model must thus be able to produce a Nx4 list where N is the number of initial states which are taken into account.
Inescapable conclusion:
|⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| = |⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩| ≤ 2
That means: No flawless local-realistic physical models can ever reproduce the experimental findings!
That last claim remains both unproven and has been debunked in this very thread.
The relation
|⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| = |⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩| ≤ 2
just results from probability theory applied to the *pre-programmed instructions* which are produced by local-realistic physical models. Nobody has debunked it so far. Some merely *believe* that they have debunked it, that’s all.
On the contrary. The equality
|⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| = |⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩|
has a sum of disjoint correlations with the bound of 4 on its left-hand side, and a quantity with a bound of 2 on its right-hand side that cannot be “violated” by anything, let alone quantum mechanical predictions or experimental observations. It thus debunks itself.
Ok, so the values for
A(0, -1), A(90, -1), A(0, 1), A(90, 1), B(45, -1), B(135, -1), B(45, 1), B(135, 1)
are
-1, -1, 1, 1, -1, -1, 1, 1
Please correct me if I am wrong here (and also please provide the correct values if so).
Is that what quantum mechanics predicts? Is that what is observed in the experiments?
I thought I have already provided you the correct values in the following post:
Eight numbers is all I humbly ask for.
And you are given the eight numbers you asked for.
In case the selected sub-ensembles which you use to determine ⟨A₁B₁⟩, ⟨A’₂B₂⟩, ⟨A₃B’₃⟩ and ⟨A’₄B’₄⟩ roughly “mirror” the hidden variable’s probability distribution which you use to determine ⟨A₁B₁⟩, ⟨A’₁B₁⟩, ⟨A₁B’₁⟩ and ⟨A’₁B’₁⟩, you will always have
|⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| = |⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩| ≤ 2
Perhaps you missed my following post. Experimental facts seem to contradict your claim:
We are talking about the CHSH inequality, not about experiments. For any local-realistic physical model – without any exception – one will always have:
|⟨A₁B₁⟩ + ⟨A’₂B₂⟩ + ⟨A₃B’₃⟩ – ⟨A’₄B’₄⟩| = |⟨A₁B₁⟩ + ⟨A’₁B₁⟩ + ⟨A₁B’₁⟩ – ⟨A’₁B’₁⟩| ≤ 2
There is no way out.
Your claim has been disproven explicitly in this published paper:
https://arxiv.org/abs/1211.0784
(see also http://link.springer.com/article/10.1007%2Fs10773-014-2412-2 ).
Please see the derivation in the last appendix — especially the last equation — eq. (C15).
See also the following paper on the arXiv for further explanations of the above derivation:
https://arxiv.org/abs/1501.03393 .
Yes, and they are
1, 1, -1, -1, 1, 1, -1, -1
Correct?
(there are only 2^8=256 combinations to go through here, so finally we will have to hit the right one).
Good.
But these are not what is predicted by quantum mechanics, or observed in the experiments. So please try another combination from the remaining 255 possibilities.
Alternatively, just look up the experimental paper I provide and find the right numbers.
Well, this is the nice thing about computers. You can try every 256 combinations in less than a second. Not one of those combinations could reproduce all four QM correlations. Sorry.
And why is that a surprise? Those are not the combinations that are either predicted by QM or observed in the experiment I linked to above. Those where only your presumed numbers.
You think that there are more combinations than 256?
You tell me.
You asked: “… what are the 8 values A(0, -1), A(90, -1), A(0, 1), A(90, 1), B(45, -1), B(135, -1), B(45, 1), B(135, 1)?”
I answered: “Those eight values predicted by my model are exactly the same as those predicted by quantum mechanics and observed in Aspect-type photon-based experiments.”
Do you now understand my answer? Or do you still require my model to make predictions beyond what is predicted by quantum mechanics and observed in the experiments?
QM does not predict any values for those eight values, because in QM they simlply doesn’t exist. But they obviously exist in any local realistic model like yours. So for the n’th time, what are eight values?
So you do demand my model to make predictions beyond what is predicted by quantum mechanics and observed in the actual experiments. Well, my model is under no obligation to satisfy such unphysical demand. It predicts exactly what is predicted by quantum mechanics and what is observed in the experiments. Moreover, it is manifestly local and realistic.
In QM, the A and B functions don’t exist. If they don’t exist in your model either, why do they appear in your papers?
The purpose of the manifestly local, realistic, and deterministic functions A and B in my model is to reproduce the statistical predictions of quantum mechanics and explain what is actually observed in the experiments. It is easy to verify that they do this job impeccably.
But do these functions have values? And if they do, what are the values for
A(0, -1), A(90, -1), A(0, 1), A(90, 1), B(45, -1), B(135, -1), B(45, 1), B(135, 1)?
Or maybe they are a complete novelty; functions without values?
As an experimental physicist, I really wonder what all these discussions are about.
Let us assume we have a classical “local-realistic” physical model to describe the evolution of “two entangled spin1/2 particles in the singlet state”, a set of initial states, h_i(0), which “realize” the singlet state at time t = 0, and a “local-realistic” simulation algorithm which produces a set of *pre-programmed instructions*, {A(a; h_i(0), t), A(a’; h_i(0), t), B(b; h_i(0), t), B(b’; h_i(0), t)}, for what spin to exhibit at measurement angles {a, a’, b, b’} and at time t.
When using the algorithm, one should be able to produce the following outputs for every initial state h_i(0):
Output_1: {A(a; h_i(0), t), B(b; h_i(0), t)} – when you input (a, b; h_i(0), t)
Output_2: {A(a; h_i(0), t), B(b’; h_i(0), t)} – when you input (a, b’; h_i(0), t)
Output_3: {A(a’; h_i(0), t), B(b; h_i(0), t)} – when you input (a’, b; h_i(0), t)
Output_4: {A(a’; h_i(0), t), B(b’; h_i(0), t)} – when you input (a’, b’; h_i(0), t)
Here, A(a; h_i(0), t) is either +1 or -1 etc.
One can now simply cross-check the outputs for any given h_i(0) to check the consistency and coherence of the “local-realistic” physical model. That means, the model must be able to produce *one* unambiguous output set {A(a; h_i(0), t), A(a’; h_i(0), t), B(b; h_i(0), t), B(b’; h_i(0), t)} for every h_i(t=0). If this is the case, one has done it.
Determinism is one fundament to set up such *pre-programmed instructions*. In classical physics, theories are really deterministic. If you tell me the initial positions and spins of both particles, I can — with enough computational power — tell you how every one of them will evolve, move, and where they will be located at any point in time. And it doesn’t matter whether one considers a R^3 or a S^3 spacetime model!
In case one gets inconsistent outputs, one would simply conclude that the *pre-programmed instructions* interfere with each other in some strange way; maybe, the simulation algorithm contains some “bugs” or the claimed local-realistic physical model is in itself flawed.
By this way, one can simply check whether the retraction of the paper “Local causality in a Friedmann–Robertson–Walker spacetime” was justified or not.
Richard’s comment above is where I think we should start the discussion as to whether Joy’s papers has any fatal “flaws” or not. And to keep things focused, let me start with what we should NOT discuss:
I suggest that we set aside for now any discussion of the CHSH sum, and focus on the correlations E(AB)=-a.b and whether Joy has succeeded in explaining these using a LRHV in which A=+1 or -1 and B=+1 or -1 are binary-valued correlations (denoted B), i.e., on whether Joy has in fact used geometric algebra, flawlessly, to explain the correlations on a BLRHV foundation whereby:
C = E(AB) = E(sum_N (+1 or -1)*(+1 or -1) ) = –a.b (1)
and the first A=(+1 or -1) is local and realistic to Alice and the second B=(+1 or -1) is local an realistic to Bob, using some hidden variable(s). I have already shown in (1.10) of https://jayryablon.files.wordpress.com/2016/12/unitary-operators-and-quantum-correlations.pdf and my posts on December 28, 2016 at 11:13 am and December 28, 2016 at 4:24 pm that the correlations E(AB)=-a.b can be explained by an ordinary LRHV theory without binary correlation values; the trick is to do this WITH binary-valued correlations.
The reason why I suggest that we set aside any CHSH discussions is that they are not logically necessary for the task at hand. If we can obtain C = E(AB) = –a.b using any sort of theory, BLRHV or otherwise, then we all know by using settings for b’, a, b, a’ which are 0, 45, 90 and 135 respectively, given cos(45 degrees)=1/sqrt(2), that one can break the CHSH bound of 2 and reach 2 sqrt(2). So let’s stay focused on the correlations and whether Joy derives those with a BLRHV theory which is free of fatal flaws.
And to be clear, this is not to say that CHSH-Bell is unimportant. If we are explicit about the binary-valuedness, CHSH-Bell holds that the correlations E(AB)=-a.b cannot be explained by ANY type of BLRHV theory, and that any theory which purports to do so must have a flaw. So by looking for the ostensible flaw in Joy’s theory, we are already following Bell-CHSH: The power of Bell is that it divines that there must be a flaw in Joy’s theory, without knowing anything about the specifics of his theory, simply because it claims to be a BLRHV theory. So we now want to pinpoint the flaw which Bell predicts can be found, and the place to look would be in the correlation derivation.
And that is what Richard has brought us to by pointing to equations (16) and (17) of https://arxiv.org/pdf/1106.0748v6.pdf. Richard takes the position that these equations constrain the Alice and Bob measurement functions to be:
A(alpha,mu)B(beta,mu)=-1 (2)
and regards this as constraining Alice and Bob to use the same setting a=b for their detectors. This is precisely where Richard was when I first got involved in this discussion, see below:
And Joy’s response at the time was:
followed by:
So this is the crux of where Joy and Richard do not agree. Richard believes that Joy’s logic constrains the detectors. Joy believes this is a terribly wrong interpretation of the geometry.
I should also note that Joy has been consistent in this position from the start with his first paper in 2007 at https://arxiv.org/pdf/quant-ph/0703179v3.pdf, and did not simply start taking this position as a “cleanup” measure in order to reply to critiques. See Joy’s (8) in that original paper which already was clearly stating that A=-B but referring to these as “dynamical variables” and not Alice and Bob “measurement functions.” And he said at the time following his (8) that “This simply means that A_n(lambda) = + 1 if the two unit vectors n and lambda happen to point through the same hemisphere centered at the origin of lambda, and A_n(lambda)=−1 otherwise.”
This is the part of Joy’s work that needs to be made impeccably clear, because this looks to be what Joy’s critics believe is the fatal “flaw” that Bell’s Theorem divines, and because Joy insists that this is not a flaw at all but endemic to the geometry owing to the “profound difference” between (6) and (12) in that first paper, which he says translate into differences between R^3 and S^3.
Can we all set aside everything else, pick up the discussion right at this point, and only examine that which is needed to understand and clarify this issue?
Jay
I think we can simplify this even further, down to eigth numbers in {-1, 1}, actually.
In Christian’s model there are two functions A(a, lambda) and B(b, lambda). Lambda can take two values (let’s call them -1 and 1). Now, what are the function values
A(0, -1), A(90, -1), A(0, 1), A(90, 1), B(45, -1), B(135, -1), B(45, 1), B(135, 1)?
All four correations can be computed and checked from these eight numbers.
Are those eight values a part of the statistical predictions of quantum mechanics, or are they what is actually observed in the experiments?
Correct me if I am wrong, but I believe that Joy would determine these eight numbers using the “standard scores” in his equations (27) and (28) of https://arxiv.org/pdf/1106.0748v6.pdf, which is the paper Richard has pointed to. And he would also point to the CHSH identity that he obtains in (41). Though as I said earlier, the first step is to explain the correlations in a “flawless” manner, so Joy’s raw scores (16) and (17) and his standard scores (27) and (28) and how he obtains and justifies them leading to (32) and (33) must be the focal point.
But why doesn’t he simply post the eight numbers here? After all, there are only 256 possible alternatves?
Why don’t you post the eight numbers here yourself if these are indeed the eight numbers predicted by quantum mechanics and/or observed in the actual experiments?
Jay,
Nowhere in any of my papers will you find the following pair of equations:
A(a, lambda^k) = lambda^k
and
B(b, lambda^k) = – lambda^k .
I have no idea where these equations have come from, or what they are supposed to mean.
As I have already noted, the functions A(a, lambda^k) and B(b, lambda^k) on the LHS of the above equations are outcomes of measurements by Alice and Bob, occurring at their space-like separated detectors. On the other hand, lambda^k on the RHS of the above equations is the hidden variable, or an initial state of the singlet system, originating at the source, in the overlap of the backward light-cones of Alice and Bob. Therefore the outcome functions A(a, lambda^k) and B(b, lambda^k), or their specific values +1 or -1, should never be confused with the random variable lambda^k shared by Alice and Bob, even if they happen to be numerically identical. It is quite puzzling to me why anyone would want to equate these two completely different physical concepts in the manner of the above pair of equations.
Now it is true that for any given detection vector “n”, quantum mechanics predicts perfect anti-correlation, which in the notation of my model can be written as:
A(n, lambda^k) B(n, lambda^k) = -1.
But this holds only for the very special case in which both Alice and Bob chose exactly the same detector setting “n.” For all other directions “a” and “b”, chosen freely and independently by Alice and Bob, the product of their outcome functions would be
A(a, lambda^k) B(b, lambda^k) = +1 or -1.
The above is the prediction of my local model, and it has been rigorously derived in equations (67) to (79) of this paper: https://arxiv.org/abs/1405.2355 .
Let us first address the above issues, Jay, before moving on to the remaining of your queries.
Joy, let me tell you where I think they have come from, and perhaps Richard or someone else can confirm.
Your variable mu=+/-I=lambda*I. Keep in in mind also that I=e_x^e_y^e_z is the unit cube trivector. So in (16) of https://arxiv.org/pdf/1106.0748v6.pdf. for example, one may deduce:
A(alpha,mu) = +1 if mu=+I (1)
which mean that:
A(alpha,mu) = +1 if lambda=+1 (2)
because if mu=+I then lambda=+1 . Then, a leap of logic is taken, whereby from (2) somebody says that because if A(alpha,mu) = +1 if lambda=+1, then:
A(alpha,mu) = lambda (3)
You, I guess, would argue that this is an illegitimate step, along the line of going from “if I have one apple then I also have one orange” because perhaps the grocery store packages apples and oranges together, to deducing that “1 apple = 1 orange.”
You, Joy, have only said that A=-B, for example, in (8) of your https://arxiv.org/pdf/quant-ph/0703179v3.pdf. Would everyone agree that (3) is an illegitimate deduction, and that the discussion should turn to what it means to say A=-B?
Jay
Indeed, Jay, 1 apple = 1 orange does not mean apple = orange. So let us move on.
Note, however, that I do not say A = -B in my equation (8) you mention. What I say is:
for any given detection vector “n”, if it happens to be chosen by both Alice and Bob, then
A(n, lambda^k) B(n, lambda^k) = -1.
That is not quite the same as what you seem to be reading from my equation (8).
OK, Joy, agreed that AB=-1 when the detectors both point in the same direction, which is the special case a.b=1.
So, let’s go back to (16) and (17) of https://arxiv.org/pdf/1106.0748v6.pdf. Please lay out, when you look at those two equations, exactly what you have in mind insofar as the physics they represent.
Jay
The physics captured by eqs. (16) and (17) is straightforward and described in the paper. A and B define two independent detection processes, happening at a space-like separated distance from each other. The spin mu.s approaches Alice’s detector I.a, and its normalized component along I.a is then mu.a, with the geometric product (I.a)(mu.a) resulting in A = +/-1, depending on whether mu = +I or -I. And likewise for Bob. That is all there is to it.
Jay,
one has the feeling that your last comments
Jay,
one has the feeling that your comment only serves to obscure the “debate” again.
Jay,
you will find the answer in:
https://jayryablon.files.wordpress.com/2016/10/jcrg-2.pdf
Jay,
A(a, lambda^k) = lambda^k
and
B(b, lambda^k) = – lambda^k
are nothing else than the outcomes of eq. (59) and eq. (63) in
https://arxiv.org/abs/1405.2355 .
Carry out the mathematical limes operations in in the correct way and it’s done!
As Jay appreciated and agreed earlier, 1 apple = 1 orange does not mean apple = orange, even in the “limit”, especially when the apple (A) is a detection process happening at a remote detector and the orange (lambda) is an uncontrollable random variable originating from the source located in the overlap of the backward light-cones of Alice and Bob.
Joy,
Let me ask my question differently.
I am a very visual person, and geometry is one of my favorite things. Probably quite a few others here are the same way. All we are talking about here is a three dimensional space with vectors pointing in certain directions with certain magnitudes and I want to see pictures of those. For example, we can draw x, y, x axes and then draw the unit volume I=e_x e_y e_z and we know if we flip the order of any two axes we get, e.g., -I=e_y e_x e_z which signifies a parity inversion because it is equivalent to flipping the z axis alone, then rotating x and y into their original positions.
In the “vector” spaces I and many other are used to, we might, for example, in two dimensions, write a= (a_x, a_y) = (4, 3) and we know that this is a 3-4-5 triangle with sin theta = 3/5 and cos theta = 4/5. But in geometric algebra we say — as you have clarified to me — that we define a “vector” as:
a == a_x e_x + a_y e_y = 4 e_x + 3 e_y (1).
This means that I start at the origin, draw a vector of length 4 along the x axis, then, starting at the head of the first vector, I draw a second vector of length 3 along the y axis. Then, If I draw a vector from the origin to the head of the second vector, I have a vector of length 5 with the same sin theta = 3/5 and cos theta = 4/5. It is a different but equivalent way to write a vector using vector addition, and such operations are at the heart of geometric algebra. So (1) is just a single number 5 with a direction attached.
So, looking at your (16) of https://arxiv.org/pdf/1106.0748v6.pdf, it would be helpful if everyone including me can visualize this very precisely, using the geometry that it represents. So when we see:
A(alpha,mu) = (-I.a)(mu.a)=+/-1 (Joy 16)
you should be able to talk us through the process of geometric construction, or better yet, draw us a picture starting with x, y and x axes which have the trivector I pasted on them. In this picture, we should be able to see 1) the vector that represents a for Bob’s detector, 2) the vector that represents the spin direction s of the doublet member that Bob detects, 3) the angle between a and s, and 4) a value of +1 if a and s are in the same hemisphere and a value of -1 if a and s are in opposite hemispheres. Every element of (joy 16), A, I, a and mu=lambda I should be labelled on that figure to illustrate its contribution to that figure.
Due to what may be unfamiliarity with geometric algebra, I think many of us would appreciate such a picture, either in words, or in a figure that show each element of (Joy 16) represented on the figure.
Jay
Jay, you will find a visual picture of a bivector mu.a in the Figure 1.2 on page 5 of this paper: https://arxiv.org/pdf/1201.0775.pdf (which is the first chapter of my book).
As we can see in the above picture, the unit bivector mu.a = lambda x I.a is just a number, +/-1, about the direction a (i.e., a shapeless directed plane orthogonal to the vector a), where lambda is the orientation (or parity) of mu, relative to the standard trivector “I” you mention. Thus mu.a is simply +/-1 about the direction a, relative to I.a, which by convention is chosen to be +1 about the direction a.
Now mu.s is a random spin-bivector, whereas I.a is a fixed detector-bivector. The component of the spin mu.s along the direction of the detector I.a is then = mu.s x cos(a, s), just like the component of the vector s along the vector a is s x cos(a, s). The normalized component of mu.s along the direction of the bivector I.a is thus simply mu.a, just like the normalized component of s along the direction of the vector a is simply a, because s and a are unit vectors and mu.s and I.a are unit bivectors by construction, or assumption.
Since all unit bivectors square to -1, the product (-I.a)(mu.a) = +/-1 (i.e., a scalar number), independent of the direction a. Therefore we have A(a) = +/-1. That is all there is to it.
You should download GAViewer and play around with it. Here are some pictures of oriented volumes.
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=115&p=4021&hilit=gaviewer#p4027
Unfortunately the site for GAViewer seems to be down right now.
http://www.geometricalgebra.net/
If you are interested, send me an email and I will send to you.
When consequently performing the math – especially the limes operations – in “Local Causality in a Friedmann-Robertson-Walker Spacetime” ( https://arxiv.org/abs/1405.2355 ) one ends up with nothing else than
E(a, b) = -1 for all a = b,
a direct mathematical consequence of explicit mathematical assumptions. At least, this is a correct finding regarding the singlet state.
There is no way to obtain the result E(a, b) = -1 in my model without either setting a = b by hand or by violating the conservation of zero spin angular momentum. For arbitrary a and b the conservation of zero spin angular momentum necessitates the correlation E(a, b) = -a.b.
Would you then, please, explain the step from eq. (59) to (60) or from eq. (63) to (64) in https://arxiv.org/abs/1405.2355 in plain words. Jay R. Yablon is – more or less – asking the same.
As noted in the preamble to those equations, the sign-functions appearing in the equations (60) and (64) are not a part of my Clifford-algebraic model. What is demonstrated in the equations from (57) to (64) is that my measurement functions defined in (54) and (55) are only effectively the same as the sign-function-based measurement functions proposed by Bell in his own local model of 1964. Eqns (60) and (64) play no role in my 3-sphere model.
In case I understand you right, there is no need to carry out the diverse mathematical limes operations which appear in your paper.
The limits are essential in equations (54) and (55), which play fundamental role in my model.
But equations (57) to (64) are not essential, apart from being pedagogically helpful.
But with equations (54) and (55) in https://arxiv.org/abs/1405.2355 equation (68) ends up with E(a,b) = -1 for all a and all b.
Not without violating the conservation of zero spin angular momentum. For arbitrary a and b the conservation of zero spin angular momentum necessitates the correlation E(a, b) = -a.b.
Yes, that is part of what I have been asking. But last night, after three months of being involved in these discussions, and a week or so after having taken the time to really study geometric algebra, I finally understand what Joy is doing, and I also realize that it is impossible to understand what Joy is doing without understanding geometric algebra.
What Joy means by the “similarity” sign in the step from eq. (59) to (60) or from eq. (63) to (64) which has also stumped me for weeks, is that (59) and (63) “play the same role in Joy’s theory” as “the sign-function-based measurement functions proposed by Bell in his own local model of 1964.” Just as the sign function is a “hemisphere detector” which returns the value +1 if the spin and the detector are pointed in the same hemisphere a.k.a. correlated and returns the value -1 if they are pointed in the opposite hemisphere a.k.a. anti-correlated, so too to (59) and (63) return +1 and -1 under identical circumstances.
But mathematically, because of the nature of numbers in geometric algebra which also have intrinsic direction and parity, these +1 and -1 simply have different mathematical properties and consequences from those that pop out of the sign function. And those different properties make all the difference in the world when it comes to the strong correlations. In essence, Joy has found a better “machine” than the sign function for producing the correlation numbers +1 and -1, and that better machine can multiply two “ones” together and get a result which can be schematically represented as:
E(AB) = E(sum_N (+1 or -1)*(+1 or -1) ) = –a.b (1)
and is exactly what we want for the strong correlations. But anybody who does not really take the time to understand geometric algebra in a careful as with any course of earnest study, will find themselves and Joy talking past one another, which from what I see has been happening for years.
Because the study of what for many may be a new subject is a time commitment, what I will do is prepare a several page tutorial on geometric algebra which I hope will out an end to people talking past each other. This may take a day or so, and given that I am leaving the US on Thursday for ten days away, I will have to fit it in between other things. But I will make every effort to get this tutorial posted before I leave. And I will make it clear and transparent and visual and geometrically comprehensible, so that anybody who reviews this can assimilate the subject matter quickly without much heavy lifting.
Jay
Jay,
when I consequently use the rhs of equation (54) and (55) in https://arxiv.org/abs/1405.2355 in order to evaluate the rhs of equation (68), I end up with nothing else than
E(a, b) = -1 for all a = b (as s1 = s2 according to eq, (65)).
From my side, the story has thus come to an end! That’s all, nothing more to say.
Your conclusion is incorrect. The RHS of eqs. (54) and (55) in https://arxiv.org/abs/1405.2355 , when substituted in the RHS of eq. (68), together with the conservation of the zero spin angular momentum (66), immediately equates eq. (68) to eq. (72), which then trivially leads to the result (75), which is the strong correlation
E(a, b) = -a.b
for arbitrary directions a and b, freely and independently chosen by Alice and Bob. This theoretical result has been verified in a computer simulation using the GAViewer, the code and a plot of which can be found here:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=296#p7322 .
As promised yesterday, I have prepared what turned out to be a seven-page “tutorial” summarizing geometric algebra, and uploaded this to https://jayryablon.files.wordpress.com/2017/01/geometric-algebra-tutorial-2.pdf. This summary has an obvious eye toward the discussions of Bell’s theorem that we have been having here, but I have tried to present this from a purely mathematical standpoint so that it stands independently of these discussions.
I hope that the participants in this discussion who really have not studied geometric algebra before and so have been winging it (and you know who you are) will take the time to assimilate these materials. I did the work so you all would not have to. If there are any errors, certainly point those out, I also like to produce “flawless” work if I can. 🙂
With that, I am off midday tomorrow to an unplugged vacation outside the US until mid-January. In the mean time, keep it cool, and have a good week!
Jay
Jay, might I suggest you duplicate your equation (13) using the opposite bivector basis, and compare the two bivector result portions as bivectors avoiding the cross product representation?
Why, Rich? Are (11) or (12) erroneous in some way? If so, How? And dot and cross products are closest to how we observe these things, so why not use them, as long as they are used correctly? The whole point of a bi-vector and my Figures 1(a) and 1(b) is to show that there are two cross-product vectors associated with a bi-vector, which is why in this context they are called bi-vectors. In the more familiar context to many, the EM field strength is called a bi-vector; in that case because you gave an electric and a magnetic field vector. So why would we avoid talking about and using the two vectors? Jay
I ask you to do this because it will demonstrate that you will arrive at the same result for both basis choices, using only the rules of GA in your derivation, since you have the misconception lambda is not a real number, or does not participate as such in GA expressions. If you understood lambda is a real number, you could have avoided the somewhat lengthy derivations of your (13) done both ways by understanding it does come down to (-1)(-1) = (+1)(+1).
Why not use the cross product? Because it leads you down the path of treating the bivector portion as a triplet of real valued coefficients detached from their proper basis elements. This is where Christian and his supporter have gone off the path. The coefficients do change signs, but the ignored basis elements do also.
Please, everyone, do the math for Jay’s (13) using alternatively the right and left bases, compare the bivector result portions with their bivector basis elements still attached. You will find them to be identical.
Quickly, in the middle of packing: I agree that my (13) will yield identical results down to the seventh line (after the fourth equality). Then I use (12) which is (pm = + or -):
a ^ b = mu . (a x b) = pm I . (a x b) (12)
and various expanded expressions for this. The sole question is whether this equation (12) is correct or not. If not, it would have to be replaced by:
a ^ b = I . (a x b) (12′)
Either (12) is correct, or (12′) is correct. Everything hinges on which one is correct.
Jay
Sorry but no. The final word on the subject is the equality of your (13) when done both ways. Any analysis that contradicts this equality must, since math is consistent, be in error.
The typical paradigm for math/physics is picking one chiral choice for the algebra and staying within its singular definition. Only then can you use something like a cross product, since a x b can be defined singularly and unambiguously. When you involve two bases in one expression, you must be careful of your choice of terminology, and must always carry the basis elements along the way, and map to the same basis before adding coefficients, otherwise you have nonsense.
Or… say you are stuck in the right-handed basis, what does the left-handed basis look like from the one you are stuck in? And the simple answer is,
(I.b)(I.a)
Now… I wonder what that is equal to?
Jay,
thanks for your manuscript https://jayryablon.files.wordpress.com/2017/01/geometric-algebra-tutorial-2.pdf . I have checked the math. I am confused with your equation (14) which seems to be the essential one.
mu = ± I is a shared property of the vectors a and b.
I get, considering the case mu = +I :
(mu . a)(mu . b) = (+I . a)(+I . b) = +(I . a)(I . b) = −a . b – a ^ b = −a . b − I . (a x b)
I get, considering the case mu = -I :
(mu . a)(mu . b) = (-I . a)(-I . b) = +(I . a)(I . b) = −a . b – a ^ b = −a . b − I . (a x b)
When averaging, I get never rid of the term “I . (a x b)”.
Addendum:
Jay, maybe you have overlooked somewhere in your lengthy derivations, that “minus times minus” is equal to “plus”.
I have checked Jay’s calculations. He has not overlooked ““minus times minus” is equal to “plus”.” On the other hand, there is a sign mistake in your attempt to calculate Jay’s eq. (14).
I have done nothing else than using the „Hodge duality“. Is there a re-definition I am not aware of.
You have misapplied Hodge duality in your attempt to calculate Jay’s equation (14).
Some program lines from http://rpubs.com/jjc/233477
f = -1 + (2/sqrt(1 + ((3 * s)/pi)))
p = function(u,v){colSums(u*v)}
q = function(u,v,s){ifelse(abs(p(u,v)) > f, 1, 0)}
g = function(u,v,s){p(u,v)*q(u,v,s)}
A = +sign(p(a,e))*q(a,e,s) # Alice’s results A(a, e, s) = +/-1
B = -sign(p(b,e))*q(b,e,s) # Bob’s results B(b, e, s) = -/+1
Considering the q function, I would write
A = +sign(p(a,e))*q(a,e,s) # Alice’s results A(a, e, s) = +1, 0, -1
B = -sign(p(b,e))*q(b,e,s) # Bob’s results B(b, e, s) = +1, 0, -1
Addendum:
Physically, measurement outcomes are either +1 or -1. In case one has a straightforward „local-realistic“ physical model, an „ifelse“ program code {ifelse(abs(p(u,v)) > f, 1, 0)} must not be used and makes no sense. This is a forbidden action.
I have done nothing else than using the „Hodge duality“. Is there a re-definition I am not aware of.
You have misapplied Hodge duality in your attempt to calculate Jay’s eq. (14) and ended up making a sign mistake.
No!
For all mu I get always
(mu . a)(mu . b) = −a . b − I . (a x b) !
And you do not see the mismatch of trivectors on the LHS and RHS of your equation?
From equation (6) in Jay R. Yablon’s manuscript /1/ one gets:
(I . a) = a_x (e_y ^ e_z) + a_y (e_z ^ e_x) + a_z (e_x ^ e_y)
From equation (7) in Jay R. Yablon’s manuscript /1/ one gets:
(-I . a) = a_x (e_z ^ e _y) + a_y (e_x ^ e_z) + a_z (e_y ^ e_x)
When using e_i ^ e_j = -e_j ^ e_i one ends up with
(-I . a) = -a_x (e_y ^ e _z) – a_y (e_z ^ e_x) – a_z (e_x ^ e_y) = -1*(I . a)
With the same reasoning one has:
(-I . b) = -1*(I . b)
Thus,
(-I . a)(-I . b) = -1*-1*(I . a)(I . b) = (I . a)(I . b) = −a . b − I . (a x b)
/1/ https://jayryablon.files.wordpress.com/2017/01/geometric-algebra-tutorial-2.pdf
You have a sign mistake in your last equation. It is easy to spot. You can see it from the mismatch of trivectors on the LHS and RHS of your last equation, which incorrectly reads
(-I . a)(-I . b) = −a . b − I . (a x b).
The correct equation with matching trivectors on both the LHS and RHS should read:
(-I . a)(-I . b) = −a . b + I . (a x b).
Thus you have ended up misapplying the Hodge duality in the left-handed system.
Jay’s calculations are correct. In particular, his equations (11), (12) and (14) are correct.
Then you are claiming that
(I . a)(I . b) = −a . b − I . (a x b)
is not correct. The rest is simple algebra.
I am claiming no such thing.
In my own experience writing my own papers, I have often found that getting an overall sign correctly implemented is more that 10% of the work and time that goes into writing a paper. For example, in my paper https://jayryablon.files.wordpress.com/2016/10/lorentz-force-geodesics-brief-4-2.pdf, section 2 is devoted entirely to getting the signs all lined up correctly, and section 10 is largely devoted to making sure that the signs carry properly all the way through. It is not a trivial matter, because you can get everything else right as to magnitudes, but if you predict that bodies will fall up in a gravitational field, you will be wrong. And here, correlations are all about signs.
LJ has pointed to my equation (14) in https://jayryablon.files.wordpress.com/2017/01/geometric-algebra-tutorial-2.pdf as the place he is confused. And the problem can be pinpointed in the calculation he laid out:
Up to the final step, LJ’s calculation is correct for both mu = +I and mu = -I. In the final step, he is saying that
a ^ b = I . (a x b) (LJ 12)
In my equation (12) I am saying that:
a ^ b = mu . (a x b) (JRY 12)
That is where the difference is. One has to be correct, the other in error. So let’s discuss this, before I have to pack it in and get on the road and in the air.
a ^ b is a bi-vector. Let’s adopt a right-hand-rule (RHR) coordinate system. Irrespective of coordinate handedness, there are two cross-product uni-vectors originating from the vertex at which the tails of a and b meet. The first is right handed and thus positively-signed on the orthogonal axis in the RHR system, the second is left handed and thus negatively signed in this system. If we instead adopt a LHR coordinate system, there will still be two vectors orthogonal to the a and b plane. That never changes. But now the first will be negatively signed on the new coordinates and the second positively signed on the new coordinates. That is all that is happening.
If you would, please go back and read the last seven lines right before my figure 1, beginning with “All this really means…” and think on this for awhile. I had to. And it took me some time to write it carefully. The confusion lies in thinking as if the choice of RHR or LHR (lambda = +1 or -1) affects the two vectors associated with the a ^ b bi-vector. It does not. There are still two vectors, one pointing in each direction. All this means is that you are changing the sign of the orthogonal coordinates for each of the two vectors. But the physics must be unaffected by, i.e., invariant with regard to, your choice of coordinates. You cannot have one physical result for one choice of coordinates and another result for a different choice of coordinates. The parameter which represents your coordinate system — in this case lambda for handedness of the coordinates — must therefore be physically un-observable except on a relative basis, just like, say, a gauge / phase angle in gauge theory, an absolute velocity in SR, and a general system of coordinates in GR. That is why it is a “hidden” variable.
So as to the two equations above, I believe it is (JRY12) which properly says, in words, that “there are two vectors emanating from the vertex tail of a and b orthogonal to the plane defined by the bi-vector a ^ b. One points toward + in the coordinate handedness you choose, whether right or left, the other points toward – in the handedness you choose, whether right or left.” There is a confusion between two choices of handedness for the coordinates and two vector directions orthogonal to the bi-vector. This confusion needs to be crystallized and resolved in people’s minds.
If all this is so, then the following would be the correct results for my (12) and variants of that (pm = + or -):
(mu . a)(mu . b) = (+I . a)(+I . b) = +(I . a)(I . b) = −a . b – a ^ b = −a . b − mu . (a x b) (2a)
(mu . a)(+I . b) = (+I . a)(mu . b) = pm (I . a)(mu . b) = −pm a . b – pm a ^ b = −pm a . b − pm mu . (a x b) = −pm a . b − I . (a x b) (12b)
Lots of effort to get the sign done correctly, so that gravity makes things fall down. 🙂
I think that’s about it, see y’all in about ten days.
Jay
From equation (6) in your manuscript /1/ one gets:
(I . a) = a_x (e_y ^ e_z) + a_y (e_z ^ e_x) + a_z (e_x ^ e_y)
From equation (7) in your manuscript /1/ one gets:
(-I . a) = a_x (e_z ^ e _y) + a_y (e_x ^ e_z) + a_z (e_y ^ e_x)
When using e_i ^ e_j = -e_j ^ e_i one ends up with
(-I . a) = -a_x (e_y ^ e _z) – a_y (e_z ^ e_x) – a_z (e_x ^ e_y) = -1*(I . a)
With the same reasoning one has:
(-I . b) = -1*(I . b)
Thus,
(-I . a)(-I . b) = -1*-1*(I . a)(I . b) = (I . a)(I . b) = −a . b − I . (a x b)
Maybe, you have to recheck equations (6) and (7). What I am doing is essential math using your recipes.
/1/ https://jayryablon.files.wordpress.com/2017/01/geometric-algebra-tutorial-2.pdf
Jay cannot reply until he returns back from his trip. But his equations (6) and (7) are just fine. I have pointed out the mistake you are making. Please go back and read my responses above.
What mistake?
(-I . a)(-I . b) = -1*-1*(I . a)(I . b) = (I . a)(I . b)
That’s all. Derived simply on base of equations (6) and (7) presented in Jay R. Yablon’s tutorial https://jayryablon.files.wordpress.com/2017/01/geometric-algebra-tutorial-2.pdf .
When presenting recipes to do the math, one shouldn’t accept when they do not work to get the results one would like to get. “Confirmation bias” seems to be the essential problem in this whole story.
Sorry, it should read:
When presenting recipes to do the math, one *shouldn* accept when they do not work to get the results one would like to get. “Confirmation bias” seems to be the essential problem in this whole story.
Addendum:
Maybe, the disagreement is related to the fact that the 3-sphere model is based on a non-trivial Mobius-like twist in the Hopf bundle of S^3.
Still waiting for an answer!
Some program lines from Christian’s local-realistic simulation http://rpubs.com/jjc/233477
f = -1 + (2/sqrt(1 + ((3 * s)/pi)))
p = function(u,v){colSums(u*v)}
q = function(u,v,s){ifelse(abs(p(u,v)) > f, 1, 0)}
g = function(u,v,s){p(u,v)*q(u,v,s)}
A = +sign(p(a,e))*q(a,e,s) # Alice’s results A(a, e, s) = +/-1
B = -sign(p(b,e))*q(b,e,s) # Bob’s results B(b, e, s) = -/+1
Considering the q function and the „ifelse“ statement, I would write
A = +sign(p(a,e))*q(a,e,s) # Alice’s results A(a, e, s) = +1, 0, -1
B = -sign(p(b,e))*q(b,e,s) # Bob’s results B(b, e, s) = +1, 0, -1
Physically, measurement outcomes are either +1 or -1. In case one has a straightforward „local-realistic“ physical model, an „ifelse“ statement like {ifelse(abs(p(u,v)) > f, 1, 0)} is not needed. Why is it there?
I am still missing a plain answer!
From equation (6) in Jay R. Yablon’s tutorial /1/ one gets:
(I . a) = a_x (e_y ^ e_z) + a_y (e_z ^ e_x) + a_z (e_x ^ e_y)
From equation (7) in Jay R. Yablon’s tutorial /1/ one gets:
(-I . a) = a_x (e_z ^ e _y) + a_y (e_x ^ e_z) + a_z (e_y ^ e_x)
When using e_i ^ e_j = -e_j ^ e_i one ends up with:
(-I . a) = -a_x (e_y ^ e _z) – a_y (e_z ^ e_x) – a_z (e_x ^ e_y) = -1*(I . a) (1a)
or
(-I . a) = -1*(I . a) (1b)
With the same reasoning one ends up with:
(-I . b) = -1*(I . b) (2)
Thus, one has finally:
(-I . a)(-I . b) = -1*-1*(I . a)(I . b) = (I . a)(I . b) (3a)
or
(-I . a)(-I . b) = (I . a)(I . b) (3b)
Inescapable conclusion: (-I . a)(-I . b) and (I . a)(I . b) must be the same!
/1/ https://jayryablon.files.wordpress.com/2017/01/geometric-algebra-tutorial-2.pdf
I could not agree more with both of the above observations regarding Bell’s theorem.
Since Jay is off to enjoy his well-deserved vacation and cannot reply until he returns, let me reassure the readers that his calculations in his latest PDF are just fine. In particular, his equations (6), (7), (12), (13) and (14) are just fine. There is nothing wrong with them at all.
If a, b and c are unit vectors and “I” is a unit trivector (or a volume form), then in Geometric Algebra the right-handed and left-handed bivector bases can be defined very simply as
(I.a) (I.b) (I.c) = +1 ————– for the right-handed bivector basis,
and
(I.a) (I.b) (I.c) = -1 ————– for the left-handed bivector basis.
It is then easy to verify that the above two conditions lead to the following two equations
(I.a) (I.b) = -a.b – I.(a x b) ————– in the right-handed bivector basis,
and
(I.a) (I.b) = -a.b + I.(a x b) ————– in the left-handed bivector basis.
For a detailed discussion on the above two conditions and the corresponding equations please see eqs. (6) to (16) in the following paper: https://arxiv.org/pdf/1203.2529.pdf .
Jay’s equation (14) is then simply a short-hand way of writing the above two alternative equations, and it is exactly the same as equation (22) of the above paper. The two alternative possibilities are what constitute the only hidden variable lambda in my 3-sphere model.
I am still missing a plain answer!
From equation (6) in Jay R. Yablon’s tutorial/1/ one gets:
(I . a) = a_x (e_y ^ e_z) + a_y (e_z ^ e_x) + a_z (e_x ^ e_y)
From equation (7) in Jay R. Yablon’s tutorial /1/ one gets:
(-I . a) = a_x (e_z ^ e _y) + a_y (e_x ^ e_z) + a_z (e_y ^ e_x)
Is there any disagreement?
When using e_i ^ e_j = -e_j ^ e_i one ends up with:
(-I . a) = -a_x (e_y ^ e _z) – a_y (e_z ^ e_x) – a_z (e_x ^ e_y) = -1*(I . a) (1a)
or
(-I . a) = -1*(I . a) (1b)
With the same reasoning one ends up with:
(-I . b) = -1*(I . b) (2)
Thus, one has finally:
(-I . a)(-I . b) = -1*-1*(I . a)(I . b) = (I . a)(I . b) (3a)
or
(-I . a)(-I . b) = (I . a)(I . b) (3b)
This leads to the inescapable conclusion: (-I . a)(-I . b) and (I . a)(I . b) are the same!
/1/ https://jayryablon.files.wordpress.com/2017/01/geometric-algebra-tutorial-2.pdf
Since we live in a 3-dimensional space, the question you should really be asking is whether
(+I . a)(+I . b)(+I . c) and (-I . a)(-I . b)(-I . c) are the same.
I refer you to my comment back on October 27 11:31pm, my response to Jay’s Jan 4 7:04pm post and his final response to the same before he left on his trip, where he has concluded for himself there is equality when you stay within the confines of GA, and not attempt to (incorrectly) introduce the cross product.
One more time, let me say that for any algebra defining a product ‘*” with algebraic elements A and B,
(-A)*(-B) = (+A)*(+B)
Be clear, I do mean any legitimate algebra you could possibly dream up.
Since there remains some confusion on this point, let me point out that in the 3-dimensional space the product
(+I . a)(+I . b) = (-I . a)(-I . b)
is not sufficient to set apart the right-handed bivector basis from the left-handed bivector basis. For that purpose one must recognize that the triplet
(+I . a)(+I . b)(+I . c)
is not the same as the triplet
(-I . a)(-I . b)(-I . c).
While the triplet
(+I . a)(+I . b)(+I . c) = +1 ,
the triplet
(-I . a)(-I . b)(-I . c) = -1 ,
and this difference thus distinguishes the right-handed basis from the left-handed basis.
Joy, I agree with Donald Graft and Lord Jestocost above that this is a model of the detection loophole. The constraint (32) in your paper depends on the measurement choices, so it is really a threshold for whether a detection occurs. I know that just before equation (40) you assert that the detection rate is 100%, but in the simulation
… but in your simulation it is not 100%, as your q() function generates 0’s that are thrown out. (And q takes th detector setting as an input.)
I disagree with your reading of my model and its simulation. There are no zero outcomes in either of them. There is a strict one-to-one correspondence between the initial states (e, s) of the singlet system and the simultaneous measurement outcomes (A, B). Thus there is no question of non-detected events or missed initial states. Consequently neither my theoretical model nor its event-by-event simulation has anything to do with the detection loophole.
It seems q() would have to have no dependence on the detector settings a and b for that to be true. But it does in the simulation.
How else could you determine which are valid states in S^3? You probably should stop thinking in R^3.
I’m not uncomfortable with theowing out an initial state in the simulation. It would be like throwing out a 6 when trying to get a uniform random integer between 1 and 5 by rolling a 6 sided die. But you use q() to throw out trials (when it returns 0), and it depends on the LATER detector setting choice.
No trials are thrown out from S^3 itself. As Fred says (and as anticipated in the preamble of the two simulations), the confusion arises only by thinking in terms of R^3 instead of S^3.
As a physicist, I would say: What is the role of a {ifelse(abs(p(u,v)) > f, 1, 0)} statement in the q() function when one has a manifest local-realistic simulation algorithm for a LHV-model. Very strange!
As a physicist you may know the concept of the physically meaningful “reduced phase space” from gauge field theories. The correct physical degrees of freedom are obtained in gauge theories by quotienting out the unphysical mathematical degrees of freedom. The role of the if-else statement in the simulations is exactly the same. It quotient-outs the unphysical degrees of freedom from R^3 canvas to isolate the correct physical degrees of freedom in S^3.
I would say: Some combinations (a,e,s) and (b,e,s) in A = +sign(p(a,e))*q(a,e,s) or B = -sign(p(b,e))*q(b,e,s) with q = function(u,v,s){ifelse(abs(p(u,v)) > f, 1, 0)} do not work.
Joy,
a {ifelse(abs(p(u,v)) > f, 1, 0)} statement in the q() function within a manifest local-realistic simulation algorithm (http://rpubs.com/jjc/233477) has nothing to do with S^3 or R^3. It produces “Zero”-outputs regarding A = +sign(p(a,e))*q(a,e,s) or B = -sign(p(b,e))*q(b,e,s). I think some (e,s) states cannot be projected and must therefore be discarded, somehow a “detection loophole”. Otherwise, there would be no need for such a q()-function.
I have already answered that question:
Why you are then using a {ifelse(abs(p(u,v)) > f, 1, 0)} statement in the q() function?
I would say:
Some combinations (a,e,s) and (b,e,s) in A = +sign(p(a,e))*q(a,e,s) or B = -sign(p(b,e))*q(b,e,s) with q = function(u,v,s){ifelse(abs(p(u,v)) > f, 1, 0)} do not work.
Of course they don’t work because those states never existed in S^3 in the first place. That function is how you determine that.
Two concluding remarks regarding the paper “Local causality in a Friedmann–Robertson–Walker spacetime”, retracted by the editors of “Annals of Physics”.
1) The inescapable result using Geometric Algebra, that is to say (-I . a)(-I . b) = (I . a)(I . b), shows that the model underlying “Local causality in a Friedmann–Robertson–Walker spacetime” is based on a mathematical flaw.
2) There exists no working local-realistic simulation algorithm. The claimed implementation suffers from the fact, that some combinations (a,e,s) and (b,e,s) in A = +sign(p(a,e))*q(a,e,s) or B = -sign(p(b,e))*q(b,e,s) with q = function(u,v,s){ifelse(abs(p(u,v)) > f, 1, 0)} do not work and lead thus to indefinite results.
To my mind, the retraction of the paper “Local causality in a Friedmann–Robertson–Walker spacetime” was thus justified by scientific reasons and was in no respect *politically* motivated.
Joy, I think a rewrite of the paper along these lines would not receive such resistance:
I have a simulation which I consider local reastic. The trick is in the q() function where the detector setting choice and initial state determine if a trial should be counted or not. It might look like a detection loophole, but I view it another way because… It might alternatively be viewed as backwards causation, but I see it differently because…
Both of your claims are false, as I have explained in this thread. Moreover, your reason for your conclusions is not the reason provided for the retraction of my paper on the publisher’s website. There is not a single scientific, mathematical, or physical flaw in my model or paper.
I invite you to publish your criticisms of my model and its simulation on the arXiv for the benefit of the physics community, where I can then respond to them in full scientific details.
Jay, got an email on your question to me, apparently you linked it as a response to an unrelated comment by Diether. Let me paraphrase your question as what my opinion of your (12) and the correctness of mapping between the wedge and cross product is. The short answer is it is inappropriate to try it in the current setting. Let’s try a more pedagogical approach. If you had x component electric and magnetic field strengths of Ex = 5 and Bx = -5, would you conclude Ex+Bx=0? I would hope not. If your (13) additionally had the GA vector c1 e1 + c2 e2 + c3 e3, would you attempt to put it in the same physical xyz space you put the bivectors in and attempt to add c components to your cross product coefficients? I would hope not.
In the latter example, what tells you it would be wrong to add the c coefficients to the cross product coefficients? You could after all associate c1 e1 with the physical x direction you bring up, so why not add c1 to +/- (a2b3 – a3b2)? It becomes abundantly clear why you can’t when the basis elements are included, for you then have for example c1 e1 – (a2b3 – a3b2) e2^e3. Easy to see the issue.
Bringing in the cross product representation, you are inserting only the coefficients +(a2b3 – a3b2) and -(a2b3 – a3b2) into the physical x direction, as you claim as physical vectors pointing in opposite directions, and therefore assuming they sum to zero. Doing so, you jumped outside GA holding on to some of its remnants, incorrectly applying significance in the new scheme.
When you did as I asked, calculating your (13) in both GA orientations stopping short of the invalid cross product insertion, you correctly found that (-I.a)(-I.b) = (+I.a)(+I.b) by not making the invalid jump outside of GA. This is reality.
The conclusion is Christian’s derivation in S^3 that E(a,b) = -a.b is incorrect as it requires
(-I.a)(-I.b) + (I.a)(I.b) = -2 a.b, which clearly is not the case.
This paper should have never been published because of this math error as well as others
The experimental outcomes +1, -1 are defined differently at Alice and Bob. By your logic, it is mathematically incorrect to multiply those two numbers and calculate an average. In fact, you can perform an experiment in which Alice defines clockwise = +1, and Bob defines clockwise as -1, then you can send a series of pairs of clocks to both and after a while you get a list of numbers from both and calculate the average of the paired product. But you appear to be arguing that it is mathematically illegitimate unless their definitions are harmonized and all the data is transformed to the same basis.
The experimental results +1 and -1 are and can only be real numbers. As such, the basis issues you refer to are not relevant to any post measurement data manipulations. The basis issues are only relevant to the mathematical model floated to explain the generation of the experimental results.
That is a different argument. But your problem with Joy’s model does not appear to be that his results are not real. Your argument is that he is multiplying results defined within two different basis inconsistently. If you are right, then what I say is correct. Alice and Bob will not be allowed to multiply their results together unless they are transformed to the same basis irrespective of whether the results are real or not.
If you want to argue that the results must be real, do so. The basis arguments are unconvincing.
There is no transformation required as you claim. Real numbers add, subtract, multiply and divide by the same rules independent of whether or not the algebra in question is limited to the real numbers by themselves, or if the real numbers are part of a higher dimension hypercomplex algebra.
There is no information available in the measurement values +1 and -1 that can possibly guide you to how these values were generated, or what algebra that process was governed by.
The basis issues and Christian’s math errors are related to his E(a,b) = -a.b derivation, not the sum of products of Alice and Bob +1 or -1 measurements.
The above claim by Lockyer is false. Nowhere in any of my papers have I used, required, or derived the equation
(-I.a)(-I.b) + (I.a)(I.b) = -2 a.b
as he claims. Anyone can verify this from this paper: https://arxiv.org/abs/1405.2355
The actual derivation of the correlations E(a, b) = -a.b is now published in the following paper in the International Journal of Theoretical Physics: https://arxiv.org/abs/1211.0784 .
The same result is also published in several other papers by me on the physics arXiv:
https://arxiv.org/find/all/1/au:+Christian_Joy/0/1/0/all/0/1 .
Lockyer’s mathematical and conceptual mistakes in his equation above have been pointed out to him before in considerable detail, both by me as well as several other physicists:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=226#p5859
In your first reference, the paper in question, you have the following
(50) L(a, lambda) = lambda I.a
(51) L(b, lambda) = lambda I.b
In your (72) the infinite sum can be replaced by the sum of 2 values since the distribution of lambda = +1 or -1 is 50%/50%, one with lambda = +1 and one with lambda = -1. You then have
(72) 1/2 { L(a,-1)L(b,-1) + L(a,+1)L(b,+1) }
This by (50) and (51) can be written as
(72) 1/2 { (-I.a)(-I.b) + (I.a)(I.b) }
My claim you dispute follows directly from your position this somehow equals -a.b.
I am thinking this is fairly clear now to most observers, and should be to you also. Why don’t you do what I asked Jay to do, derive the results exclusively carrying the GA basis, and not inserting erroneous cross product substitutions?
The above misrepresentation by Lockyer of the equations used in my paper to derive the correlation E(a, b) = -a.b brings out his mathematical and conceptual mistakes quite clearly.
Since the orientation lambda is a fair coin, my equation (72) can indeed be simplified to
1/2 { L(a,-1)L(b,-1) + L(a,+1)L(b,+1) }
as Lockyer has done. But its following misrepresentation is not from my paper at all:
1/2 { (-I.a)(-I.b) + (I.a)(I.b) } .
It is simply equal to (I.a)(I.b), and does not follow from my equations (50) and (51) as Lockyer has claimed. This is because L(a, lambda) and L(b, lambda) are non-commuting bivectors. They do not commute, as Lockyer seems to have unwittingly assumed. Moreover, lambda = +1 and -1 is a toss of the orientation of S^3. Therefore the correct equations are in fact the equations (77), (78) and (79) of my paper, which can be simplified in a single equation as
1/2 { (+I.a)(+I.b) + (+I.b)(+I.a) } = -a.b .
Note that there is no need of using the cross product or the duality relation between vectors and bivectors to derive the correlation E(a, b) = -a.b. They follow at once from the definition of the inner product in geometric algebra itself, as in equation (79) of my paper.
I think everyone can see what I did was direct substitution.
Indeed. As if L(a, +I) and L(b, +I) were a commuting vectors and lambda was just a number.
But it is clear from the definitions of spins, detectors, and lambda in my paper that while
L(a, +1) L(b, +1) = D(a) D(b) ,
L(a, -1) L(b, -1) = D(b) D(a) .
As if nothing!! Direct substitution with no qualifiers, yielding the following equivance as you previously conceded to:
Your admission
(72) equivalent to 1/2 { L(a,-1)L(b,-1) + L(a,+1)L(b,+1) }
Direct substitution using your (50) and (51)
(72) 1/2 { (-I.a)(-I.b) + (I.a)(I.b) }
But it is easily demonstrated that (I.a)(I.b) = (-I.a)(-I.b) by the rule of real number multiplication indicating the 2 minus signs cancel. So, what are we left with? Obviously
(72) 1/2 { (I.a)(I.b) + (I.a)(I.b) } = (I.a)(I.b)
Thus (72) = -a.b if and only if a=b, surely not a restriction anyone can live with.
My mistake, if and only if a=b or a=-b, same unacceptable restriction
I have already pointed out Lockyer’s mathematical and conceptual mistakes above which he keeps repeating.
L(a, lambda) and L(b, lambda) are not commuting vectors, and lambda is not just a number.
L(a, lambda) and L(b, lambda) are non-commuting bivectors, and lambda is the orientation (or handedness) of the 3-sphere. Moreover, as stressed in my paper, an orientation of a manifold is a relative concept (see also the explicit definition of orientation, Definition V.1, in this published paper: https://arxiv.org/pdf/1211.0784.pdf ).
Thus handedness of the spin basis L(a, lambda) has meaning only with respect to the handedness of the detector basis D(a) and vice versa, as explicitly noted in equation (56).
The correct derivation of the correlation E(a, b) = -a.b is thus given in equations (77), (78) and (79) of the paper, as an alternative to the derivation in equations (73), (74) and (75).
Now, as noted in my paper, since lambda is a fair coin, equation (72) can be simplified to
1/2 { L(a, lambda = +1) L(b, lambda = +1) + L(a, lambda = -1) L(b, lambda = -1) } ,
where the two cases of right-handed and left-handed orientation are necessarily equal to
L(a, lambda = +1) L(b, lambda = +1) = D(a) D(b) = (+I.a) (+I.b)
and
L(a, lambda = -1) L(b, lambda = -1) = D(b) D(a) = (+I.b) (+I.a) .
The last two identities follow straightforwardly from the definitions (48) to (51), as demonstrated in the Appendix A of this paper https://arxiv.org/abs/1501.03393 .
If one does not use the above pair of relations to derive the correlation, then one is working within a fixed basis for S^3, and therefore not working within my model in the first place.
Substituting now the last two equations into the first equation immediately gives
E(a, b) = 1/2 { (+I.a)(+I.b) + (+I.b)(+I.a) } = -1/2 { a b + b a } = -a.b .
Lockyer’s incorrect assertions and mistaken inferences stem from his failure to recognize that lambda is a relative concept and that L(a, lambda) and L(b, lambda) are non-commuting numbers. If one does not respect that, then one is not working within my model to begin with.
In case I understand you correctly, lambda is not a real number like +/-1 but merely a „relative concept“. Suppose then that we now regard this shared lambda as a random event which has a 50%-50% chance to belong to a +1-„relative concept“ or -1-„relative concept“ for any given one of N trials; in case we conduct then a very large number of trials, this would lead finally – on the average -to no concept at all.
i have wondered for a long time how Joy Christian could argue forcefully he has made no math errors despite overwhelming evidence to the contrary. The revelation he believes lambda is not a real number explains a lot. It is this misconception that got him off on the wrong foot.
It had to be something like this, which is why in the past elsewhere and here I have attempted to back up to common ground all parties could agree on, and try working together from there. Whether or not lambda is a “relative concept”, we should all agree there are 2 bivector basis choices:
Basis 1: e2^e3, e3^e1, e1^e2
Basis 2: e3^e2, e1^e3, e2^e1
I asked Jay to do his equation (13) using basis 1, then again with basis 2, staying within geometric algebra rules and avoiding possible confusion bringing in cross products. He revealed he found exactly what I said he would, a demonstrated equality. For Christian’s claim in S^3 E(a,b)=-a.b, the bivector result portion using basis 1 must subtract from the bivector portion using basis 2 such that the agreed on sum that follows has no bivector content, for this is precisely what the two summed values represent.
1/2 { L(a,+1)L(b,+1) + L(a,-1)L(b,-1) }
Equality between the 2 summed terms is precisely what you would expect if lambda actually is a real number, yet it does not explicitly come into play in the above analysis.
I became interested in this “Retraction Watch” blog entry when reading Joy Christian’s serious allegation: “To my eyes this fact alone proves beyond doubt that the secret removal of my paper from their website was entirely politically motivated, without any scientific basis whatsoever“ (see comment Joy Christian October 4, 2016 at 10:32 am). I was really worried about that; if the retraction was *politically* motivated, this could in the end lead to severe damage of the reputation which the hard science „Physics“ still enjoys within the interested public.
I thus decided to get involved into the debate in order to find out whether Joy Christian’s accusation might be justified to a certain extent. To my mind, the retraction is scientifically justified and there is no need to discuss subtleties.
From a classical standpoint we would imagine that each particle emerges from the singlet state with, in effect, a set of pre-programmed instructions for what spin to exhibit at each possible angle of measurement, or at least what the probability of each result should be. Thus, what ultimately counts is the response (+/-1) of the measurement apparatus as a function of the measurement angle in relation to the spin of one particle (from symmetry considerations it follows that the instructions to one particle are just an inverted copy of the instructions to the coupled particle).
From equation (54) and (55) in https://arxiv.org/abs/1405.2355 one reads out the response function of the measurement apparatus:
A(a, lambda^k) = +1 if lambda^k = +1 ……………………………………………….. (LJ1)
A(a, lambda^k) = -1 if lambda^k = -1 ………………………………………………… (LJ2)
B(b, lambda^k) = -1 if lambda^k = +1 ………………………………………………… (LJ3)
B(b, lambda^k) = +1 if lambda^k = -1 ………………………………………………… (LJ4)
This is the “probability list” for the measurement outcomes as predicted by the proposed classical local-realistic hidden variable (LHV) model.
*************************************************************************************
At this point, one meets the fundamental difference between quantum mechanical predictions and predictions made by LHV models.
In classical physics, you make *statements about the object*, here the two spin1/2 particles in the singlet state. LHV theories are really deterministic. If you tell me the initial positions and spins of both particles, I can — with enough computational power — tell you how every one of them will evolve, move, and where they will be located and what the spin’s orientations will be at any point in time; everywhere, in the Friedmann-Robertson-Walker spacetime or somewhere else. And by that, one has principally everything at hand to predict the pure *object-related* responses of the measurement apparatus and to determine the correlation E(a,b). The behavior of the object rules what the observing subject will get when performing a measurement. The subject is a passive observer.
That is why LHV models always produce probability lists, like the list (LJ1 – LJ4), for definite measurement outcomes: For every given hidden variable, one can in principle compute a unique and definite set of measurement outcomes, viz. one can make pure *statements about the object* which are completely *independent of the observing subject*. On base of such abstract reasoning, one can derive, for example, the CHSH inequality (in the considered case the list would be a Nx4 list).
Let’s look now to quantum mechanics:
“Quantum mechanics forbids statements about the object. It deals only with the object-subject relation.” (Schroedinger to Sommerfeld, 1931, this quote can be found on N. David Mermin’s homepage http://www.lassp.cornell.edu/mermin/)
It is – to my mind – impossible to translate the notion “object-subject* relation into classical notions! To say it in other words: In quantum mechanics, there exists absolutely nothing like a response list (LJ1 – LJ4). You are even not allowed to think about. In the considered case, you have nothing else than the following *object-subject* relation for the correlation E(a,b):
E_qm(a,b) = a . b …………………………………………………………………………………. (LJ5)
Following the statistical (Born) interpretation of quantum mechanics, the formalism of quantum mechanics is a theory which provides statistical predictions referring to a sufficiently large ensemble of equally prepared systems S(phi) after the measurement of the observable in question.
*************************************************************************************
Let’s go back to the list (LJ1 – LJ4). Using it, illustrates the fundamental dilemma one is caught in when trying to reproduce quantum mechanical predictions by means of classical models. From equations (LJ1 – LJ4) one has – without exception and with no further relation to a, b and lambda^k and no further relation to any definition of these quantities:
A(a, lambda^k)* B(b, lambda^k) = -1*( lambda^k)^2 = -1 ………………………….. (LJ6)
This was pointed out by Richard D. Gill in https://arxiv.org/abs/1203.1504.
Turning now to equation (67) in https://arxiv.org/abs/1405.2355 to calculate the correlation E(a,b). It reads:
E(a,b)_lhv = LIMES_n {1/n* SUM_k:1_n { A(a, lambda^k), B(b, lambda^k) } } ……… (LJ7)
where LIMES_n means the mathematical limes operation with n (number of trials) going to infinity and SUM_k:1_n means summing the terms A(a, lambda^k), B(b, lambda^k) from k = 1 up to k = n.
When inserting (LJ6) into the rhs of equation (LJ7), one obtains:
E(a,b)_lhv = -1 ……………………………………………………………………………………… (LJ8)
Conclusion: A LHV model based on a local-hidden variable lambda^k as proposed in “Local causality in a Friedmann–Robertson–Walker spacetime” (https://arxiv.org/abs/1405.2355 ) is not able to produce the quantum mechanical correlation E_qm(a,b) = a . b.
One might now try to devise more sophisticated LHV models using sets of numerous hidden variables in order to produce more extensive probability lists for *object-related* measurement outcomes. Nevertheless, there is no way out: One will always end up with lists! And by means of such lists one can always deduce Bell’s no-go theorem:
„No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics“.
The above outlined dilemma is a *fundamental* one. Quantum mechanics and its predictions, all of which have been experimentally confirmed, cannot be understood in terms of classic notions and conceptions. Quantum mechanics deals exclusively with – in our classical view often – “paradox” *object-subject* relations. There is nothing like – so to speak – observer-independent objects in the sense our “every-day” experiences might suggest us. Thus, every classical theory which treats objects as if they are independent of the observing subject, must finally fail to reproduce all of the predictions of quantum mechanics.
Classical physics has no equivalent for these *object-subject* relations, viz. there are no means to “translate” quantum mechanical features – think about double slit experiments, delayed choice experiments, quantum entanglement or the Quantum Zeno effect – into classic notions and conceptions. Try it and you will produce nothing else than weird nonsense. The body of literature is full of such stuff. The classical “A OR B” which one can notionally even read out from imaginary, classical probability lists is fundamentally different from the quantum mechanical “A AND B”. There is no way to bridge this abyss. That’s the reason why all LHV models have failed or will fail finally.
That seems to be paradox, but when following Feynman’s reasoning one might understand: The “paradox” is only a conflict between reality and your feeling of what reality “ought to be”. And to cite Feynman again: “Attempting to understand quantum mechanics causes you to fall into a black hole, never to be heard from again”.
Correction:
(LJ5) must read:
E_qm(a,b) = -a . b
Larsson and Gill, stated in their paper (https://arxiv.org/pdf/quant-ph/0312035v2.pdf, page 4):
The problem here is that the ensemble on which the correlations are evaluated changes
with the settings, while the original Bell inequality requires that they stay the same. In effect,
the Bell inequality only holds on the common part of the four different ensembles ΛAC′ , ΛAD′ ,
ΛBC′ , and ΛBD′
What is the common part of the 4 disjoint ensembles generated in experiments?
I recommend you ask Richard D. Gill.
There exists no proof in the literature of the above claim. Indeed, there cannot possibly exist a proof of such a blanket statement with undefinable terms such as “physical theory” and “local hidden variables.” In 1964 John Bell attempted to prove the above claim by claiming that it was impossible to reproduce the correlations E(a, b) = -a.b predicted by quantum mechanics for a specific quantum state, namely for the singlet state or the EPR-Bohm state. It is however easy to prove that:
(1) Bell’s supposed proof for the above specific claim is flawed, because the CHSH inequality on which his proof is based can be derived without assuming locality and realism, as done in the appendix of this paper: http://philsci-archive.pitt.edu/12655/ . There the CHSH inequality is derived by assuming only that Bob can measure along b and b’ simultaneously while Alice measures along either a or a’, and likewise Alice can measure along a and a’ simultaneously while Bob measures along either b or b’, *without assuming locality*. The experimental “violations” of CHSH inequality therefore only means impossibility of measuring along b and b’ (or along a and a’) simultaneously.
And
(2) It turns out that, contrary to Bell’s claim, the singlet correlations E(a, b) = -a.b can in fact be reproduced using purely local functions A(a, h) and B(b, h), in a manifestly local and realistic manner. A detailed and explicit local-realistic derivation of the correlations -a.b is presented in the paper withdrawn by Annals of Physics, https://arxiv.org/abs/1405.2355 , which also includes references to several event-by-event computer simulations of the correlations. The code and a plot of one such simulation is available at the following link:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=296#p7322 .
Thus Bell’s blanket claim quoted above (a) remains unprovable, (b) is intrinsically flawed for the reason explained in (1), and (c) Bell’s specific claim that it is impossible to reproduce the singlet correlations E(a, b) = -a.b local-realistically also turns out to be false, as evident from the theoretical counterexample and its event-by-event simulations presented in (2) above.
Sorry, read my comment carefully. Your papers are irrelevant with respect to the topic I am addressing.
On the contrary, my papers and the local-realistic counterexample to Bell’s claim are precisely the topic of this thread. Contrary to the claim in your post above, a comprehensive framework reproducing all quantum mechanical correlations already exists as a rigorous theorem on page 12 of this paper: https://arxiv.org/abs/1201.0775 . What is more, I am in the process of writing a detailed review of my local-realistic framework that is based entirely on the geometrical topological properties of the 3-dimensional physical space.
Deliver a physically consistent {h; A(a; h), A(a’; h), B(b; h), B(b’; h)}-list for the responses of the measurement apparatuses, and we can discuss furthermore. You are claiming that you have devised a local-realistic hidden-variable model and I have explained what such a classical model must be able to produce.
To be honest, do you really think that you have understood the origins of quantum correlations?
Yes! I have. Please read my paper entitled “On the Origins of Quantum Correlations” to find out for yourself: https://arxiv.org/abs/1201.0775 .
More mathematical details can be found here: https://arxiv.org/abs/1101.1958 .
This is all very simple. Christian’s model has two functions A(a, lambda) and B(b, lambda), where the functions give the outcomes (-1/+1) of Alice’s and Bob’s experiments respectively, given their settings a and b, and the value of the hidden variable lambda. So far so good; any LHV theory must provide functions like that.
In Christian’s model, lambda can take on two values (let’s denote them by -1 and 1).
Let’s now look at the values
A(0, -1), A(90, -1), A(0, 1), A(90, 1), B(45, -1), B(135, -1), B(45, 1), B(135, 1).
Christian will not tell us the function values for those arguments (it seems he thinks he has no “obligation to do that”). But since there is only a finite number of possibilities (2ˆ8 = 256), we can computationally check the correlations for all combinations.
Spoiler: None of the 256 possible combinations reproduce the quantum correlations.
It is indeed very simple: the correlations have been exactly reproduced, by simply applying World War II vintage radar signal detection theory, to a pair of polarized, entangled coins that exhibit only a single bit of recoverable information.
See: http://vixra.org/pdf/1609.0129v1.pdf
HR: see my recent comments about Joy’s simulation’s q() function. He is throwing out certain trials based on the initial state and later detector settings combined. So it does not fit the Bell framework. But he is asserting throwing out trials like this is not a detection loophole. Somehow these combinations of initial states and detector settings are supposed to be impossible because of spacetime geometry. I don’t see it, unless the order of events can be reversed. And then it seems one is debating the meaning of “causality”, or else it is backwards causation which Joy says it is NOT in his paper.
The above two comments contradict the claim by Bell’s theorem, which was quoted above to be the following:
In the specific case of the singlet correlations we are concerned about here, the above claim concerns reproducing the 13 different probabilities predicted by quantum mechanics. These 13 probabilities are explicitly reproduced in my paper: https://arxiv.org/abs/1405.2355 . Let me say this again: All 13 probabilities have been rigorously reproduced within a purely local-realistic setting of quaternionic 3-sphere. The local functions A(a, h) and B(b. h) used for this purpose then lead to the following correlations and the individual statistical results explicitly:
E(a) = (1/N) Sum_k { A(a, h_k) } = 0 ,
E(b) = (1/N) Sum_k { B(b, h_k) } = 0 ,
and
E(a, b) = (1/N) Sum_k { A(a, h_k) B(b, h_k) } = -a.b .
In other words, all of the experimental results observed in Nature and predicted by quantum mechanics for the singlet state (or the EPR-Bohm state) have been reproduced in the paper linked above in a purely local-realistic manner, which have then been verified in several computer simulations, including the one using the GAViewer program linked in my previous post. I find it extraordinary that all of this evidence is simply being overlooked.
The functions A(a, h_k) and B(b, h_k) defined by equations (54) and (55) in https://arxiv.org/abs/1405.2355 represent points of the Einsteinian 3-sphere. h_k is the orientation of this 3-sphere.
According to equations (54) and (55) one has for orientation h_k=+1:
A(a, h_k) = +1 ………………………………………..……………………….. (LJ1)
B(b, h_k) = -1 ………………………………………………………………….. (LJ2)
And for orientation h_k=-1:
A(a, h_k) = -1 ……………….. ………………………………………………… (LJ3)
B(b, h_k) = +1 ……………….. ………………………………………………… (LJ4)
Consequentially:
A(a, h_k=+1)* B(b, h_k=+1) = -1 ………………………………………. (LJ5)
A(a, h_k=-1)* B(b, h_k=-1) = -1 ………………………………………. (LJ6)
Consequentially:
E(a, b) = (1/N) Sum_k { A(a, h_k) B(b, h_k) } = -1 ………………………… (LJ7)
As I have pointed out to you before, there is no way to derive the correlation anything other than E(a, b) = -a.b within my 3-sphere model without violating either the conservation of zero spin angular momentum or the geometrical and topological properties of the 3-sphere.
In other words, your last equation, eq. (LJ7), is imply incorrect, because it manifestly violates the conservation of zero spin angular momentum. Please do you calculation again respecting my eq. (66), and you will find that the correlations within S^3 can only be E(a, b) = -a.b.
LJ7 is a trivial consequence of LJ5 and LJ6, by simple substitution.
How should I respect your equation (66)? On the rhs of equations (54) and (55) in https://arxiv.org/abs/1405.2355 there is – much to my regret – nothing like s1 or s2.
You can respect my equation (66) — which is nothing but a statement of the conservation of zero spin momentum — just as I have done in my equation (70). If you do not respect my equation (66) as I have done in my equation (70), then you are violating one of the most basic conservation laws of physics, in addition to abandoning the 3-sphere geometry.
When using equation (66) to evaluate equation (70) or (71) in https://arxiv.org/abs/1405.2355 one gets
E(a, b) = -1.
It seems that you have forgotten to carry out the double mathematical limes operation “s1 approaches a” and “s2 approaches b” in a correct way. A simple mathematical error!
Actually one inevitably gets E(a, b) = -a.b, unless Alice and Bob have chosen to set a = b.
Joy,
all your claims are based on a math error. When going from equation (71) to (72) in https://arxiv.org/abs/1405.2355 you have simply forgotten to “keep” the double limes function
lim[for “s1 approaches a” and “s2 approaches b”]{ term }
That’s all!
In that case I recommend that for the benefit of the larger physics community you point out my “math error” on the physics arXiv where my paper is published. Needless to say, in my opinion you are the one who has been making math errors, but that is just my opinion.
All that has been done since many years by others. What should I supplement?
I am working as an experimental physicist and I know that the responses of detector A and B can only be +/-1. So, provide a physically consistent response list {h; A(a; h), A(a’; h), B(b; h), B(b’; h)} for all initial states, h, of the singlet system you would like to consider. You are claiming that you have a classical local-realistic hidden-variable model in the Friedmann-Robertson-Walker Spacetime! So, you have everything at your hand to provide the response list. Why should I bother with the math? By means of such a response list, I can then simply perform my virtual “experimental measurements”.
Afterwards, I will evaluate the interesting correlations and submit a short paper to arXiv.org in order to show that somebody has gotten again the CHSH inequality when relying on local-realistic hidden variable models.
And please, don’t answer in your typical way: “If you provide a response list generated by, for example, quantum mechanics, then I will immediately provide you the list you are asking for.” We are here not on http://www.sciphysicsforums.com/spfbb1/index.php!
No such response list is either predicted by quantum mechanics or observed in any actual experiment. Your demand thus contradicts your previous claim: “No physical theory of local hidden variables can … reproduce all of the [statistical] predictions of quantum mechanics“. By contrast, you will find all 13 probabilities predicted by quantum mechanics reproduced explicitly in my paper, which are also observed in the actual experiments.
Dr. Christian, you are correct that such a list is neither predicted by quantum mechanics nor observed experimentally. However, the whole point of a hidden-variable theory is to go beyond what quantum mechanics predicts and provide the “hidden” deterministic values. For example, this is what the prototypical hidden-variable theory – Bohmian mechanics – does.
After all, if a hidden-variable theory does not go beyond quantum mechanics, why bother with it? Why not simply stay with quantum mechanics and save oneself all the trouble?
You are completely mistaken about all of the points you have just made. The purpose of any hidden variable theory is to simply reproduce the predictions of quantum mechanics and / or account for what is actually observed in the experiment. To think otherwise is to miss Einstein’s point. No one has explained this point (i.e., Einstein’s view) better than John Bell himself.
But it is obvious that if h_k can only take two values with 50/50 chance, then the rhs in
E(a, b) = (1/N) Sum_k { A(a, h_k) B(b, h_k) }
can only be -1, 0, or 1, for any vaues of a and b. This follows from elementary arithmetic. It does not reproduce the predictions of quantum mechanics.
Oh, but it does within S^3: https://arxiv.org/abs/1405.2355 .
It is trivial to verify that within S^3, with 50/50 orientation chance of h = +1 or h = -1,
E(a, b) = (1/N) Sum_k { A(a, h_k) B(b, h_k) } = -a.b
is inevitable, together with
E(a) = (1/N) Sum_k { A(a, h_k) } = 0
and
E(b) = (1/N) Sum_k { B(b, h_k) } = 0 ,
for A(a, h_k) = +1 or -1 and B(b, h_k) = +1 and -1, as defined in my eqs. (54) and (55).
In fact E(a, b) = -a.b holds even in a 2D Mobius world: https://arxiv.org/abs/1201.0775 .
For everyone asking Joy to supply a list, you are looking for a list like this:
h=hidden variables emitted
A=Alice’s observation with one detector setting choice
A’=Alice’s observation with the other detector setting choice
B=Bob’s observation with one detector setting choice
B’=Bob’s observation with the other detector setting choice
The data would look like this:
h,A,A’,B,B’
h_1,+1,+1,-1,+1
h_2,-1,+1,-1,-1
h_3,+1,-1,+1,+1
…
etc.
We all agree (even Joy) that what we call the “rhs calculation” cannot exceed the bound 2.
Then I think we all agree (even Joy), that if one attempts to make this look like real data, by (independently of the way the rows look) randomly choosing one Alice and one Bob observation to make an NA, then the resulting “lhs calculation” has almost no chance of exceeding the bound 2 by a statistically significant amount either. So no Bell violation unless we are really lucky. But I think Joy is imagining something which translates to a different process:
After picking the NA’s above for a trial i, his scheme would require we plug in h_i and Alice’s detector setting choice to his “q() function”. This function would either tell us to keep or discard the trial. We would similarly do this for Bob. We would only keep the rows that meet both Alice’s and Bob’s thresholds. This looks like a “detection loophole”, but it seems Joy says that it is not because spacetime geometry makes the combinations of hidden states and detector setting choices impossible. I then think it is “backwards causation”, as how else can the LATER detector setting choice make a trial have never happened? Joy’s paper says it is NOT backwards causation. I don’t see this, unless we are using some hyper-nuanced definition of “causation”.
Luck has nothing to do with it. It is mathematically impossible for anything to violate a Bell inequality. That fact should be obvious by now. All that is left is that we have the “strong” correlations predicted by QM and validated by the experiments. Now we also have a local realistic S^3 model that also predicts the same thing and is also validated by the experiments. Now what you guys are missing is that you are rejecting the S^3 postulate. But you have to admit that if S^3 topology and geometry are true, then Joy is right. It is an alternate explanation for what is seen in the experiments.
No such list is either predicted by quantum mechanics or observed in the actual experiments, so why is anyone asking for such a list in the first place?
Because according to everyone else’s definition of what a hidden-variable model is, it must be able to produce such a list.
Maybe this can be a peaceful resolution to this discussion? You are simply talking about different things.
Such a definition of a hidden variable model makes no sense. Bell himself (let alone Einstein) never demanded anything that goes beyond what is predicted by quantum mechanics and / or observed in the actual experiments. To quote from the opening paragraph of Bell’s 1965 paper: “…”hidden” because if states with prescribed values of these variables could actually be prepared, [then] quantum mechanics would be observably inadequate.” Thus a demand of a list of hidden variables and measurement outcomes of the kind being demanded here would prove quantum mechanics wrong (not just incomplete). Einstein’s goal was not to prove quantum mechanics wrong, and Bell understood that very well. But if what you are saying is right, then some followers of Bell do not understand either Einstein or Bell.
You have misunderstood what Bell means with this paragraph. He is saying that one cannot actually have control over the hidden parameters, otherwise we have simply a straightforward demonstration that quantum mechanics is incomplete.
But I would appreciate an answer to my earlier question. If you don’t think that hidden-variable models should provide predictions beyond quantum mechanics, why are you interested in them? What do they have that simple quantum mechanics does not?
I understand perfectly well what Bell means by “observably inadequate” in this quote: ”hidden” because if states with prescribed values of these variables could actually be prepared, [then] quantum mechanics would be observably inadequate.” He certainly did not mean “incomplete” by “observably inadequate.” He discusses incompleteness later in the same paper.
My interest in the 3-sphere model stems from the fact that, unlike quantum mechanics, it provides a non-mystical, locally causal explanation of the observed strong correlations, as a direct consequence of the geometrical properties of the physical space itself. Thus quantum correlations for me are simply an aspect of general relativity.
Incomplete in the sense that there is phenomenon that we can produce experimentally but that it is not accounted for by quantum mechanics. That is, a straightforward demonstration that this is the case, as opposed to a wish for parameters that would “complete” quantum mechanics in the sense of Einstein. You can call it “observably inadequate” if you want, the issue is about whether one can actually control the hidden states, not about whether they predetermine the measurement results.
I see, so what you care about is having a locally causal explanation of the correlations, not a deterministic one. Fair enough, thanks for the answer.
But my model is manifestly deterministic. It is local, realistic, and deterministic in the senses originally envisaged by Einstein and later precisely defined by Bell. See the appendix of my paper: https://arxiv.org/abs/1405.2355 .
Well then you do need to supply the list.
No I don’t. No one needs to supply such an unphysical list. Such a list has nothing to do with what is predicted by quantum mechanics and / or actually observed in the experiments.
To be fair, one can do that without backwards causation, you just need the hidden variable to determine both the outcome of the measurement and the setting. It’s called superdeterminism.
And yes, it is as ridiculous as it sounds. It is saying that the universe somehow conspires so that you always choose the settings that makes the data violate a Bell inequality, even though the data does not if you pick the settings randomly.
There is no super-determinism or backward-causation in my model. Nor are there any loopholes. There is only a solution, S^3, of Einstein’s field equations of general relativity, plus 50/50 chance of the initial orientation of S^3. The strong correlations then follow like a clockwork, without needing any loopholes, or super-determinism, or backward-causation.
It doesn’t matter whether the local hidden-variable (LHV) model is devised for the R^3- or S^3-spacetime. What Bell discovered is that the result
E_qm(a,b)= -a . b
is impossible in any local hidden-variable theory.
According to D. J. Griffiths, “Introduction to Quantum Mechanics”, the argument is stunning simple. To prove Bell’s inequality, the only assumption is that the LHV model is subject to the following equations (1) and (2):
A(a; h_k) = ±1 ; B(b; h_k) = ±1…………………………………………. (1)
A(a; h_k) and B(b; h_k) give the results of respectively the spin measurement on particle 1 in direction a and on particle 2 in direction b!
A(d; h_k) = −B(d; h_k) for all vectors d ……………………………………….. (2)
When the detectors are aligned, the results are perfectly anti-correlated!
Defining the average value of the product of the two components A(a; h_k) and B(b; h_k) by
E_lhv(a, b) = (1/N) Sum_k { A(a; h_k)*B(b; h_k) } ……………………………. (3)
one gets finally by considering another unit vector c and by doing simple math:
| E_lhv(a, b) – E_lhv(a, c) | ≤ 1 + E_lhv(b, c) …………………………………….. (4)
(see page 376 – 379 in D. J. Griffiths, “Introduction to Quantum Mechanics”)
To show that equation (4) is valid for any LHV model, needs nothing more than to consider a pair of entangled particles and that the measurement results are 2-valued and predetermined by h_k, a local hidden variable.
The response list required by me can therefore be reduced to, for example:
{h_k; A(0°; h_k), A(90°; h_k), A(45°; h_k)}
No experiment can ever observe the numbers
You are mistaken about your very first assertion above, and therefore the rest of your comments are irrelevant. Just as our planet is not flat, or R^2, but round, or S^2, our spacetime need not be flat, or R^3, but round, or S^3. And that simple fact makes all the difference in the world. For within S^3 the singlet correlations cannot possibly be anything other than
E(a, b) = -a . b ,
as I have proved in great detail in this published paper: https://arxiv.org/abs/1211.0784 .
In case you have access to D. J. Griffiths textbook “Introduction to Quantum Mechanics”, try to refute the inequality (see page 376 – 379 in “Introduction to Quantum Mechanics”):
|E_lhv(a, b) – E_lhv(a, c)| ≤ 1 + E_lhv(b, c)
A(a; h_k) = ±1 , B(b; h_k) = ±1 and A(d; h_k) = −B(d; h_k) for all measurement directions d can be retrieved from the paper “Local Causality in a Friedmann-Robertson-Walker Spacetime”.
Why should anyone be bothered with such irrelevant inqualities when I have already reproduced the exact predictions of quantum mechanics purely local-realistically?
I think Joy Christian means that his model is deterministic, but not predictive (not beyond QM, at least).
I guess that means that (say) A(90, -1), which is a specific instance of the function A(a, lambda) in his papers, has a determined value, but that value can not be predicted. So it’s unpredictable in a dertermined way.
Or, to put it in other words, it is determined in an unpredictible way.
There is nothing mysterious about that. We can predict with certainty that a deterministic coin will land on its head or its tail with 50/50 chance in a large number of tosses, but we cannot predict at all whether it will land on its head or its tail in a given specific toss.
Sorry, but this is not correct. For a model to be deterministic it means that given the inputs it gives you one, and only one, output.
If it can give two different outputs for the same input it means that the model is actually stochastic. If you can’t give an answer at all, the model is not even well-defined then.
That is not what I said. The functions A(a, h) and B(b, h) are defined by my equations (54) and (55), subject to the condition (66). Given a, b and h, the functions A(a, h) and B(b, h) are deterministic, precisely of the form specified by Bell in equation (1) of his paper of 1964.
Ok, so if you can actually produce the outputs then your model is deterministic. Can you tell me the values of A(0, -1), A(90, -1), A(0, 1), A(90, 1), B(45, -1), B(135, -1), B(45, 1), B(135, 1)?
Are these numbers predicted by quantum mechanics or observed in the actual experiments?
No, but this is irrelevant for my question. You are claiming to have a deterministic model. Can your functions actually produce one output for a given input? If you cannot give me the outputs, then you cannot claim to have a deterministic model.
You are asking me to produce something that is neither predicted by quantum mechanics nor observed in any actual experiment. I am under no obligation to comply with such an unphysical demand. As anyone can verify for themselves, my model is perfectly local, realistic, and deterministic. The problem is not with my model at all, but with the double-standards that it is being subjected to by some followers of Bell (but not by Bell himself).
Yes, but can we predict what happens *if* it lands on its head, and what happens *if* it lands on its tail? In other words, does A(90, -1) and A(90, 1) have values?
Of course they do. They are given by my equations (54) and (55), subject to condition (66).
So, according to (54) we get A(90, -1) = -1 and A(90, 1) = 1.
The quantum mechanical correlation E_qm(a,b)= -a . b cannot be reproduced – despite other claims – by any local hidden-variable model!
**************************************************************************************
From https://arxiv.org/abs/1405.2355 (“Local Causality in a Friedmann-Robertson-Walker Spacetime“) one reads:
A(a; h_k) = +/1 and B(b; h_k) = +/-1 ………………………………………………. (1)
A(d; h_k) = −B(d; h_k) for all directions d ……………………………………………………. (2)
A(a; h_k) and B(b; h_k) give the results of respectively the spin measurement on particle 1 in direction a and on particle 2 in direction b; when the detectors are aligned, the results are perfectly anti-correlated!
With these two assumptions, it can be simply proven**:
|E_lhv(a, b) – E_lhv(a, c)| ≤ 1 + E_lhv(b, c) …………………………………………………… (3)
(here, E_lhv(a, b) = (1/N) Sum_k { A(a; h_k)*B(b; h_k) } etc.)
Maybe, some do not understand the meaning of the inequality (3). If equations (1) and (2) hold true, one must end up with inequality (3).
In case a and b are orthogonal and c makes an angle of 45 degrees with both a and b, one gets, for example, for the quantum mechanical correlations:
E_qm(a, b) = 0, E_qm(a, c) = E_qm(b, c) = −0.707.
These results are clearly inconsistent with the inequality (3) because one would have 0.707 ≤ 0.293.
**************************************************************************************
Conclusion:
Either E_lhv(a, b) ≠ E_qm(a, b) or E_lhv(a, c) ≠ E_qm(a, c) or E_lhv(b, c) ≠ E_qm(b, c) !
Quod erat demonstrandum!
** an easy mathematical exercise; see: D. J. Griffiths, “Introduction to Quantum Mechanics”
After months of discussion here it should be clear that any Bell-type mathematical inequality has nothing to do with physics. Since they cannot be violated by anything at all — let alone by anything in physics, such inequalities are nothing more than mathematical curiosities. Moreover, the popular Bell-CHSH inequality can be derived without assuming locality or realism: http://philsci-archive.pitt.edu/12655/ . Therefore it does not have any significance for the question of local-realism that Bell and his followers mistakenly thought it did.
What is significant for physics is what is predicted by quantum mechanics and what is actually observed in the experiments. What is predicted by quantum mechanics and what is actually observed in the experiments for the singlet state have been comprehensively reproduced in my paper in a purely local-realistic manner: https://arxiv.org/abs/1405.2355 . This has then been verified independently in several numerical simulations, including using the GAViewer program appropriate for my Clifford-algebraic framework:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=296#p7322 .
The evidence presented above is overwhelming. There is no point in denying this evidence.
What is predicted by quantum mechanics and what is actually observed in the experiments for the singlet state cannot be reproduced by your purely local-realistic model https://arxiv.org/abs/1405.2355. All your “magic” math has been debunked since many years by Richard Gill and others.
In case your math is correct, you would get on base of your model:
|E_lhv(a, b) – E_lhv(a, c)| ≤ 1 + E_lhv(b, c)
And thus, either E_lhv(a, b) ≠ E_qm(a, b) or E_lhv(a, c) ≠ E_qm(a, c) or E_lhv(b, c) ≠ E_qm(b, c) !
No way out!
To date no one has found a single genuine error in either my mathematics or my physics. The arguments by Gill and others against my model have been repeatedly refuted by me and several other knowledgeable physicists. See, for example, these papers and blog-posts:
https://arxiv.org/abs/1501.03393
https://arxiv.org/abs/1301.1653
https://arxiv.org/abs/1203.2529
https://arxiv.org/abs/1110.5876
https://arxiv.org/abs/quant-ph/0703244
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=271#p6808
The Bell-type inequality you are defending has nothing whatsoever to do with physics.
Joy, are you saying something analogous to this? Alice and Bob’s locations are like two dots far away from one another on a sheet of paper. But for the purposes of what happens when their simultaneous measurements are made, the geometry is as if the paper is folded so that the two dots are touching? (And then I guess we need to be able to “leave the sheet” very briefly, or somehow they are “glued together”.)
A better analogy is a 2D Mobius world. See the first appendix of this paper for pictures and intuitive understanding of what is going on physically: https://arxiv.org/abs/1201.0775 .
Then I agree all this talk of lists is irrelevant. One could agree with Bell entirely, except at the end when one might say “therefore nonlocality” one would say something like “therefore the relevant info is not travelling along the same path as the particles, but instead ‘jumps a layer’ of space”. Whether the spacetime geometry arguments you make are compelling is beyond my scope. Whether the paper should have been pulled or not needs to be battled out among spacetime geometers.
That can’t be correct. Let’s set a=0 and b=45. Then any experiment with these settings for Alice and Bob would only result in two combinations:
A(0, -1), B(45, -1) or
A(0, 1), B(45, 1)
In any actual experiment, four combinations are observed.
That can’t be correct. Let’s set a=0 and b=45. Then any experiment with these settings for Alice and Bob would only result in two combinations:
A(0, -1), B(45, -1) or
A(0, 1), B(45, 1)
But in any actual experiment, four combinations are observed (with settings a=0 and b=45).
If so, then please provide a reference to the experimental paper where observations of such individual numbers have been reported.
Let me just list the QM predictions for those numbers:
1, 1 (7.5% frequency)
-1,-1 (7.5% frequency)
1, -1 (42.5% frequency)
-1, 1 (42.5% frequency)
These are the QM predictions for the resulst of Alice, Bob (with relative frequencies rounded to one decimal point in parentheses), for settings 0, 45.
As you can see, all four combinations are represented.
If these are the predictions of quantum mechanics, then they are also the predictions of my 3-sphere model, since its predictions are identical to those of quantum mechanics.
Demonstating this is a patently false statement is extremely simple, for lambda is a real number, thus for your reduced (72)
(72) 1/2 {L(a,+1)L(b,+1) + L(a,-1)L(b,-1)} = 1/2 {(I.a)(I.b) + (-I.a)(-I.b)} = (I.a)(I.b)
This is only equal to -a.b if a = +/- b, for when this is not the case, there is bivector content.
This is not disputable. You have been asked to do Jay’s (13) which is = L(a,lambda)L(b,lambda) for lambda = +1 then -1, using ONLY the rules of straight up geometric algebra carrying the bivector basis elements, as a demonstration that the bivector content does not subtract out in (72) above for arbitrary a and b. You have yet to demonstrate it does to back up your claims of no math errors. Why?
In the years since this was first pointed out to you, you have modified your approach, each time committing another math error. Each of the computer simulations you say have verified your claims have intrinsically duplicated your math error, and having done so proved absolutely nothing. The latest, the GAViewer program, which natively does geometric algebra in one orientation only, achieves this by doing a commutation of bivector product order when lambda = -1. Since the bivector product with bivector result anti-commutes this indeed does negate the bivector result portion, but it is inserted by hand to duplicate the results of your math error outlined above. There is no legitimate GA math justification for it, and yet you incorporated it by falsely claiming L(a,-1)L(b,-1) = (I.b)(I.a) to “respect” a lack of commutation. Not respect, disrespecting.
I have already pointed out Lockyer’s mathematical and conceptual mistakes above:
http://retractionwatch.com/2016/09/30/physicist-threatens-legal-action-after-journal-mysteriously-removed-study/#comment-1246033
The cross product need not be used to derive the strong correlation. The difference between the mistaken claims by Lockyer and the actual derivation within my model is the following.
My result E(a, b) = -a.b follows at once from the identities
L(a, lambda = +1) L(b, lambda = +1) = D(a) D(b)
and
L(a, lambda = -1) L(b, lambda = -1) = D(b) D(a) .
These identities are derived in the equations (77) to (79) of my paper, which do not use the cross product. They use only the geometric product (I.a) (I.b) = -a.b – a /\ b , and follow straightforwardly from the definitions (48) to (51), as demonstrated in the Appendix A of this paper https://arxiv.org/abs/1501.03393 . Substituting the above two identities into
1/2 { L(a, lambda = +1) L(b, lambda = +1) + L(a, lambda = -1) L(b, lambda = -1) }
then immediately gives the strong correlation:
E(a, b) = 1/2 { (+I.a)(+I.b) + (+I.b)(+I.a) } = -1/2 { a b + b a } = -a.b .
If one does not use the above pair of identities to derive the strong correlation, then one is working within a fixed basis for S^3, and therefore not working within my model to begin with. Lockyer incorrectly assumes fixed basis for S^3 and claims that the second identity above should be
L(a, lambda = -1) L(b, lambda = -1) = D(a) D(b) ,
thus missing the very point of my model. He insists on this incorrect equation to get his incorrect result instead of E(a, b) = -a.b, but I fail to understand why anyone would want to work with a hidden variable model that purposely does not produce the strong correlation.
The flaw in that argument has already been exposed here. In your expression,
|E_lhv(a, b) – E_lhv(a, c)| – E_lhv(b, c) ≤ 1…………………………………………………… (3)
a,c and b, c can’t happen at the same time as a,b so the inequality one has to use for QM is,
|E_lhv(a, b) – E_lhv(a, c)| – E_lhv(b, c) ≤ 3…………………………………………………… (4)
No violation of that one.
I think you will have to produce a more convincing argument than that. I wrote:
That can’t be correct. Let’s set a=0 and b=45. Then any experiment with these settings for Alice and Bob would only result in two combinations:
A(0, -1), B(45, -1) or
A(0, 1), B(45, 1)
QM predicts that all four combinations will be present in experiments. How can this be reconciled with your determiistic functions A and B?
Yes.
QM predicts 13 different probabilities, and my deterministic model predicts exactly the same 13 probabilities. The evidence is already present here: https://arxiv.org/abs/1405.2355 .
Your (78) is demonstrably false. The commutation you show is not supported by the rules of geometric algebra. This is nothing more than your totally unsubstantiated claim, hardly proof of anything. Show us the derivation, which is identical to my repeated request for you to do Jay’s (13) both ways. So why don’t you set us all straight by showing your proof of (78)? Please stick to GA rules and explicitly carrying the basis elements, no insertions of your misconceptions, no geometric hand waving or fingers/thumb on right vs. left hand justifications.
I’d like to register here that Dr. Christian is consistently refusing to produce the outputs of his functions. He cannot therefore claim to have a deterministic model.
a,c and b, c must not happen at the same time as a,b. Why?
In case you are considering a classical local-realistic hidden-variable model, you are able to predict everything that will happen at a given time. You are claiming to have the corresponding local-realistic simulation algorithm. Feed in {h_k; a, b, c} and you get what you are looking for. Or do I have to suppose that the algorithm doesn’t work properly?
The rest is nothing else than simple math.
With (which holds true for any classical local-realistic hidden-variable model, even in the S^3 spacetime)
A(a; h_k) = +/1 and B(b; h_k) = +/-1 …………………………………………….. (1)
A(d; h_k) = −B(d; h_k) for all directions d ………………………………………………. (2)
you will get:
|E_lhv(a, b) – E_lhv(a, c)| ≤ 1 + E_lhv(b, c) …………………………………………….. (3)
Nothing else, no way out!
Every claim, that you have devised a classical local-realistic hidden-variable model which is able to reproduce quantum correlations, can easily be rebutted by means of inequality (3). Either you accept it or you will waste your time further on with – so to speak – inventing a “perpetual motion machine of the first kind”.
As I have already noted, equations (77) to (79) of my paper follow straightforwardly from the definitions (48) to (51) given in the paper, as demonstrated in the Appendix A of this paper:
https://arxiv.org/abs/1501.03393 .
In particular, equation (78) is proved in the Appendix A of the above paper, which is
L(a, lambda = -1) L(b, lambda = -1) = D(b) D(a) .
Further detail are also given in Section V of this paper: https://arxiv.org/abs/1211.0784 .
As I have already noted, the measurement functions A(a, h) and B(b, h) of my local model are defined by equations (54) and (55) in the paper, subject to the condition (66). Given a, b and h, the functions A(a, h) and B(b, h) produce unique outputs, precisely of the form specified by Bell in equation (1) of his 1964 paper. If my model is not deterministic, then neither is the model defined by Bell in equation (1) of his 1964 paper.
Such a definition is perfect. A deterministic model is exactly one that when given a, b, and h produces unique outputs A(a, h) and B(b, h). The problem is that I’m giving you a, b, and h, and you are refusing to give me the outputs A(a, h) and B(b, h).
The outputs A(a, h) and B(b, h) for your given a, b, and h are given by my equations (54) and (55), subject to the condition (66). I have even provided an explicit plot for all a, b and h:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=296#p7322 .
Excellent! Then you will have no difficulty at all in telling me the values of A(0, -1), A(90, -1), A(0, 1), A(90, 1), B(45, -1), B(135, -1), B(45, 1), and B(135, 1).
I have already provided full details of a complete deterministic model, exactly as defined by Bell in his 1964 paper, which reproduces every statistical prediction of quantum mechanics. What you are asking for is (1) not Bell’s requirement for a deterministic model, (2) not predicted by quantum mechanics, (3) not observed in any experiments, and (4) amounts to asking me to prove quantum mechanics wrong, which was not what either Einstein or Bell wanted us to do.
I think that nobody will ever get any consistent set {h_k; A(a; h_k), A(a’; h_k), B(b; h_k), B(b’; h_k)} from Joy Christian because the manifest local-realistic simulation algorithm is flawed.
For QM (or experiments) they can’t happen at the same time so the terms in the inequality become independent thus an absolute bound of 3. You can’t use the same inequality to evaluated both LHV and QM. There is no violation of the Bell inequality by QM so what good is it?
Explaining fundamental classical physics to the staff of the “Einstein Centre for Local-Realistic Physics” is becoming a very hard job.
In classical physics, the theories are really deterministic. If you tell me the initial positions and momenta of all the particles in the Universe, I can — with enough computational power — tell you how every one of them will evolve, move, and where they will be located at any point in time. All quantities you are interested in can be known in advance. Classical theories make predictions. By means of measurements, you simply check whether your theory is valid or not.
Local-realistic hidden-variable (LHV) theories are classical theories. If you tell me the initial spin orientations of both spin1/2 particles in the singlet state, I can predict the results of respectively the spin measurements on particle 1 in every direction a and on particle 2 in every direction b, at every time. That’s why all LHV theories can finally be implemented as computer program code!
Generally, one has (which holds true for any classical LHV model, even in the S^3 spacetime):
A(a; h_k) = +/1 and B(b; h_k) = +/-1 …………………………………………….. (1)
A(a; h_k) and B(b; h_k) are the pre-programmed instructions for what spin to exhibit at each possible angle of measurement for initial state h_k.
A(d; h_k) = −B(d; h_k) for all directions d ……………..……………………………. (2)
When the detectors are aligned, the exhibited spins are perfectly anti-correlated.
With equations (1) and (2) you get (D. J. Griffiths, “Introduction to Quantum Mechanics”):
|E_lhv_calc(a, b) – E_lhv_calc(a, c)| ≤ 1 + E_lhv_calc(b, c) …………………………….. (3)
Here, for example, E_lhv_calc(a, b) reads:
E_lhv_calc(a, b) = (1/N) Sum_k { A(a; h_k)*B(b; h_k) } ……………………………… (4)
(Quantum mechanics knows nothing like the rhs of equation (4), so one should not compare apples with pears!)
E_lhv_calc(a, b), E_lhv_calc(a, c) and E_lhv_calc(b, c) have nothing to do with measurements; they are *calculated* expectation values on base of the considered LHV model.
Having a local-realistic simulation algorithm, it doesn’t matter whether you calculate today the set {h_k; A(a; h_k), B(b; h_k)}, tomorrow the set {h_k; A(a; h_k), B(c; h_k)} and the day after tomorrow the set {h_k; A(b; h_k), B(c; h_k)} for all considered initial states h_k. When you are impatient, you can calculate the sets all at once and you must get a consistent master set {h_k; A(a; h_k), A(b; h_k), B(b; h_k), B(c; h_k)}.
Any local-realistic simulation algorithm must be able to deliver a consistent set {h_k; A(a; h_k), A(b; h_k), B(b; h_k), B(c; h_k)}. Unless, the algorithm doesn’t work properly and is flawed.
You are deflecting my deflection. It is mathematically impossible for anything to violate a Bell inequality. So that makes a problem of properly comparing QM with LHV. In fact, the Bell inequalities can be derived without any consideration of “local-realistic”. So what possible good are they for physics?
At the heart of your misunderstanding of geometric algebra is not appreciating lambda is not something mystical like your ‘relative concept”, it is simply a real number, and behaves as such in all of your equations.
If you understood this, you would immediately see the LHS of your two Appendix equations (A2) and (A3) in the first reference are equal. However, the two RHS differ by sign on (in this case) your cross products. This is a clear indication there is a math error, and your (A5) (-I.a)(-I.b) = (I.b)(I.a) dependent on these inconsistent equations thus does not follow.
In each of your so called derivations/proofs, you make the very same error. It just appears in a different guise, incorrect epsilon formulation, incorrect cross product formulation, incorrect wedge product formulation. Yet you trot them out as justification you have not made a single math error.
Valid math is completely consistent. Once you realize from exceedingly simple math, your reduced
(72) 1/2 {L(a,+1)L(b,+1)+L(a,-1)L(b,-1)} = 1/2{(I.a)(I.b)+(-I.a)(-I.b)} = 1/2{(I.a)(I.b)+(I.a)(I.b)} = (I.a)(I.b)
it is over, since this generally is not equal to -a.b as your claim. Nothing using valid math will change this conclusion, only by committing math errors can you possibly arrive at E(a,b)=-a.b using the sum of different orientation bivector products div 2 for arbitrary a and b.
What you are describing is not Joy’s model.
I am using his equations and properly applying geometric algebra correcting his math errors. If your point is the meaningless tautology if I do not use his (mathematically incorrect) model, I will not arrive at the same conclusions then I would agree. That is not the point of the discussion.
(72) Is the foundation of his model’s equivalence to QM expectations. It does not equate to -a.b.
The correct equation of my model is
L(a, lambda = -1) L(b, lambda = -1) = D(b) D(a) .
Any variant of the above equation is not my model.
This claim is demonstrably false. My result E(a, b) = -a.b follows at once from the identities
L(a, lambda = +1) L(b, lambda = +1) = D(a) D(b)
and
L(a, lambda = -1) L(b, lambda = -1) = D(b) D(a) ,
which are proved in the Appendix A of this paper https://arxiv.org/abs/1501.03393 .
Substituting the identities into eq. (72) of this paper https://arxiv.org/abs/1405.2355 , i.e.,
1/2 { L(a, lambda = +1) L(b, lambda = +1) + L(a, lambda = -1) L(b, lambda = -1) } ,
immediately leads to the strong correlation:
E(a, b) = 1/2 { D(a) D(b) + D(b) D(a) } = -1/2 { a b + b a } = -a.b .
Lockyer incorrectly assumes fixed basis and claims that the second identity above should be
L(a, lambda = -1) L(b, lambda = -1) = D(a) D(b) .
But this incorrect identity is not of my model at all and misses the very point of my model.
*******************************************
Explaining fundamental classical physics to the staff of the “Einstein Centre for Local-Realistic Physics” is becoming a very hard job.
In classical physics, the theories are really deterministic. If you tell me the initial positions and momenta of all the particles in the Universe, I can — with enough computational power — tell you how every one of them will evolve, move, and where they will be located at any point in time. All quantities you are interested in can be known in advance. Classical theories make predictions. By means of measurements, you simply check whether your theory is valid or not.
Local-realistic hidden-variable (LHV) theories are classical theories. If you tell me the initial spin orientations of both spin1/2 particles in the singlet state, I can predict the results of respectively the spin measurements on particle 1 in every direction a and on particle 2 in every direction b, at every time. That’s why all LHV theories can finally be implemented as computer program code!
Generally, one has (which holds true for any classical LHV model, even in the S^3 spacetime):
A(a; h_k) = +/1 and B(b; h_k) = +/-1 …………………………………………….. (1)
A(a; h_k) and B(b; h_k) are the pre-programmed instructions for what spin to exhibit at each possible angle of measurement for initial state h_k.
A(d; h_k) = −B(d; h_k) for all directions d ……………..……………………………. (2)
When the detectors are aligned, the exhibited spins are perfectly anti-correlated.
With equations (1) and (2) you get (D. J. Griffiths, “Introduction to Quantum Mechanics”):
|E_lhv_calc(a, b) – E_lhv_calc(a, c)| ≤ 1 + E_lhv_calc(b, c) …………………………….. (3)
Here, for example, E_lhv_calc(a, b) reads:
E_lhv_calc(a, b) = (1/N) Sum_k { A(a; h_k)*B(b; h_k) } ……………………………… (4)
(Quantum mechanics knows nothing like the rhs of equation (4), so one should not compare apples with pears!)
E_lhv_calc(a, b), E_lhv_calc(a, c) and E_lhv_calc(b, c) have nothing to do with measurements; they are *calculated* expectation values on base of the considered LHV model.
Having a local-realistic simulation algorithm, it doesn’t matter whether you calculate today the set {h_k; A(a; h_k), B(b; h_k)}, tomorrow the set {h_k; A(a; h_k), B(c; h_k)} and the day after tomorrow the set {h_k; A(b; h_k), B(c; h_k)} for all considered initial states h_k. When you are impatient, you can calculate the sets all at once and you must get a consistent master set {h_k; A(a; h_k), A(b; h_k), B(b; h_k), B(c; h_k)}.
Any local-realistic simulation algorithm must be able to deliver a consistent set {h_k; A(a; h_k), A(b; h_k), B(b; h_k), B(c; h_k)}. Unless, the algorithm doesn’t work properly and is flawed.
Classical statistical mechanics is local realistic but not deterministic.
It is deterministic in the same way as a random number generator in a computer program is deterministic. It is deterministic, but so complex that it is better analysed statistically.
Here is a paper by Abner Shimony (the “S” in the CHSH), who is widely regarded to be a prominent authority on all matters of Bell’s theorem: https://arxiv.org/abs/physics/0105046
In this paper Shimony writes: “The outcome of a measurement on 1 is a function
A(a, λ) = ±1
regardless of what quantity is measured on 2; and likewise the outcome of a measurement
on 2 is a function
B(b, λ) = ±1
regardless of what quantity is measured on 1. Since the outcomes are definite when the
quantity specified by a (respectively b) is given along with λ, the hidden variables theory is
deterministic.”
The above are the only required conditions on the functions A(a, λ) and B(b, λ) for a model to be local, realistic, and deterministic. Therefore my model is local, realistic, and deterministic.
I should also mention that I did my PhD under the guidance of Professor Abner Shimony.
Dr. Christian, I’m not asking you to do anything that you didn’t say yourself you could do. Namely, you said that “Given a, b and h, the functions A(a, h) and B(b, h) produce unique outputs”. I give you a = 0, b = 45, and h = 1. Can you tell me then what are the unique outputs A(a,h) and B(b,h)?
A(a, h) and B(b, h) are given by my equations (54) and (55), subject to the condition (66). They are either +1 or -1. But that is all that can be, need be, and should be said about them.
It cannot be that hard. Why can’t you simply tell me the answer? What is A(0,1)? And what is B(45,1)?
A(0,1) = +/-1 and B(45,1) = +/-1.
These are two outcomes. Your model is therefore not deterministic.
A(0,1) = +/-1 and B(45,1) = +/-1 are perfectly deterministic outcomes, just like any outcome of a coin-toss — heads or tail. According to your definition any coin-toss is not deterministic.
According to your own definition, a function must give a unique output to be deterministic. And yes, according to my definition a coin-toss is in general not deterministic.
That settles then. For me a coin-toss is deterministic, and for you it is not.
“Einstein Centre for Local-Realistic Physics”
Thanks for the laugh.
“Bell Centre for Local-Realistic Physics” would have been more funny, but we decided to be kind to Bell.
As long as your model has any part of lambda represents a choice of orientation for S^3, and the sum of the same geometric product of bivectors done in the two orientations always yields a scalar result, you will need to preface your “model” with “mathematically incorrect model”, for this claim which you require to reach -a.b is not supported by the rules of geometric algebra.
I am a bit perplexed by your continued claim I incorrectly assume a fixed basis, when every post talks about doing the math in both bases, and your misconceptions about what this means mathematically is the source of your math errors. If your complaint springs from your desire to directly add the coefficients generated by the multiplication rules of one orientation to the coefficients generated by the other, without first harmonizing the basis differences, guilty as charged. But this is N Dimensional Algebra 101, taught in the introductory class; you can’t legitimately add the coefficient attached to one basis element to the coefficient attached to another different basis element when you do algebraic element addition. If you would do what I have asked you to do since you seem to be incapable of accepting the lambda is a real number argument, do Jay’s (13) in one basis then the other, maintaining the bivector bases, add the results. You will end up with bivector results that look like
-(a1b2-a2b1) e1^e2 + (a1b2-a2b1) e2^e1
You want to combine -(a1b2-a2b1) and +(a1b2-a2b1) and end up with a zero result. The problem with this is the bases are different. But by the rules of geometric algebra e1^e2 = -e2^e1, and when you apply this map to one basis element or the other, the two coefficients will have the same sign, demonstrating the equality.
Just as (I.a)(I.b) = (-I.a)(-I.b) because the RHS minus signs are real numbers that cancel each other out, you could use the above orientation map in a geometric algebra product of basis elements as in
(e1^e2)(e2^e3) = (-e2^e1)(-e3^e2) = (e2^e1)(e3^e2)
The first GA product is left(right) orientation and the last is right(left) orientation depending on your definition of right and left. Obviously we have equality, the coefficient here would be a3b1 in both cases.
Ok, that is enough for me, I don’t think there is any point in continuing this discussion. Have fun, those whose stay.
For me, it is enough, too. I know that simple questions like „What is A(0,1)?” and “What is B(45,1)?” will never be answered.
To be fair, he did answer:
Unfortunately, this makes his model non-deterministic, and the whole discussion on this thread moot.
My model is perfectly deterministic, and — for the sake of argument — even if it were not deterministic, it still would be a perfectly valid counterexample to Bell’s theorem. Determinism was not a prerequisite for either Einstein’s (or EPR’s) argument or Bell’s argument. But my model happens to be manifestly deterministic, as anyone can verify from equations (54), (55) and (66), if not from the numerical simulations I have linked above.
Wow! I was away for ten days and there are over 100 new posts here!! But not all of them kept it cool, so I hope I can again remind people to tone it down and try to be objective.
Anyway, when last I was here, I posted a tutorial on geometric algebra (GA), which I had only learned myself in the preceding week or two. While I was away, I thought more about the issue of the left- versus right-handed basis vectors and the cross-product relation, and was not fully happy with the way I dealt with those in that tutorial. So after getting home today, I whipped together a new document at https://jayryablon.files.wordpress.com/2017/01/basis-invariance-2.pdf, which I believe cleans up some of the deficiencies in my last go-round.
I have kept this entirely focused on the formalism of geometric algebra, and made no comments at all regarding how Joy’s theory makes use of the GA. That is deliberate. It should be possible to gain agreement here about the underlying formal mathematics and geometry, before we venture out into whether Joy’s properly applies this. And it should be able to do this without all the emotion that people seem to have invested in either supporting or refuting Joy’s theory.
Jay
Looks good, Jay. The L and R subscripts are helpful. I would just like to point out that the reason (4R) and (4L) seem to be equivalent is that they are “viewed” from their own handed perspective. If the left-handed system is viewed from the right-handed perspective its order is reversed to (I.b)(I.a). Which is easily seen from the cross product relation
(b x a) = -(a x b).
And we also get the sign change. A little GA trick we learned from figuring out how to get GAViewer to work correctly since it is in a fixed right-handed basis.
(4R) and (4L) appear to be identical because they are identical, fully mathematically identical. Your “viewed from their own handed perspective” is non-mathematical commentary I would presume comes from the way you visualize vectors within your mind. You have a stated position that since the left thumb points in the opposite direction the right thumb does when simulating the vector product in left and right handed systems, if you add them the result is zero. Actual math does not support this.
It is called “applying physics to the math”. More specifically “applying physical geometry to the math”. This is geometric algebra by the way.
No good physics requires bad math. GA is firstly an algebra, with very definite mathematical rules that drive the moniker “geometric”, not the other way around. These rules show Christian reached his conclusion by not following them. They show you your right and left thumbs do not subtract out as you think they should when you simulate right and left handed bivector products.
Heads = Tails is impeccable math, but nonsensical physics. All arguments by Lockyer amount to setting Heads = Tails, which is impeccable math but nonsensical physics. Physics requires more than rules of algebra. It requires understanding of the physical problem to which math is applied correctly, as done in my model.
I corrected a few typographical errors in what I posted yesterday, and made a few other minor changes. An updated document is now posted here https://jayryablon.files.wordpress.com/2017/01/basis-invariance-3.pdf. Jay
Just to be clear, and for the record, Rick, are you admitting that (8) in https://jayryablon.files.wordpress.com/2017/01/basis-invariance-3.pdf is correct? That is a yes or a no. Once your agreement with (8) is established as a matter of record, we can discuss what comes after (8). Jay
Your use of the cross product is equivalent to mixing metaphors, it is not a native structure of GA, which is the algebra of record. This is why I mentioned to you earlier you jumped out of GA when you brought it in. The cross product structure is somewhat mirrored but in introducing it, you are dropping the notion of basis elements, running the risk of crossing wires when different handedness algebraic sums are made. Things remain crystal clear when you stick to the knitting of GA and carry the bases in the math. When you do this, the right thumb does not subtract out the left thumb in that naive mental picture of what is actually going on instead of using the rules of algebra.
Why are you going beyond demonstrating equality between right and left handed bivector products anyway? Surely you understand no valid math is going to break that equality.
Jay, let me explain why your equation (8) is absolutely correct. It does not matter whether one uses cross products or the intrinsic duality relation between vectors and bivectors. What matters is that your equation (8) describes a pair of equations, one for the right-handed system and the other for the left-handed system. Moreover, this pair of equations involve three bivectors, because we are in 3-dimensional physical space [ we are using Cl(3,0) ]. Let A, B and C be the three bivectors [ no cross product, no duality relation ]. Then the pair of equations contained in your equation (8) are simply manifesting the pair of right-handed and left-handed systems defined by ABC = +1 and ABC = -1, respectively. The rest are details.
Thanks for agreeing the right handed bivector product equals the left handed bivector product since this, unfortunate use of the cross product aside, is what (8) is saying. Can we please finally dispense with the inaccurate claim (-I.a)(-I.b) = (I.b)(I.a)? By your statement here, (-I.a)(-I.b) = (I.a)(I.b).
Jay, let me note also that the duality relation between vectors and bivectors employing cross products is routinely used in geometric algebra, and it is quite easy to prove:
Since e_x, e_y and e_z are orthonormal vectors, we can write I = e_x e_y e_z , and then
-I . (e_y /\ e_z) = – I e_y e_z = – (e_x e_y e_z) (e_y e_z) = + (e_x e_y e_z e_z e_y) = + e_x ,
since e_y e_y = 1 and e_z e_z = 1 . But e_x = e_y x e_z , so using I^2 = -1 we have
e_y /\ e_z = I . e_x = I . (e_y x e_z) .
Thus there is nothing wrong at all with your use of cross products and duality relations.
Rick, as someone who is still learning the finer points of geometric algebra, perhaps while I am doing so, rather than my paraphrase what I think Joy is arguing, let me simply refer you to the way Joy has argued regarding this issue, to arrive at (15) and (16) in his paper at https://arxiv.org/pdf/1203.2529v1.pdf. Presumably, you believe those equations (15) and (16) are in error. Would you please pinpoint the exact source(s) of the error(s) which you think leads to these presumably erroneous (15) and (16)? I think that would be helpful to anybody, including me, trying to sort out competing claims about whether Joy is taking unjustified / erroneous liberties with the math signs, or not. Thanks, Jay
Jay, the two minus signs on the LHS of (16) are real number scaling by -1, as such they cancel out, making the LHS of both (15) and (16) equal. Christian’s error is clearly visible by the fact the RHS of (15) and (16) are not equal. The inequality here is a sign difference on his unnecessary cross product representation, but would be also shown as a sign difference on the bivector portion of the geometric product. The error is common to both, not inserted by the switch to the cross product.
The right and left handed basis geometric product used by Christian are correctly distinguished by the two representations (-I.a)(-I.b) and (I.a)(I.b). Realizing the two minus signs cancel on the former clearly indicates there is a mathematical equivalence between the two. This is completely consistent with what you would find if you did your original (13) explicitly using the right bivector basis, then explicitly using the left hand bivector basis, both step by step. The short-cut here is similar to the minus sign cancellation above, shown by me in another post repeated here
(e1^e2)(e2^e3) = (-e2^e1)(-e3^e2) = (e2^e1)(e3^e2)
This demonstrates the left and right handed partial bivector product attached to coefficient a3b1 are equal.
At every turn, treating lambda as the real numbers +/- 1 leads to total consistency with the rules of GA.
What does this all mean for Christian? Simply his derivation of E(a,b)=-a.b is invalid. His (72) reduces instead to (I.a)(I.b) which has no bivector content only when a = =/- b.
Rick,
You would likely agree that (15) and (16) of Joy’s https://arxiv.org/pdf/1203.2529v1.pdf are obtained respectively from his (1) and (11), which I reproduce below:
beta_j beta_k = – delta_{jk} – eta_{jkl} beta_l (1)
beta_j beta_k = – delta_{jk} + eta_{jkl} beta_l (11)
These are probably a better place to start, because they strip out all superfluous issues and represent the bivectors beta_i = eta_{ijk} e_j ^ e_k at the most elemental level.
Open ended question: what would you say about the relationship between (1) and (11)? Is there an “error” in either of these? Or are these simply two different sub algebras, the former with a right handed tri-bi-vector (Joy’s equation numbers):
beta_j beta_k beta_l = +1 (6)
and the latter with a left-handed:
beta_j beta_k beta_l = -1 (9)?
Jay
Jay, you should not rely on my stated opinions/definitions nor Christian’s unless you want to extend the argument.
The accepted rules for geometric algebra are widely available online and in textbooks.
Do the math on (I.a)(I.b), (-I.a)(-I.b) and (I.b)(I.a) explicitly using only the rules for GA basis products, keeping the math representations the sum of (coefficient)(basis element). After all reductions and combining is complete, determine any equalities between the three, which requires all of the coefficients attached to like basis elements to all be equal for the two compared results. Doing things the long way takes whether or not lambda is a real number or Christian’s “relative concept” out of the picture.
My position is you will find (-I.a)(-I.b) = (I.a)(I.b)
Christian’s position is (-I.a)(-I.b) = (I.b)(I.a)
Only one of us can be correct.
That is not my position. It has nothing to do with my model.
Jay, equations (15) and (16) of my paper you have linked are simply two alternative hidden variable possibilities. As such, they are two perfectly valid equations, describing a right and left oriented 3-sphere, respectively. Lockyer’s mistake is to confuse the two hidden variable possibilities, as in Heads = Tails, without realizing that Heads and Tails do not occur at the same time. They correspond to two different, successive runs of the EPR-Bohm experiment.
Jay, let me bring out Lockyer’s mistake using only basis elements of right- and left-handed bivector subalgebra without getting us distracted by cross products or duality relations.
Let x, y, and z be three orthonormal vectors. From these three vectors let us construct the right-handed basis bivectors defined by A = yz, B = zx, and C = xy. Then we have
ABC = (yz) (zx) (xy) = y (zz) (xx) y = yy = +1 ,
where all products are geometric products. Thus, the right-handed subalgebra is defined by
ABC = +1 ……………………………………………………………. (1)
Similarly, define the left-handed basis bivectors by changing the signs of A, B, and C, so that we have A = -yz = zy, B = -zx = xz, and C = -xy = yx. Then the product ABC will be
ABC = (-yz) (-zx) (-xy) = -y (zz) (xx) y = -yy = -1 .
Thus, for the left-handed subalgebra we have
ABC = -1 ……………………………………………………………. (2)
These two alternative possibilities, ABC = +1 and ABC = -1, are the two hidden variable possibilities in my local model. For each run of the EPR-Bohm experiment the spin basis are supposed to satisfy either ABC = +1 or ABC = -1 (but not both for the same run).
By demanding equality on the RHS’s of equations (15) and (16) of my paper you have linked Lockyer is demanding that the product ABC defined above must be equal to +1 in both (1) and (2), thus missing the very point of my local-realistic model.
Let us begin with conditions (1) and (2) above, which are very easy to understand, without getting distracted by cross products and duality relations. Do we agree that the definitions (1) and (2) are correct? And do we understand that they constitute the two hidden variable alternatives in my local-realistic model? If the answers are “yes” to both of these questions, then we can move to equations (15) and (16) of my paper: https://arxiv.org/abs/1203.2529 .
Please refrain from using straw men arguments. You did it here and several times earlier.
The only important thing is whether or not (-I.a)(-I.b) = (I.b)(I.a) or not.
In fact, (-I.a)(-I.b) = (I.b)(I.a) is the real straw man. It has nothing to do with my model.
This is directly contradicted by your own words:
Equation (A5) of the appendix of your referenced paper is precisely the equality you claim has nothing to do with your model.
(-I.a)(-I.b) = (I.b)(I.a) has nothing to do with my model.
Rick and all, my two page reply is at https://jayryablon.files.wordpress.com/2017/01/sign-ambiguity-3.pdf. Jay
Jay, in one basis you have (I.a)(I.b) for the bivector product in question. The very same product in the other basis is (-I.a)(-I.b). There is no equality between these qualified by “in like bases”, the equality is BETWEEN the two basis representations.
You will see this equality if you do what I have asked you to. Doing the math in GA sums of (coefficient)(basis element) as suggested keeps things very straight forward and unambiguous. The basis elements have well defined properties for both handedness choices. You are adding an unnecessary layer of complication with the cross product, and the lack of attached bivector bases makes this representation not GA.
Please do it as I suggested.
Rick, maybe I am missing your point. Perhaps you would be so kind as to “do it as you suggested,” then we can all see what you are trying to get at.
But again, the root of this is in the relations (1R) and (1L) in https://jayryablon.files.wordpress.com/2017/01/sign-ambiguity-3.pdf. You are talking about tree branches, I am talking about the root and trunk of the tree. If we answer the root question the branch questions will resolve as well.
So for the third time I ask, please discuss (1R) and (1L) all by themselves, and what you see as their interrelationship, as well as your thoughts (or those in the textbooks you have in mind) as to whether those can coexist in the same theory. I am giving you an open-ended question about (1R) and (1L). As much as some may find it annoying when Joy says “read my papers,” some may also find it annoying when someone says “widely available online and in textbooks.” Send a link, or post a scan. Let’s get it all on the table, here.
Jay
Also, you asked me to calculate variants of (I.a)(I.b). Those are necessarily calculated to be a.b + or – I times a cross product. And the Levi-Civita tensor in (1R) and (1L) ensures there will be a cross product as soon as you contract these with a and b. So I thought by focusing on (1L) and (1R) that I would stay away from cross products as you seemed to wish; now you are dragging me back to expressions with cross products in them while telling me to avoid cross products. If you believe the cross products are an unnecessary layer of complication, then you believe that the Levi-Civita tensors in (1R) and (1L) are an unnecessary layer of complication, which means that you have a problem with (1R) and (1L). So I’d like to know what that problem us, and since I have now asked three times, maybe the third time will be the charm. Talk about (1R) and (1L), say whatever you wish, but do not change the focus to somewhere else until you have talked about (1R) and (1L).
Final point for the evening, because I do not want you to think I am avoiding your questions.
Using (1R), we always have (I.a)(I.b)=(-I.b)(-1.b). Using (1L), we also always have (I.a)(I.b)=(-I.b)(-1.b).
As to (I.a)(I.b) versus (I.b)(I.a), that is best answered by the commutator, which is why I added that in (4) of https://jayryablon.files.wordpress.com/2017/01/sign-ambiguity-3.pdf. Commutators are used by physicists all the time to answer a broad range of fundamental questions.
The commutator (4R) tells us that when we use (1R):
[beta_i, beta_j] = -2 eta_{ijk} beta_k (4R)
The commutator (4L) tells us that when we use (1L):
[beta_i, beta_j] = +2 eta_{ijk} beta_k (4L)
So when you contract in a_i and b_j, you get
[beta_i a_i, beta_j b_j] = -2 eta_{ijk} beta_k a_i b_j (4R)’
and
[beta_i a_i, beta_j b_j] = +2 eta_{ijk} beta_k a_i b_j (4L)’
These reduce to:
[ (I.a), (I.b) ] = -2 I. (axb) (4R)”
[ (I.a), (I.b) ] = +2 I. (axb) (4L)”
Hopefully that puts enough information on the table for you to make your point.
This also makes the interesting point that if I label (4R)” with an R subscript and (4L)” with an L subscript as a reminder of whether we are using (1R) or (1L), then:
[ (I.a), (I.b) ]_R + [ (I.a), (I.b) ]_L = 0 (5)
which is alternative way to state Joy’s claimed result that:
E ( (I.a) (I.b) ) = -a.b (6)
Equation (5) is reminiscent of other expressions for commutator = 0 which are litmus tests for conservation principles, Abelian gauge theories, simultaneous observable, and the like.
I would like to bring to a head, this discussion about the sign of the tri-vector, and the left versus right handed basis, and Joy’s supposed “sign error,” by discussing some important concepts in geometric algebra that I have figured out for myself. I do not know whether you will find these discussed in the GA textbooks. If it is, great, I am repeating. If it is not, then they should be, and the texts need to be updated and the GA rules refined to include this. Because I need to get back to my day job today after ten days away, and have several clients who want me to write patents for them, to save time I have made a freehand sketch and posted it at https://jayryablon.files.wordpress.com/2017/01/geometric-invariance1.pdf, which I shall use for this discussion. When I have some time again, I will draw this cleanly with my CAD program.
Basically, GA is a way of using algebraic scalar numbers with embedded vector properties to represent geometry. What is important is the geometry; the algebra is just a language that we use to express the geometry. But it is the geometry that matters, no matter what language we use to discuss that geometry. And specifically, whatever physics we describe using the geometry must remain invariant no matter what language we use to represent that geometry. And this means as well that the algebraic language must be employed in such a way that the physics in the geometry is not affected by how we choose our algebraic language.
So in this regard, it is very important — if this has not already been done elsewhere — to establish the concepts of a) “geometric equivalence” paired with “algebraic inequivalence,” and of b) “algebraic equivalence” paired with “geometric inequivalence.” Concept a) is perfectly permissible in our use of GA language, because it leaves the geometric / physical happenings invariant even under different algebraic descriptions. Concept b) is forbidden in our use of GA language, because even though our algebraic descriptions may appear to be equivalent, these supposedly equivalent descriptions alter the geometric / physical happenings.
Let me now give some examples, using https://jayryablon.files.wordpress.com/2017/01/geometric-invariance1.pdf as a pictorial guide.
Suppose I use the tri-vector
+I=e_x^e_y^e_z (1)
as the orthonormal basis for my use of GA. So I draw the x, y, z axes and the basis vector trio as shown in the left hand figure. Then, in this space, suppose that I have a vector V which represents the physics in this geometry, and that this vector V happens to point up along the z axis.
Now, it may be that somebody else will employ a different tri-vector basis than I did, but no matter what they do, the direction of the vector V must remain invariant, because that represents the physics. So I need some rules for acceptable uses of the GA language that allow me to leave the physics unaffected, which means in this case, that the direction of the vector V must not be changed by a change in the GA language.
So the first rule I impose is that no matter what other language I or other people use, we must all agree that whenever we talk about the direction of a vector as being orthonormal to two other vectors, we will always use the right hand rule as our convention. So I draw a right hand with the thumb pointed toward e_x and the fingers toward e_y so the palm lines up with the vector V along e_z.
Now, suppose another person, such as Rick or LJ or Richard, decides they want to use -I for their tri-vector basis. And let us suppose that the particular -I they use is the following:
-I=(-e_x)^(-e_y)^(-e_z) (2)
In this particular rendering of -I, I get the minus sign because of the three minus signs used on each of the x, y, z basis vectors. That is the algebra. Geometrically, this means that I flip all three of my e_x, e_y and e_z basis vectors as shown in the right hand figure. Then, I point my thumb to e_x and my fingers to e_y and find that V will still point up along the z axis, even though it now points oppositely to e_z. Because (2) is algebraically unequal to (1) because of their opposite signs, there is an “algebraic inequality” between (1) and (2). Yet, so long as I adhere to the RHR for orthonormal relations, using (1) and (2) together with this RHR results in a “geometric equality” because the direction of V does not get changed.
If you think this through, you will see that this remains the case, no matter which direction V may be pointing, because (2) turns everything “inside out” along all three axes. And therefore, choosing (2) rather than (1) has no effect whatsoever on the physics, so that the physics remains invariant under the transformation between I and -1 as defined by (1) and (2). If I use (1) and someone else uses (2), so long as we all agree to the RHR, there is no change in the physics.
There is also an important converse to this: Suppose that someone else decides to use:
+I=(-e_x)^(-e_y)^(e_z) (3)
which remains equal to +1 because I have flipped the signs / directions of e_x and e_y but not of e_z. Now, if I adhere to the RHR, then as to the xy plane the vector V will have its direction unchanged. However, if I introduce a second vector V2 pointing toward the +x direction, and apply the right hand rule, then the vector will have its direction changed by 180 degrees in relation to the yz plane. So here we have an example of “algebraic equality” paired with “geometric inequality.” So although it looks like (1) and (3) are the same because they are algebraically equal, switching from (1) to (3) and applying the RHR will in fact cause the physics to change, and this is impermissible.
The moral of the story is that in GA, algebraic equivalence is not king. Two expressions can be algebraically equivalent such as (1) and (3), yet their blind interchange can cause the physics to change, which is unacceptable. On the other hand, two expressions can be algebraically inequivalent, such as (1) and (2), yet they can be interchanged without altering the physics. So long as the physics is invariant, this is acceptable.
The second moral is that you cannot simply say “because a=b” everything is good, or because “a ne b” something is wrong. (1) ne (2) yet all is good. (1) = (3) yet something is not good. In geometric algebra you must always keep in mind the geometry being represented, and not only look at algebraic equality or inequality. In GA, an equal sign may disguise a geometric inequality and give wrong result that we think is right, and a not equal sign may mask a geometric invariance that gives a correct result that we think is wrong. All equal signs are not created equal!
Finally, this means that there are a set of rules which essentially state that a) any algebraic equality or inequality which does not change the physics in the geometry is permissible, while b) any algebraic equality or inequality which does change the physics in the geometry is impermissible. These are the rules that need to be added to the GA textbooks, if they are not there already.
Finally, coming back to my https://jayryablon.files.wordpress.com/2017/01/sign-ambiguity-3.pdf from yesterday, so long as Joy uses (2) for his -I, the geometry will be invariant with respect to his use of (1R) versus (1L). And, whatever he does use for his -I, must leave the geometry invariant under the interchange between I and -I. So long as these rules are understood and applied, Joy is on firm ground, IMHO, to be using (1R) versus (1L) together in the same theory because the geometry will remain invariant under the sign change represented by his hidden variable.
Now I have to get back to my day job writing patents.
Jay
Clarification:
In the final paragraph where I say “Finally, coming back to my https://jayryablon.files.wordpress.com/2017/01/sign-ambiguity-3.pdf from yesterday, so long as Joy uses (2) for his -I,” the (2) I am referring to is
-I=(-e_x)^(-e_y)^(-e_z) (2)
in today’s post, not (2) in yesterday’s PDF file. I am only referring to (1R) and (1L) in the PDF.
Very good pedagogy, Jay. Let me assure you that all good books on Geometric Algebra incorporate the importance of geometry you have stressed. Historically, the name “Geometric Algebra” was preferred by Clifford himself, precisely for the reasons you spell out. In mathematical circles that name was replaced by the name “Clifford algebra”, with more emphasis on algebra than geometry. But in modern times David Hestenes has revived the old name preferred by Clifford, with decades of pedagogical efforts like yours to stress that it is geometry (and thus physics) that is the King, not algebra, which is just a shorthand tool.
Jay,
Your (1L) and (1R) are perfectly good definitions for quaternion algebra isomorphic to the scalar- bivector subalgebra of GA, but only when taken individually. When you form an expression using both algebras, just what Christian is asking for, both (1R) and (1L) become simultaneously true statements.
One thus is free to combine them. Add the two equations then divide by 2. You get the true story of what is happening, that being the product of two betas is -1 if the indexes are the same, and zero otherwise. This certainly will give the product of any two “beta vectors” as minus their inner product, but this is not geometric algebra. But it is what Christian was looking for and was his initial mistake years ago.
The issue was created by calling the basis elements beta in both cases when in reality they are not the same. For you to achieve some clarity on what the truth is, please replace beta in (1R) with alpha. The map between alpha and beta is alpha_i = -beta_i. You can use this mapping to replace alpha with beta in (1R) duplicating (1L) and replace beta in (1L) with alpha duplicating (1R) showing you now have a consistent basis for both chiral basis choices.
If you care to return to your (13), I would suggest doing the geometric product first with the alpha basis, then with the beta basis, less typing. Then map either alpha to beta or beta to alpha, you will demonstrate the mathematical equivalence between right and left handed geometric products.
Commentary on your last post: Valid geometry, physics and algebra are never in conflict. Math is king, and it is always consistent.
I think I am done here.
There are no mistakes in my work, initial or final. As anyone can verify, the geometry and physics of my model have been very clearly spelled out already in my very first paper on the subject: https://arxiv.org/abs/quant-ph/0703179 . Unfortunately some people did not understand what I was doing, and wrongly deduced that I had made a mistake of some kind. Later some of them recognized their own mistakes and understood what I was doing. As Jay has stressed, as long as one maintains correct bookkeeping, there is no rule against using both left- and right-handed basis within geometric algebra to construct a physical model. In my model the correct bookkeeping is rigorously maintained. The left and right subalgebras are never mixed, as clearly demonstrated in this paper: https://arxiv.org/abs/1203.2529 .
Rick, is this your suggested calculation? https://jayryablon.files.wordpress.com/2017/01/lockyer-calculation.pdf.
If so, then you should explain why you think it is impermissible to do what Joy is doing in the context of his model and I am sure Joy will explain why he is permitted to do so in the context of his model. Jay
You equations (1R) and (1L) are the same as the pair of equations in the equation (8) of your earlier post, and they transform into each other the same way. I see no problem with that.
I assume you means (8) in https://jayryablon.files.wordpress.com/2017/01/basis-invariance-3.pdf?
Yes. But more importantly my identity in question, namely
L(a, lambda = -1) L(b, lambda = -1) = D(b) D(a) ,
can be very easily proven in your current notation by noting that
alpha_j alpha_k = beta_k beta_j ,
and multiplying both sides with a_j b_k. It is as simple as that.
I really want to be done here. Your last PDF correctly captures what I was suggesting.
I have no expectations Christian will admit he made a math error after what, 5 years of his denial?
There is no “in the context of his model” that can permit bad math to become acceptable.
Christian’s reduced (72) requires 1/2 the sum of a bivector geometric product done first in one basis then in the other basis to equal minus the inner product of the two GA vectors used to form the bivectors. GA does not support this conclusion.
I have told you how you could prove this, and now that you have the proper basis set in hand, I will repeat it. Do the bivector product, your original (13) first in the alpha basis and then in the beta basis. Map one or the other, does not matter which, to the other basis then compare apples to apples mathematically. You will find them to be identical. Therefore Christian did indeed commit a math error in the derivation of perhaps the most important equation of his model. QED.
I beg to differ. My equation (72) at once leads to E(a, b) = -a.b from the identities
L(a, lambda = +1) L(b, lambda = +1) = D(a) D(b)
and
L(a, lambda = -1) L(b, lambda = -1) = D(b) D(a) ,
which are proved in the Appendix A of this paper https://arxiv.org/abs/1501.03393 .
Substituting them into eq. (72) of the paper https://arxiv.org/abs/1405.2355 , i.e., into
1/2 { L(a, lambda = +1) L(b, lambda = +1) + L(a, lambda = -1) L(b, lambda = -1) } ,
immediately leads to the strong correlation:
E(a, b) = 1/2 { D(a) D(b) + D(b) D(a) } = -1/2 { a b + b a } = -a.b .
Evidently the above calculation is quite straightforward as well as completely error-free.
Jay, in the notation of your current equations (1R) and (1L) the second identity I mentioned above can be seen as alpha_j alpha_k = beta_k beta_j. You will find no error in my derivation.
I have now formally proved the above assertion to remove any doubt that it is indeed easy to prove: http://libertesphilosophica.info/blog/wp-content/uploads/2017/01/4-vector.pdf .
Sorry but no.
You come off the path every time you move away from explicitly showing the bases. You want to make vector wedge products out of the far right on your (4) and (5). Trouble is, the bases are different. If you make them the same to have a consistent move to the wedge, you will incur a sign change that pretty much spoils it from there. And since you get to your conclusion once again by committing a sign error, you should come to the conclusion you will never get there with legitimate math. Math is consistent.
There are no errors in my model. Physics and geometry are consistent. There is only one 4-vector, ab, and it exists independently of any choice of basis. The errors are yours, not mine.
Rather than blanket statements that the error is mine not yours, please get specific about how my analysis about the wedge products from your (4) and (5) was not correct. I was very specific with you, as I have always been.
What is not specific about “There is only one 4-vector, ab, and it exists independently of any choice of a basis.” ? The RHS’s of (4) and (5) represent one and the same 4-vector in R^4, namely ab, which exists independently of any choice of a basis in which it may be expanded.
That is a very fair request. I have several meetings related to my day job throughout the day, but the next time I have a chance, unless you all are singing kumbaya by then :-), I will try to write out the calculation of Joy’s (6) and (7) of http://libertesphilosophica.info/blog/wp-content/uploads/2017/01/4-vector.pdf, or a correct (6) and (7) if Joy is wrong, in gory detail. Jay
Yes, Jay, I claim that my (6) and (7) are trivially correct, and you are welcome to verify that.
(4) and (5) of http://libertesphilosophica.info/blog/wp-content/uploads/2017/01/4-vector.pdf can be rewritten as
alpha(a) alpha(b) = – a.b – alpha(a x b)
beta(b) beta(a) = – a.b – beta(a x b)
Since beta_i = – alpha_i for each i, beta(a x b) = – alpha(a x b)
So (4) and (5) can only equal one another if a x b = 0
And which algebra or basis the equation beta(b) beta(a) = – a.b + alpha(a x b) belongs to?
It contains representations from both, which is the paradigm your model specifies by summing fair coin choices between right and left orientations used to form the same geometric product.
Rather than obfuscation, tell us how -e_ijk a_i b_j alpha_k = -e_ijk a_i b_j beta_k as they must for both to lead to -a^b.
This will only be the case if a^b = 0, since -0 = +0. This is precisely what Richard is saying
Nowhere will you find anything like ” beta(b) beta(a) = – a.b + alpha(a x b) ” in any of my papers. What it represents is a misunderstanding and misrepresentation of my model.
Neither have I claimed ” -e_ijk a_i b_j alpha_k = -e_ijk a_i b_j beta_k ” anywhere in my papers or on this thread. I have no idea where that comes from.
I am quite sure I fully understand what your model is about, as are most if not all of your critics. You should be thankful people have pointed out your mistakes so you can correct them if possible and constructively move forward.
You may not have the exact equations but you most certainly are implying them. Richard’s is using the same cross product form you have repeatedly used. Mine is following your wedge use in the referenced document. Doesn’t really matter, you have it wrong either way.
Knowing you have a Phd in physics, I am puzzled by your comment you “have no idea where that came from”. Each side of my equation was used to produce -a^b by you, so they would have to be equal. You don’t understand this?
I understand your comments as nothing more than your mistaken readings of my papers and my recent document. Neither you nor anyone else who has attempted to criticize my model has actually understood what it is all about. That is not unusual in science, especially with regard to novel and original ideas. Your misunderstandings and mistakes about my model have been repeatedly pointed out to you by me and others for the past six years, but you have refused to accept your mistakes. As far as I am concerned there is nothing wrong with my model, or with my recent document.
First, thanks for your participation here. Are you planning to follow through with your commitment here? We would all like to know your conclusions here as well as on your (13) worked out properly with GA bivector bases or the alpha/beta bases.
As with Bell, if the assumptions are accepted the math is certain. The assumptions and their cover of physics is a matter of opinion. We have mathematical certainty here. Which side do you come down on?
Rick, thank you for your participation as well, and thanks for asking.
I see recent silence here, but no kumbaya.
So yes, I absolutely will follow through on my commitment, and have been preparing a gory-detailed document to make certain that I get it right. Other-work permitting, I hope this will be ready to post here in the next day or two.
The diplomat in me hates to take sides; my preference is to lay out facts and hope that people of scientific disposition all come to the right conclusions, and via mathematical certainty, to the same conclusions. But I also owe everybody the courtesy of not waffling once I reach my own conclusions, so I will state those as well so people can agree or disagree with me.
Stay tuned. 🙂
Jay
As promised, I have prepared a document which I hope will break the almost ten-year-old impasse over whether or not Joy has made (and persists in making) a sign error in the central equation underlying his model. This document may be obtained at https://jayryablon.files.wordpress.com/2017/01/geometric-algebra-tutorial-4.pdf, and at 19 pages it contains all the gory detail I said it would. I have tried not to take sides, but to present mathematical facts. Let me explain here, the structure and motivations of this document:
In my experience both professional and personal, I have found that the best way to reconcile disputes is to thoroughly understand the views of each side as well as they understand it themselves, and to present those views back to the parties in a way that they can a) acknowledge that their position is understood and not misconstrued or turned into a caricature or a straw man and b) understand the other parties’ view as well as they understand their own. Often, that by itself leads parties to reach the right conclusions and do the right thing, especially for scientific-minded individuals.
Toward that end, this document has three sections, titled as below:
1. Geometric Algebra in a Right-Handed basis (agreed upon by everyone?)
2. Geometric Algebra in a Left-Handed basis (according to Joy Christian’s critics)
3. Geometric Algebra in a Left-Handed basis (according to Joy Christian)
I have tried not to reach conclusions or take sides, but to present the respective viewpoints of each side to the full extent that I understand them. My preliminary questions are these:
A. To everybody: I believe that beyond any minor quibbles with language or presentation, all would agree that my handling of the right-handed basis in section 1 is correct. Please confirm whether I am correct about this, and if not, what is wrong or missing.
B. To Richard Gill, Rick Lockyer, LJ, HR, Stephen Parrott, and anybody else who believes that Joy is making a mathematical sign error, please review section 2 and let me know if this section fully represents your views accurately and without distortion. If not, what is wrong or missing.
C. To Joy Christian, Fred Diether, MF and anybody else who believes that Joy has been correct all along, please review section 3 and let me know if this section fully represents your views accurately and without distortion. If not, what is wrong or missing.
Once we can all agree on a set of underlying mathematics, and can all agree that everybody’s views have been properly presented without distortion, then we can get to the question of whether section 2 or section 3 is the correct way to get to the left-handed basis. One of these must be right, the other must be wrong.
I will conclude here with my concluding statement from this document:
This document is written and offered in the spirit that the preeminent qualification of a true scientist is the ability to acknowledge error in the face of contrary facts, and then pursue a different direction without looking back. Either Christian needs to do so, or his critics need to do so. It is my hope that this document will help each side understand the position of the other side, assess its own viewpoints, and determine whether it is Christian or his critics who have properly attended to the representation of left-handed geometric algebra in relation to its right-handed incarnation.
Jay
19 pages or simply
(-I.a)(-I.b) = (-1)(I.a)(-1)(I.b) = (I.a)(I.b) not = (I.b)(I.a) proving Cristian’s math error in his (72).
“we must instead define, not derive….” Well, there it is. If you don’t care for the results consistent application of the algebra yields, throw in ad hoc definitions. Trouble is, you no longer have GA.
Thank you, Jay, for your enormous efforts. Although I would have written your Sections 2 and 3 considerably differently (giving far less credit to the “critics” of my local-realistic model), I agree with the gist of your Section 3 summarizing my point of view. In particular, I fully agree with the pair of equations in your equation (3.9), which — from day one — has been the central pair of equations on which my local model is based (i.e., from 20 March 2007): https://arxiv.org/abs/quant-ph/0703179 . Later — around mid 2011 — some “critics” began to misinterpret these pair of equations and declared that I had made a sign error in the second equation of the pair. There is of course no error of any kind in my local model.
More relevantly for our discussion here, I agree with your equations (3.14) and (3.15), which amount to the equality
beta_i beta_j = alpha_j alpha_i ,
where alpha_i and beta_i are right- and left-handed basis bivectors within GA, representing counter-clockwise rotations by 180 degrees about the orthogonal directions in the physical space; say x, y and z. As such, it is very easy to verify that the above equation is trivially true.
The above line of equations perfectly captures the mathematical mistake being made by Lockyer and some other “critics” of my model for the past six years. It has nothing to do with my model, as Jay has now made abundantly clear in his exposition. It appears nowhere in any of my papers. And yet it is being repeatedly presented by Lockyer as if it has something to do with my model. I find that very puzzling. The correct equation representing my model is
beta_i beta_j = alpha_j alpha_i ,
which is Jay’s equation (3.14). This equation is very easy to verify, as I have done above:
http://libertesphilosophica.info/blog/wp-content/uploads/2017/01/4-vector.pdf .
OK, now I have to take a side. My view is that Joy Christian is correct, and that you are wrong.
I think I have shown that I completely understand the arguments of both sides. So using equation numbers from https://jayryablon.files.wordpress.com/2017/01/geometric-algebra-tutorial-4.pdf, here is why I believe you are wrong:
We start with a right-handed even (second grade) sub-algebra that I think we all agree with:
alpha_i alpha_j = – delta_ij – eta_ijk alpha_k (1.4)
This sits on the unit vectors e_1, e_2, e_3, and we deduce that it is associated with the 3-D tri-vector:
I = e_1 e_2 e_3 (1.6)
The task is now to find the correct specification of the left-handed sub-algebra.
Using beta_i to represent left-handed bi-vectors, we make the educated guess that the correct way to do this is to set beta_i = -alpha_i which we effectively do with the definition (2.1) in contrast to (1.1). So we insert beta_i = -alpha_i into (1.4) and deduce:
beta_i beta_j = – delta_ij + eta_ijk beta_k (2.4)
We then proclaim that (2.4) is the correct equation for the left-handed even-grade sub-algebra, based on our educated guess that all we have to do to achieve this is set beta_i = -alpha_i in (1.4).
But just to be sure of our guess, we make use of “math is king” to calculate what happens to the third-grade tri-vectors, and we deduce that they have become:
-I = (-e_1) -(e_2) -(e_3) (1.6)
So what the math teaches us that we did not know when we made our educated guess, is that when we set beta_i = -alpha_i to turn the parity of the even-grade bi-vector planes from right to left, we have also turned the 3D space inside out and thus flipped the direction of all the odd/first-grade unit vectors to –e_1, –e_2, –e_3.
Now we have to look at geometry proper. If we reverse all of the planes from right to left, and the “math is king” consequence of this is that we have also inverted all of the axes from right to left, the net effect is a wash: everything has been inverted once at even grade, then re-inverted at odd grade, and we are back in the right handed geometry where we started. We have not deduced the left-handed geometry that we thought we would, but have instead re-deduced the right-handed geometry, now expressed in the form of (2.4). This is a “double mapping” of the right-handed algebra, with no map of the left-handed algebra.
So the “math is king” has proved that our educated guess is wrong, and that setting beta_i = -alpha_i does not get us a left-handed geometry as we thought it would, but rather gives us back the right-handed geometry in a different guise. So, what do we do now?
If (2.4) describes the right-handed geometry as the math has now proven, then how do we write down the left handed geometry? If we stick with using beta_i = -alpha_i to invert the planes, then we must now set beta_i -> -beta_i in (2.4) to compensate for the double inversion which has already occurred. So when we do that, (2.4) becomes:
alpha_i alpha_j = – delta_ij – eta_ijk alpha_k
beta_i = -alpha_i (3.7)
beta_i beta_j = – delta_ij – eta_ijk beta_k
What we have done, as a result, is three parity inversions: first, defining beta_i = -alpha_i, second deducing from the math that this also flips the first grade and put us back in a right-handed geometry, third flipping beta_i -> -beta_i to recover the left-handed algebra.
But once we know that setting beta_i = -alpha_i actually does not yield a left-handed algebra, there is a far-simpler way to get to the left-handed algebra:
Forget all the confusion and triple flipping parity which happens when you write both alpha_i and beta_i and then try to define relations among those, or having to rotate objects while looking in mirrors. Instead, start with (1.4). Flip the second-grade parity by setting alpha_i -> – alpha_i, period. Then you immediately get:
alpha_i alpha_j = – delta_ij – eta_ijk alpha_k (3.9R)
alpha_i alpha_j = – delta_ij + eta_ijk alpha_k (3.9L)
and 19 pages becomes a one page derivation with only a small number of equations: Just flip the handedness of (1.4) by setting alpha_i to -alpha_i and be done with it! All the other confusion melts away. The insistence by others that Joy must first define some beta_i in relation to alpha_i before he can do anything else is what has vastly over-complicated the problem. All Joy has to do to get from the right to the left hand algebra is flip the sign in (1.4). Period! The resulting (3.9) is unequivocally correct!
Put another way, if you have the right to define beta_i == -alpha_i as a relation that (purportedly) brings about a left-handed from a right-handed algebra (although (1.6) proves that it does not), then Joy has the equivalent right to take the minus sign that you put in your beta_i == -alpha_i, by simply setting alpha_i -> – alpha_i in (1.4) to obtain (3.9).
If you want to pick out my words “we must instead define, not derive…” and play “gotcha,” then I have to throw those back at you: If YOU are allowed to “define not derive” beta_i == -alpha_i to go from right to left as you do (and which the math proves actually does not get you from right to left), then Joy is equally allowed to “define not derive” alpha_i -> – alpha_i to go from (1.4) to (3.9) and left to right using the same minus sign that you put into your definition that did not work. If you can put a minus sign in front of the second grade alpha_i in your definition not derivation beta_i == -alpha_i, then so can Joy cut out all the middlemen and simply set alpha_i -> – alpha_i in (1.4) to go from right to left.
In sum, years of debate and confusion can be resolved in one sentence: Start with the right handed sub-algebra (1.4), and turn it into a left-handed sub-algebra by taking each alpha_i and flipping it to -alpha_a.
QED.
Typo: about 1/3 of the way down, my post should read
-I = (-e_1) -(e_2) -(e_3) (2.6) [not (1.6)]
And also, this should be:
-I = (-e_1) (-e_2) (-e_3) (2.6)
minus signs were misplaced.
One other correction:
“Put another way, if you have the right to define beta_i == -alpha_i as a relation that (purportedly) brings about a left-handed from a right-handed algebra (although (2.6) proves that it does not),…” [again, (2.6) not (1.6)]
The absolute easiest way to show that something is wrong with Christian’s model is to consider a concrete example. In Christian’s model we find the two functions A(a, lambda) and B(b, lambda). Let’s the two possible values of his lambda as -1 and 1. Let’s set Alice’s detector setting to a=0 and Bob’s detector setting b=45. Then, According to Christian’s model any experiment with these settings would result in the two observed combinations:
A(0, -1), B(45, -1) or (1)
A(0, 1), B(45, 1) (2)
presumably with a 50/50 distribution of each.
But QM predicts the following results:
1, 1 (7.5% frequency) (3)
-1,-1 (7.5% frequency) (4)
1, -1 (42.5% frequency) (5)
-1, 1 (42.5% frequency) (6)
These are the results of Alice, Bob (with relative frequencies rounded to one decimal point in parentheses), for settings 0, 45 respectively. So we see that all four combinations will turn up in the experiment. Contrast this with Christian’s prediction of two combinations.
Note that this argument does not even mention correlations (which are just a computational consequence of the distribution anyway). It does not rely on the machinery of Bell’s theorem. In fact, it does not presuppose any mathematical ability beyond being able to count to four, and realizing that four does not equal two.
My 3-sphere model predicts exactly what QM predicts, including the following results:
1, 1 (7.5% frequency)
-1,-1 (7.5% frequency)
1, -1 (42.5% frequency)
-1, 1 (42.5% frequency) .
For detailed predictions of my model, please read: https://arxiv.org/abs/1405.2355 ,
as well as check out this Geometric-Algebra-based numerical simulation:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=296#p7322 .
As anyone can verify for themselves, there is absolutely nothing wrong with my model.
Let me clarify: “Joy Christian is correct about how to invert from a right to a left-handed algebra and that you [Rick] are wrong.”
We can get back to Bell and slips and how Joy uses geometric algebra to obtain correlations in another discussion. My only question right now is this:
Is Joy making an error, or is he not, when he claims that to get from the right-handed algebra
alpha_i alpha_j = – delta_ij – eta_ijk alpha_k (1.4)
to a left-handed algebra, the way to do this is to simply flip the bi-vector planes by setting alpha_i -> – alpha_i ?
As to the issue of whether Joy Christian has made a sign error in his paper (which allegation appears to be a primary if not THE primary reason why AOP retracted his paper), the above statement is all that is needed. But because this issue has been argued for years, I do not expect that the accusations of him having made a sign error will easily subside. So I will slightly elaborate what I said above in a way that should be easily understood by anybody, not just those of us who are familiar with geometric algebra.
Suppose I have some physical entity which I can see with my eyes. And in fact, suppose that the entity is my right hand. Suppose that I have established three mutually-orthogonal planes near my right hand which I will call “Plane_x,” “Plane_y,” “Plane_z.” Suppose now that I want to make my right hand look like my left hand. How do I do that?
A very simple way is to perform a transformation of the form (I point out that “transformations” are a valued and legitimate part of physics):
Plane_x -> -Plane_x
Plane_y -> -Plane_y (1)
Plane_z -> -Plane_z
Instantly, as anybody who has ever use a mirror can attest, my right hand will look like my left hand. But of course, we physicists require empirical evidence for what we assert, so I even stood in front of a mirror and took a photo: https://jayryablon.files.wordpress.com/2017/01/handedness.jpg.
As you will see, my right hand holding the camera which hand is reflected in the mirror, looks just like my left hand stationed in front of the mirror, while my nose pressed the button on the camera. 🙂
Now to be very precise, the actual mathematics we use to describe a mirror is that we start with
Plane_x Plane_y Plane_z. (2)
Then, if we take the mirror to be in Plane_y we are really only transforming Plane_y -> -Plane_y while leaving the other planes alone, so that (2) transforms to:
Plane_x (-Plane_y) Plane_z = (-Plane_x) (-Plane_y) (-Plane_z) (3)
So this is “math is king” equivalent to flipping all three planes.
So if I now have some right-handed function R (such as my right hand itself) which I denote by:
R (Plane_x, Plane_y, Plane_z) (4)
the way to determine the left handed function L (such as my left hand itself) is to obtain:
L (Plane_x, Plane_y, Plane_z) = R (-Plane_x, -Plane_y, -Plane_z) (5)
And that is that!
So I can now use the above framework to describe the PHYSICS of the photo of my hands at https://jayryablon.files.wordpress.com/2017/01/handedness.jpg by the equation:
Left hand (not in the mirror) = Right hand (in the mirror) (6)
The above (6) is just another way of saying (5) when (5) is applied to functions which are my hands and when I am using the mirror as the physical apparatus to conduct experiments in which geometric planes are inverted in accordance with (3). So (5) is the mathematical description and generalization of the physics of (6).
In fact, to avoid further confusion by anyone, let me now retract my earlier use of the word “definition” and will instead use “physics.” The above (5) does not “DEFINE” anything. Mea culpa for even using that word. The above (5) represents the actual experimental physics of (6), as empirically evidenced by the photo at https://jayryablon.files.wordpress.com/2017/01/handedness.jpg. Equation (5) is an empirically-grounded physics equation, not a definition.
With that, let’s now go back to (1.4) at https://jayryablon.files.wordpress.com/2017/01/geometric-algebra-tutorial-4.pdf which everybody agrees with:
alpha_i alpha_j = – delta_ij – eta_ijk alpha_k (1.4)
This is simply a right handed function:
R (Plane_x, Plane_y, Plane_z) = alpha_i alpha_j = – delta_ij – eta_ijk alpha_k (7)
where alpha_i = plane_i. That is, the alphas in the above are just fancied up planes. So to DERIVE the left handed function, I use (5) above to derive not define
L (Plane_x, Plane_y, Plane_z) = R (-Plane_x, -Plane_y, -Plane_z) (8)
= (-alpha_i) (-alpha_j) = – delta_ij – eta_ijk (-alpha_k)
= – delta_ij + eta_ijk alpha_k
And now, based on the experimental PHYSICS of the photo I just supplied, and its correct mathematical representation in (6) above and generalization in (5) above, I have PROVED by a “math is king” derivation that (3.9) in https://jayryablon.files.wordpress.com/2017/01/geometric-algebra-tutorial-4.pdf is in fact correct! This is what Joy has maintained for years despite tremendous backlash, retraction of his papers, and damage to his reputation.
If anyone is going to continue to argue that Joy has made a sign error, then they will have to argue that (5) and (6) above are physically- or mathematically-incorrect equations, which IMHO would be a foolish undertaking.
And anybody who claims that it is more complicated than that, is themselves guilty of complicating and obfuscating.
QED again!
When carefully performing the math in “Local Causality in a Friedmann-Robertson-Walker Spacetime” one gets: E(0, 45) = -1
That is simply false. The correct result predicted by my model is E(0, 45) = -0.707…
He defines it to be correct. Who can mathematically argue against a definition? But this amounts to *defining* that the correlation should be cosine. From a physical point of view, not very interesting.
Which is why I have never entered into any detailed discussion about the correctness of Christian’s mathematical machinery. I’ve been on this planet long enough to know that I can derive the cos-function in about a thousand different ways, and none of them has any relevance to physics.
I rather prefer to expose the very simple inconsistencies that anyone of my readers can understand, ref. my post above.
You have not exposed any inconsistency. You have simply ignored the evidence I have presented above. Ignoring the evidence is not the same as exposing inconsistency. If you think you have found any inconsistency in my work, then I invite you to publish your critique on the arXiv where my paper is published, for the benefit of the larger physics community. I extend my invitation also to Lockyer or anyone else who thinks that they have found some inconsistency in my work. Publish your critiques on the arXiv where my paper is published.
Any of Joy’s critics that claim he made a sign error are in fact claiming that a left hand is mathematically equal to a right hand physically from a neutral basis perspective. Easy to see that is not the case. However… a left hand from a left-handed basis perspective is mathematically equal to a right hand in a right-handed basis perspective. Perhaps that is where the critic’s confusion is coming from.
The argument was over at (-I.a)(-I.b)=(I.a)(I.b), which is a GA equality that cannot be disputed. Christian’s reduced (72) sums both sides of this then divides by two. This claim comes from a direct substitution of his definitions for L(a,lambda) and L(b,lambda) into (72) and prevents (72) from yielding -a.b generally. This is precisely what his sum of fair coin orientation choices for the same bivector product model concludes with when no math errors are committed. This is his model done correctly, and all of his claims to the contrary are not supported by his own equations.
The generation of the bivector bases for the two chiral choices is (+I.e) and (-I.e). This is not disputable and is widely stated by Christian. Calling one alpha and the other beta, the very same sign difference applies, so you only have alpha_n = -beta_n. It can’t be anything but this, so your claim Jay, that you can define the chiral difference alternatively with alpha_n = beta_n, or alpha_n = -beta_n is incorrect. Your further claim that it is not possible to use the chosen one to map between alpha and beta is quite incorrect. Math permits any valid equation to be used at will. The fact that your analysis does not work out if you do try mapping between alpha and beta should have sent off alarms that you have made an error. Your earlier (1R) and (1L) with the alpha substitution are completely consistent with the (+I.e) and (-I.e) bivector basis generation and the GA rules for their products. This is an inconvenient truth for all who think Christian has not committed a math error. Your 19 page analysis in support of Christian is not compatible with the rules of GA, and since his initial assumption of S^3 forces the need to comply with GA rules, it is not just incorrect, it is irrelevant.
The “double mapping” claims are factually incorrect. If you have the algebraic expression A alpha_1 – A beta_1, one can’t simply subtract A form A. The bases are different. By the rule of n-dimensional algebra addition, one can only combine coefficients if their attached basis elements are identical. If they can’t be made identical no further reduction is possible. Above, A alpha_1 could easily have been produced using the alpha rules, and A beta_1 produced using the beta rules, full respect given to both chiral choices. There is no ignoring one handedness in favor of the other as you claim. Christian’s model requires adding the result done in one basis to the result done in the other. The indisputable rule of n-dimensional algebra addition requires the bases to be harmonized, and none of Christian’s supporters are doing this. I put out the suggestion to do his bivector product first in the alpha basis and then in the beta basis, which fully accounts for and respects both chiral choices. Doing so will validate the first equation in this post. Is this why no Christian supporters to date have done it, or at least publicly stated their results?
The (alleged bu irrelevant) sign error had nothing to do with the rejection. The AoP never went to such details. They simply wrote:
“After our editorial meeting, we have concluded that your result is in obvious conflict with a proven scientific fact, i.e., violation of local realism that has been demonstrated not only theoretically but experimentally in recent experiments, and thus your result could not be generally accepted by the physics community. On this basis, we have made such a decision to withdraw your paper. “
HR: Let me try again: Do you regard (5) and (6) as arbitrary definitions that somebody is making to lay the foundation to justify a desired result? Or, as I believe, are they simple mathematical statements of observable, physical facts? Jay
OK, let us do the calculation explicitly:
alpha basis:
(e_j e_k) (e_k e_i) = e_j e_i = – deta_ij – epsilon_ijk (e_i e_j)
beta = -alpha basis:
(e_k e_j) (e_i e_k) = e_j e_i = – deta_ij + epsilon_ijk (e_j e_i)
Evidently the calculation refutes what Lockyer has been claiming for the past six years.
Let’s keep this very simple:
Left hand (not in the mirror) = Right hand (in the mirror) (6)
L (Plane_x, Plane_y, Plane_z) = R (-Plane_x, -Plane_y, -Plane_z) (5)
Physics. L and R are functions. True or false?
Not Jay’s (1R) and (1L) with my suggested change to alpha not beta in (1R). The indexes there are for the isomorphic quaternion basis elements, not the indexes internal to bivector wedge products.
Could you explain how your equations here, when j=k indicate (e_k e_i) = (e_i e_k) = 0 for all i not = j=k? That is what your equations show.
My equations do not indicate what you think they do. They are the same pair of equations in Jay’s equation (3.9). Either do the math yourself to recognize your mistakes, or look up the math in the introductory chapter of the standard textbook on GA by Doran and Lasenby.
Oh but they do. With selected i,j,k they will demonstrate all bivector results are zero by the mechanism I pointed out choosing j=k. Thanks for pointing out they are your equations and not mine, and let me point out to you they do not represent GA, very clearly because all bivector products are not identically zero as these equations show.
I have already done the math essentially as what you would get using Jay’s (1R) and (1L) corrected by replacing beta with alpha in (1R). If I did not see it demonstrated there was nothing spooky going on that would not prove left and right bivector products are mathematically equivalent, I would have gone back through it trying to find my error, for the fact that (-I.a)(-I.b)=(I.a)(I.b) says it all. Math is consistent so no valid math will ever show any different. So each time you put up a “proof”, it is just a matter of looking for your next math error. What you have put up here is simply the latest example.
You on the other hand have not explicitly demonstrated your claim right and left handed bivector products sum to twice -a.b explicitly using and showing the basis elements since your initial beta identical to Jay’s original (1R) and (1L) where you not coincidentally made the same mistake calling both beta. Instead, you pull in (a x b) and (a^b), representations that do not have a singular definition across chiral orientation changes. You then proceed to misapply them committing another math error, then call it proof you have not made an error.
You could have put this math error issue to bed years ago by explicitly deriving your claims carrying the bases. So indeed, do the math and show us your work. What is it going to take, 15-20 minutes of your time?
Yes, (-I.a)(-I.b) = (I.a)(I.b), or Heads = Tails, says it all about the mathematical error you have been making for the past six years. Jay and I have now exposed your error sufficiently for all to see.
The mathematical mistakes Lockyer keeps making has already been pointed out to him for many years, by me as well as others, in great detail, at the following thread:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=226#p5891 .
Well, unfortunately they did not say. Which is too bad. But the accusation of a sign error has been made for years, so it sure would be good to clear that up. And so I repeat:
HR: Let me try again: Do you regard (5) and (6) as arbitrary definitions that somebody is making to lay the foundation to justify a desired result? Or, as I believe, are they simple mathematical statements of observable, physical facts? Jay
Let’s get the “sign error” issue off the table so we can look at the other issues.
Yep, it really just boils down to,
(b x a) = -(a x b)
There is no sign error in Joy’s model. Nothing else should need to be said.
Yes. It is as simple as that. My two equations you quote are the same pair of equations in Jay’s equation (3.9). They can be found in Chapter 2 of the GA book by Doran and Lasenby.
Joy, maybe the simple way to bring this part of the discussion to an end, is for you to scan and post the pertinent pages from that textbook. Jay
OK, Jay, here are the relevant pages from the Doran-Lasenby book on Geometric Algebra:
http://libertesphilosophica.info/blog/wp-content/uploads/2017/01/IMG.pdf .
I have highlighted the crucial equations. They have been cited in my papers from day one.
Everybody should take a close look at equation (2.69) and the discussion surrounding. And in particular see how he is flipping the sign in the j quaternion. It should be easy to see that you can do the same thing by flipping the signs in all three quaternions, which means you achieve the same thing by flipping the signs in all three bi-vectors. So my (3.9) is absolutely correct, which Joy has been saying for years.
HR: Let me try again: Do you regard (5) and (6) as arbitrary definitions that somebody is making to lay the foundation to justify a desired result? Or, as I believe, are they simple mathematical statements of observable, physical facts? Jay
Let’s get the “sign error” issue off the table so we can look at the other issues.
The only reason you can tell the difference between your hand and it’s mirror image, is because only part of the space shifts handedness (the virtual space you see in the mirror). If the entire space shifted handedness (including yourself), there would be no way you could tell the difference. So as to your question, I would go with the former.
But it makes no difference, since it’s not like Christian has a great physical theory, but unfortunately there is a single sign error that topples everything. His theory could very well be mathematically impeccable, and still completely irrelevant.
He claims he has a local hidden variable model that reproduces all quantum predictions. Well, because of Bell’s theorem either it’s not a local hidden variable theory (in the sense of Bell), or it doesn’t reproduce all quantum results, or both. So let’s start with the first (and most elementary) hurdle, which indicates that the model does not reproduce all quantum results:
In Christian’s model we find the two functions A(a, lambda) and B(b, lambda). Let’s denote the two possible values of his lambda as -1 and 1. Let’s set Alice’s detector setting to a=0 and Bob’s detector setting b=45. Then, According to Christian’s model any experiment with these settings would result in the two observed combinations:
A(0, -1), B(45, -1) or
A(0, 1), B(45, 1)
presumably with a 50/50 distribution of each.
But QM predicts the following results:
1, 1 (7.5% frequency)
-1,-1 (7.5% frequency)
1, -1 (42.5% frequency)
-1, 1 (42.5% frequency)
These are the results of Alice, Bob (with relative frequencies rounded to one decimal point in parentheses), for settings 0, 45 respectively. So we see that all four combinations will turn up in the experiment. Contrast this with Christian’s prediction of two combinations.
Why do you keep repeating the same arguments over and over again even though they have been repeatedly refuted on this very thread? And why are you ignoring the overwhelming evidence I have repeatedly provided with links and detailed references on this very thread?
Refuted? You say “read my papers”, and “my model predicts that”. That is not a refutation.
Why can’t such an elementary objection I have given have an equally clear and elementary response, without any references?
And nowhere in your papers do you derive the QM distribution of combinations that I listed in my post above. How about posting the derivation here?
I have already posted the derivation several times in my previous replies to you.
Actually, Rick, that is exactly where the trouble begins, and where you go wrong. It is not necessary to call one alpha and one beta. That over-complicating the problem from the start. And I do not really care who thinks we have to do that; they are misguided. I spent four days writing out 19 pages as an accommodation the the way you approached the problem with alpha and beta, came to understood your logic and the underlying math perfectly, and learned in the process that you were making the problem way too complicated and leading yourself into a mistaken conclusion. And I will admit, for a long time I believed you were right, until it dawned on we that you were wrong and why you were wrong. Section 2 was going to be my original conclusion, until I realized that section 3 was actually the correct approach. But I kept section 2 to document he misguided path that I also traveled on. So I understand your view, but I disagree, and I am convinced you are wrong.
The derivation could have been done on one page and half a dozen equations based on:
L (Plane_x, Plane_y, Plane_z) = R (-Plane_x, -Plane_y, -Plane_z) (5)
Left hand (not in the mirror) = Right hand (in the mirror) (6)
These are physics facts. As I said earlier, I retract my use of the word “definition,” because if you accept (5) and (6) as physical facts, which they are, then nothing needs to be “defined” to go from right to left, and you can get from (1.4) of https://jayryablon.files.wordpress.com/2017/01/geometric-algebra-tutorial-4.pdf to (3.9) which Christian has claimed under heavy fire for nine years, without any betas at all, in a single step, and in under one page.
If I am not making myself clear on the physics and the mathematics, I will lay it out precisely in another PDF in the next 48 hours, and the whole problem will be encapsulated in 1-2 pages. I wrote a 19 page document that could have been done in a page or two because you and others insist on taking a circuitous route to derive left from right handedness which has added enormous confusion to a problem that is actually eminently simple. But, I will admit, I never may have seen how simple the solution actually is, had I not gone through the rabbit hole that you insisted I go thorough. So thank you for the adventure in Wonderland. 🙂
I ask you to please think as seriously about (5) and (6) above as I thought about your suggested approaches for days and week through pages of calculations, and consider the possibility that you really may be mistaken. I did not reach my conclusions hastily or with prejudice. I reached them as a serious scientist with pretty fair mathematics skills, seeking the truth.
You want us to believe that (b x a) = (a x b).
But sorry, that is never going to be true.
OK, kiddos, we are going to take a fun break from the intense discussion of the last several days, for an extra credit problem:
From the standpoint of deep fundamental physics, what is the singlemost important paragraph of the entire document I posted at https://jayryablon.files.wordpress.com/2017/01/geometric-algebra-tutorial-4.pdf? Hint: there is nothing else in that document that even comes in a close second. You may respond by simply telling us the equation number which is contained within that paragraph. I will also say this will be the north star Polaris for the discussions throughout the rest of the coursework here, to which one can always return when they lose their way.
This is open to anybody and everybody, active participants and the lurkers alike. The first person to answer correctly will have their grade for this course raised by an entire letter, or they can skip the final exam and head out to spring break two weeks early . 🙂
Jay
My answer: equation (1.11).
Equation 1.27 from Section 1. It would help if there were exactly equivalent equations in sections 2 and 3.
Any other entrants? I will keep the early spring break contest open until Monday morning. 🙂
I guess Christian missed the comment in the excellent Doran-Lasenby book following equation (2.72), where here alpha, whose sign changes the orientation is stated as being a scalar, not Christian’s “relative concept”. Since it is nothing more than the choice between +1 or -1, as I have been saying all along
(-I.a)(-I.b) = (I.a)(I.b)
Christian’s derivation of E(a,b) yields (I.a)(I.b) not -a.b
Jay, you claim I am overcomplicating things? It does not get simpler than this.
This can be said with certainty: ANY math concluding Christian made no math errors will contain at least one math error. Proper math is completely consistent, and will never refute the above which is so obviously true.
Physics can allow it? Sorry, but no.
Heads = Tails? This is an equation not a physical object.
The equation (-I.a)(-I.b) = (I.a)(I.b), which is equivalent to setting Heads = Tails, has nothing to do with my model. It is entirely your invention. Consequently, your attempted criticism of my model is a criticism of your invention. Your mathematical and conceptual mistakes have been pointed out to you for the past six years by many people, independently of Jay’s latest efforts. And it is a well known mathematical fact that an orientation of any manifold, or that of a vector space like Cl(3, 0), which can be represented by a scalar number lambda = +/-1, is a relative concept. Any student knows this fact from the first lesson in general relativity.
You say it is equivalent to setting heads = tails. Should we take from this you do not think the equality is correct? Sure seems that way. We should ALL now understand your lambda is a scalar number, and it is taught in grade school math that (-A)(-B)=(+A)(+B). GA is no different, for this is just real number scaling.
Your claim this has nothing to do with your model is disingenuous politely speaking. We have already been over this, but let’s do it one more time
Your (50) L(a, lambda) = lambda I.a
Your (51) L(b, lambda) = lambda I.b
My reduction of your (72) agreed upon by you
(72) 1/2 {L(a,+1)L(b,+1) + L(a,-1)L(b,-1)}
Direct substitution of your (50) and (51) into (72)
(72) 1/2 {(+I.a)(+I.b) + (-I.a)(-I.b)} = (I.a)(I.b) and not -a.b
For this not to be the case, you would need to prove (I.a)(I.b) is not = (-I.a)(-I.b).
That would be a fool’s errand because the equality is obvious.
Not sure whether everybody agrees on this now, but in GA -I is never equal to -e1-e2-e3. The sign always only reflects the handedness relative to I (the value of the scalar represents the volume relative to the unit volume of I).
http://challengingbell.blogspot.nl/2015/04/joy-christian-for-dummies-like-me.html
Of course initially one can choose either a left handed system or a right handed system.
Yes, that is correct Albert Jan. That is what I mean when I assert that “the orientation of a manifold [in our case the vector space Cl(3, 0), or the 3-sphere S^3] is a relative concept.”
I have seen Jay writing -I=-e1-e2-e3 at several places. Not sure whether he is expressing his own logic of that of others there. In any case, this is not correct.
I presume Jay meant I = (e1 e2 e3) –> (-e1) (-e2) (-e3) = – (e1 e2 e3) = -I.
Exactly so.
As a reminder to everyone that physics is an empirical subject involving configurations of matter described in reference to the human-invented concept we call space, evolving in reference to the human-invented concept we call time, today I am assigning a lab project to everyone. Keep in mind that lab-work is 1/3 of your grade, so it is important to make sure you complete this part of your coursework. 🙂
For this lab project you will need two pieces of equipment which will represent “configurations of matter.” First, you will need a glove which fits on what you call your right hand. Second, you will need a glove which fits on what you call your left hand. Each of these gloves represents a configuration of matter. Your left and right hands will each be your measurement instruments. Establish a baseline measurement for the right hand glove by putting it on your right hand, and for the left hand glove by putting it on your left hand, and confirm that each glove fits. (I am not intending to sound like Johnnie Cochran for the OJ Simpson trial.) 🙂
Because we will only deal today with configurations (for which we invent the space concept) and not evolution (for which we invent the time concept), we shall invent three “space” coordinates which we call x, y, z. When we put them in any of the orderings xyz or yzx or zxy or xy or yz or zx we shall call this a right-handed ordering. When we put them in any of the orderings zyx or yxz or xzy or yx or zy or xz we shall call this a left-handed ordering. We shall also define plane_x==yz, plane_y==zx and plane_z==xy, so that these are all planes to which the third axis is orthogonal in a right-handed manner.
The right-handed glove is a configuration of matter that may be described as a function of x, y, z. We shall call this function R(x,y,z). The left-handed glove is also a configuration of matter that may be described as a function of x, y, z. We shall call this function L(x,y,z). But to be a little fancier about it, let us instead write these as functions of the three right-handed planes, so that these are:
R (xy, yz, zx) and L (xy, yz, zx) (1)
As your first measurement, try to put your right hand into your left glove. Or, try to put your left hand into your right glove. If you are doing this experiment correctly, you will find that this does not work. As as a result of this measurement your have determined that:
R (xy, yz, zx) not = L (xy, yz, zx) (2)
These are not equal.
The next thing we wish to do is to find out what IS equal. So now, take your right hand glove and turn it entirely “inside out.” That is, transform your right hand glove in a way that we may be described using the space concept as (transformations are very important operations in physics):
R (xy, yz, zx) -> R (-xy, -yz, -zx) (3)
It should be easy to understand that the minus signs on the RHS of (3) represent the fact that this glove is now “inside out.”
Your next measurement is to take this inside-out right-handed glove and try to place it onto your left hand. If you have done this experiment correctly, you will see that this R (-xy, -yz, -zx) fits onto your left hand in exactly the same way as L (xy, yz, zx).
So from this experiment, if your have done everything correctly, you have now physically validated the empirical equality:
L (xy, yz, zx) = R (-xy, -yz, -zx) (4)
As a final measurement, try to place this inside our right-handed glove onto your right hand. You will find that it does not fit. So you have now also physically validated the empirical inequality:
R (xy, yz, zx) not = R (-xy, -yz, -zx) (4)
And that concludes today’s lab project. If anybody obtains results other than (4) and (5), please let us know how you managed to do so, and we will try to help you correct whatever mistake you made.
Jay
Turning a glove inside out: https://www.youtube.com/watch?v=kD0vzKlcG2g
I think one should never express I as (e1 e2 e3) in GA because the only correct definition for I is e1^e2^e3. Or am I wrong on this?
Albert, very good question, let me try to help:
The expression:
I = e1 e2 e3 = e1^e2^e3 (1)
IS OK because you have the explicit indexes i=1, j=2, k=3 showing in the equation. The concern you are expressing comes about when we have the i, j, k indexes in the equation, and each of these can be any of 1, 2, or 3. If any two indexes are the same, then we have:
e1 e1 = e2 e2 = e3 e3 = e1.e1 = e2.e2 = e3.e3 =1 (2)
But if they are different we have:
e1.e2 = e2.e3 = e3.e1 = e2.e1 = e3.e2 = e1.e3 = 0 (3)
Likewise:
e1^e1 = e2^e2 = e3^e3 = 0 (4)
but:
e1^e2 = e1 e2 = – e2^e1 = -e2 e1
e2^e3 = e2 e3 = – e3^e2 = -e3 e2 (5)
e3^e1 = e3 e1 = – e1^e3 = -e1 e3
So for tri-vectors, for example:
e1 e1 e2 = e1.e1^e2 = 1 ^ e2 = 1 e2 = e2 (6)
but
e1 e2 e3 = e1^e2^e3 (7)
So when you finally put i, j, k into an equation then you MUST write:
I = ei ^ ej ^ ek (8)
because the ^ is subtle way of telling you that i, j, k have to all be different, and that if i=j=1 and k=2, for example, then
ei ^ ej ^ ek = 0 (9)
however
ei . ej ^ ek = 1 ^ ek = ek (10)
So the dot and the wedge are a good way of reminding us, especially when i, j, k are in the equations, what combinations of indexes we have in mind for each of i, j, k.
Does that help?
Jay
Let me be even more concise:
ei ^ ej is an expression that will always be zero when i=j
ei . ej is an expression that will always be zero when i not = j.
We cannot be sure what ei ej means, unless we are told the indexes, or unless we show a dot or a wedge.
You might misunderstand my remark. I am saying that the I in GA never ever represents the cube spanned by the 3 base vectors. In GA I is a complete different object. It can as well be presented by a sphere, as pictured in http://challengingbell.blogspot.nl/2015/04/joy-christian-for-dummies-like-me.html.
So yes, they are both volumetric objects, and yes, they can both be seen as being composed by 3 orthogonal unit vectors, but the GA ‘I’ can only be composed by the wedge product.
Seeing this differently leads imo. to extra confusion, because one name (‘I’) is being used for different objects.
I think that Rick would say that if you change handedness in GA, you have to change it for the entire space, not only the glove. So even the hands would have to be turned inside out, and then the right-handed glove would still fit on the right hand.
But personally, I have no problems with Joy changing handedness for only parts of the space. Then your analysis is correct. I really hope we can put this handedness thing behind us, because it is a red herring. The problems with Joy’s model lie elsewhere.
I would say the problems with various attempts to misrepresent my model lie elsewhere. My model itself has no physical or mathematical problems. It does have political and sociological problems, however.
Not a red herring. Although your interest lies elsewhere, in the products of +/-1 combinations, his foundation is built on the claim the fundamental flaw in Bell is not using S^3 with summed fair coin orientation choices. The math errors demonstrate he can’t reproduce the expected cosine function QM does. That makes any further arguments on LHV models generating your interest meaningless.
Your earlier comments you could dream up many ways to get to the desired cosine function was a straw man argument, not on point or relevant.
Your earlier comment the math errors were not why the paper was rejected reflect a stated reason so the math errors are not important is also a bit off. No paper with math errors should ever be published. The fact that this one was demonstrated there was little effort put in by the publisher to check its math. My conclusion on this, is the fact that a paper is published means nothing. It could easily be garbage.
Can you provide a reference to your published paper by peer-review in either Clifford algebra or general relativity so we can see what you mean by a garbage versus non-garbage paper?
I think that Rick would say that if you change handedness in GA, you have to change it for the entire space, not only the glove. So even the hands would have to be turned inside out, and then the right-handed glove would still fit on the right hand.
Do I even have to say that turning one’s hand inside out is fiction not physics?
But personally, I have no problems with Joy changing handedness for only parts of the space. Then your analysis is correct. I really hope we can put this handedness thing behind us, because it is a red herring. The problems with Joy’s model lie elsewhere.
Good, HR, we agree. By early next week I plan to be done with this red herring, whether or not the people who have been making a sign mistake themselves because they think that physics allows hands to turn inside out, and for more than half a decade have made false claims that their sign mistake was Joy’s, admit their mistake….
The problems with Joy’s model lie elsewhere.
Here I would say that once we put the sign mistake red herring behind us, that is when the physics and mathematics challenges that Joy has taken on with his model really begin. It is legitimate to discuss what it means to “add” left-handed functions to right-handed functions with a statistically-equal number of each to obtain the dot product -a.b which of course is the main step on the road to quantum correlations which break the CHSH bounds. It is legitimate to discuss whether this -a.b so-obtained is connected with a theoretical framework which is “local” and “realistic” with suitable “hidden variables” (LRHV) according to commonly accepted understandings of those. And most importantly, it is legitimate to discuss whether that theoretical framework is “local” and “realistic” with “hidden variable” for individual trial counts which are binary-valued, BLRHV.
Those are the matters I plan to discuss starting this coming week, once we are done (as I intend to be) with this red herring.
Always a pleasure doing business with you, HR. 🙂
Jay
Everyone should realize I have never resorted to hand turning, the use of cross products or wedge products to justify my position, just exceptionally simple algebra.
It is not “what it means to add left-handed functions to right-handed functions… to obtain the dot product -a.b” but instead whether or not this indeed happens that is important. (I.a)(I.b) = (-I.a)(-I.b) says it doesn’t. This makes S^3 summed fair-coin choices on orientation for bivector products a non-starter for Bell, making it pointless to discuss further within this topic. Red herring? Obviously not.
Red herring? Manifestly yes! Rick you are just plain wrong, but you are working from your own “top drawer” by clinging to the illusion that you know better than everybody else and growing more and more insistent. Sorry, you are wrong! And if I have to spoon feed each step of the way how you are wrong and show six different ways why you are wrong, then I will do that because other fair-minded people may listen even if you do not. Jay
Manifestly not.
I have never said I know more than everyone else, just obviously those in the very small group of Christian supporters when it comes to algebra.
You have finally come to the realization that the left and right handed bivector bases are negation of each other, e1^e2 = -e2^e1. And yet, you fail to grasp that when two are multiplied, the two negations cancel leaving an equality with corresponding non-negated pair.
Equivalently, the full product of pure bivectors for right and left orientations are on either side of this equality
(+I.a)(+I.b) = (-I.a)(-I.b)
To prove me wrong, you will need to disprove this equality. To invalidate this equation you will need to disprove the algebraic rule of multiplication by real numbers. Good luck with that.
This may not correspond to what any of you see in your mind’s eye, visualizing vectors as actually having heads and tails giving you a visual representation of direction, where cancellation happens when you “see” them going in opposite directions. What you visualize is not reality, the above algebra is.
But as I have already pointed out (+I.a)(+I.b) = (-I.a)(-I.b) has nothing to do with my model.
That perfectly illustrates why Lockyer is wrong when applying the physics to Joy’s EPRB model. There is no change in handedness in the overall entire space; there is only a possible change in handedness for each particle pair system. The GAViewer program proves that he is wrong since it is in a fixed right-handed basis.
Your GAViewer program did nothing more than implement the math error by design. It did nothing of an independent nature to prove anything. Why should anyone be surprised it came up with the same flawed results?
The above claim by Lockyer brings out his mathematical and conceptual mistakes quite clearly. My result E(a, b) = -a.b follows at once from the identities
L(a, lambda = +1) L(b, lambda = +1) = D(a) D(b)
and
L(a, lambda = -1) L(b, lambda = -1) = D(b) D(a) ,
which are proven in the Appendix A of this paper https://arxiv.org/abs/1501.03393 . They are also proven again differently in my recent document posted above on this thread:
http://libertesphilosophica.info/blog/wp-content/uploads/2017/01/4-vector.pdf .
Substituting the identities into (72) of this paper https://arxiv.org/abs/1405.2355 , i.e.,
1/2 { L(a, lambda = +1) L(b, lambda = +1) + L(a, lambda = -1) L(b, lambda = -1) } ,
immediately leads to the strong correlation derived in (79):
E(a, b) = 1/2 { D(a) D(b) + D(b) D(a) } = -1/2 { a b + b a } = -a.b .
Lockyer mistake is to ignore the fact that beta_i beta_j = alpha_j alpha_i , as Jay has proved.
So which one of your equations (50), (51), (72) are you retracting?
If none, what you show here is in error since you arrived at a different result.
Please stop repeating flawed math, it proves nothing. Your continued obfuscation is completely transparent.
Here is your chance, then, to publish your critique of my model on the arXiv where my paper is published, providing a list of citations to your peer-reviewed and published papers on Clifford algebra and general relativity, to give the necessary credibility to your critique.
I will elaborate with some more detailed calculation when I have time, but let me refer to the contrasting equations (2.20) and (3.15) of https://jayryablon.files.wordpress.com/2017/01/geometric-algebra-tutorial-4.pdf to clarify a few points. Rick has insisted that the former is correct; Joy that the latter is correct, and I have concluded independently that the latter is correct.
It is extremely important to understand that in (4) above we are speaking about left and right handed functions of the coordinates, not globally flipping coordinates for everybody and everything. When we say R (xy, yz, zx) -> R (-xy, -yz, -zx) in (3) we are saying that the right-handed glove which is a function R of the planar orientations xy, yz, zx is being turned inside out with respect to the planar orientations. We are NOT saying that the planes are globally flipping. There is a huge difference between flipping the coordinates globally and flipping a particular configuration of matter which is a mathematical function of those coordinate. Just because I turn a glove inside out, does not mean that I am turning or must turn everything in the room or everything in the world inside out. Equations have a meaning and you must understand their meaning in order to to and discuss them properly. And when I reverse the disposition of some function on the coordinates, I am not reversing disposition of the whole world.
So the better way to write (2.20) and (3.15) in https://jayryablon.files.wordpress.com/2017/01/geometric-algebra-tutorial-4.pdf is:
a(alpha) b(alpha) = a(beta) b(beta) (2.20)
a(alpha) b(alpha) = b(beta) a(beta) (3.15)
Here, the vectors a and b represent “configurations of matter” represented as functions of the human invention that we call space coordinates. In the above, we are now defining v(alpha) == alpha_i v_i and v(beta) == beta_i v_i to express “configuration of matter as a function of space” rather than “application of space coordinates to configuration of matter.” Both are valid, but conceptually, the later makes it easier to not to lose sight of the fact that we are always talking about configurations of matter as functions of the human invention we call space.
Jay
I get your point, but there is a problem with your notation in
a(alpha) b(alpha) = a(beta) b(beta) (2.20)
and
a(alpha) b(alpha) = b(beta) a(beta) (3.15) .
The problem is that v(alpha) = alpha_i v_i and v(beta) = beta_i v_i are bivectors of grade-2, not vectors of grade-1. And therefore a(alpha) etc. are also bivectors of grade-2.
It is false statements like this that drag out the discussion. You have not demonstrated it has nothing to do with your model, because it most certainly does. You just repeat the statement seemingly to deflect its ramifications. Anyone can glance at your (50), (51) and (72) to see for themselves this is a false statement, and since this has been brought to your attention, that you should know the statement is false, yet you continue to repeat it.
(+I.a)(+I.b) = (-I.a)(-I.b) forces your derivation of E(a,b) to = (I.a)(I.b) and not -a.b.
This seems be the main point which might be of interest for the readers of the “Retraction Watch” blog (unlike the very particular discussion about the math in “Local Causality in a Friedmann-Robertson-Walker Spacetime”):
It is a fact, that your local-hidden variable model finds no *scientific* recognition within the scientific community. So what are the *political* and *sociological* problems which you are thinking of?
I have discussed the political and sociological problems I have faced for the past 10 years at several places, in considerable detail. See, for example, the following links:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=188#p5129
http://libertesphilosophica.info/blog/
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=222&start=20#p6386
Sorry, I messed up one of the links above. I meant to point to the following link:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=115#p3763 .
Forgot one thing. Since in the past participants here have falsely stated my absence from the discussion indicated my concession that they were correct and I saw my errors but just did not want to admit it, let me make this perfectly clear.
I am done with this discussion. I have demonstrated in no uncertain terms, with exceedingly simple math Christian does not derive the strong correlation cosine function from his premise of using S^3 and fair coin orientation choices for the product of two bivectors. Any presentation that contradicts this is assured to contain math errors since proper math cannot come to two disparate conclusions.
My proof only requires the rule of scalar real number multiplication. It is thus obvious. I feel no obligation to continue to point out the math errors others make in their futile attempts to save Christian’s work.
Before you leave the discussion can you please answer the following question I asked earlier so that we can put your assessment of my work in proper context. Just one reference will do:
OK, if you have never published anything in Clifford algebra or general relativity, then please give a reference to any peer-reviewed paper you have published in any subject in physics or in mathematics. Reference to just one of your peer-reviewed papers in any subject will do.
The fact is that (+I.a)(+I.b) = (-I.a)(-I.b) has nothing to do with my model. Anyone who thinks that it has, has either not understood my model at all, or fundamentally misunderstood it.
Before you go, please answer yes or no, clearly: Is
L (xy, yz, zx) = R (-xy, -yz, -zx) (4)
relating the configuration of matter function R which is a right-handed glove turned inside out, to an intact left-handed glove L, correct? Is an inside-out right handed glove equivalent to an intact left-handed glove? Yes or no? Your answering this question rather than not, could help avoid speculation about your views and attribution to you of views you do not hold.
Let me ask you a couple questions that might point out you are on the wrong path.
Are your gloves special S^3 gloves?
If you add a right glove to a left glove do you end up with no gloves?
As I said earlier you are on a fool’s errand here. This is a pure algebraic play, not visualizing your two hands, gloves, R^3 planes which is all one can visualize, naive mental pictures of vectors as arrows.
Christian’s (50) and (51) are perfectly acceptable definitions for bivectors cast in the form where the orientation can be cleanly modified and the ultimate goal of -a.b can pop out later. There is clearly no need for him to retract them.
The fundamental premise of Christian’s model is to form the geometric product of two bivectors and allow this calculation to be done alternatively in a right hand basis and a left handed basis, summing results from each with a fair coin choice of orientation, dividing by the number of tries, and this produces the strong correlation cosine function. This is precisely what his (72) attempts to do, so we should not expect him to retract this equation. He needs it to get -a.b.
As said before, direct substitution of (50) and (51) into (72) reduces to
(72) 1/2 {(+I.a)(+I.b) + (-I.a)(-I.b)}
Next question Jay. Is this a representation of his model, or does it have nothing to do with his model? Please answer without equivocation, one or the other.
The operation of addition is defined by algebra, not gloves, hands or mental pictures. To prove me wrong you will need to demonstrate algebraically (+I.a)(+I.b) is not = (-I.a)(-I.b). You will not be able to do this without making a math error.
Physics is empirical, but valid physics is never in conflict with valid mathematics.
Jay, Lockyer’s opinion on (4) is irrelevant. L (xy, yz, zx) = R (-xy, -yz, -zx) is an established physical fact, consistently implemented within my model mathematically via the identity
L(a, lambda = -1) L(b, lambda = -1) = D(b) D(a) .
Replacing my model with something else or denying the above physical fact is not an option.
Let me prove the above identity explicitly here to remove any doubt. We begin with equations (49) and (56) of this paper: https://arxiv.org/abs/1405.2355 , together with equations (50) and (51). Equation (49) defines a pair of equations, which can be written as Heads or Tails:
L(a, +) L(b, +) = -a.b – L(a x b, +)
or
L(a, -) L(b, -) = -a.b – L(a x b, -).
Using equation (56) this pair of equations can be rewritten in terms of D(a x b) as
L(a, +) L(b, +) = -a.b – D(a x b) = D(a) D(b)
or
L(a, -) L(b, -) = -a.b + D(a x b) = -a.b – D(b x a) = D(b) D(a).
Thus we have the following identities used in deriving the result (79) of the above paper:
L(a, +) L(b, +) = D(a) D(b)
and
L(a, -) L(b, -) = D(b) D(a).
I, for one, do not find anything difficult, or mysterious, or wrong in these identities.
Rick,ll I want to know is whether you can agree that L (xy, yz, zx) = R (-xy, -yz, -zx) (4) is a correct expression, or not, for describing the relationship between two like-physical objects, one with a given parity, and the other with a relatively-inverted parity. Once I have an answer to whether L (xy, yz, zx) = R (-xy, -yz, -zx) is right or wrong, then YES, without equivocation, I will try to connect it to Christian’s model and use it to prove that he has not made any sign error. If you have a problem with my proof once I post it then you can argue it at that time. If you think that my proof is destined to have an error you can point out the error out at the time I make it. But I am not going to jump ten steps ahead right now.
The first equation in my proof will be L (xy, yz, zx) = R (-xy, -yz, -zx). So I want to know if you think that this is a valid equation for what I described it to represent, namely, two identical physical objects which differ only in their parity. If you decide to, you can always argue that I am misusing L (xy, yz, zx) = R (-xy, -yz, -zx) or taking it out of context once I post my proof. But you should be able to say even now, in advance of my proof, whether L (xy, yz, zx) = R (-xy, -yz, -zx) on its own to describe two identical objects which differ only in their parity, is correct or not, without trying in advance to put words into my mouth or equations into my proof.
Joy Christian;
Jay, Lockyer’s opinion on (4) is irrelevant. L (xy, yz, zx) = R (-xy, -yz, -zx) is an established physical fact…
…denying the above physical fact is not an option.
Joy, I agree. But Rick should still have an opportunity to state whether he thinks something which is so obviously-true, namely that L (xy, yz, zx) = R (-xy, -yz, -zx) correctly represents two identical physical objects which differ only in their parity, is true.
Jay
Here is my promised mathematical and physical proof that Joy Christian did not make any sign error in his treatment of right- versus left-handed bi-vector geometric algebra. It is two pages. The proof is in one page. The refutation of the contrary position is also in one page.
https://jayryablon.files.wordpress.com/2017/01/proof-of-no-sign-error-by-christian.pdf
In fact, it is Christian’s critics who have clearly made an error, and who have insistently persisted in that error for more than half a decade, because their claim is based on the physical fiction that “left hand = right hand,” with precise specifics in the above-linked file.
TO BE CLEAR: I am not claiming or asserting that the rest of Joy’s theory claiming to contradict Bell’s Theorem and explain strong correlations using a binary local realistic hidden variables model is correct. Assessing that proposition is the next stage of this discussion, and that is where the real, legitimate questions rest.
But I do claim and assert as forcefully as I can, that the assertion that Christian made a sign error in treating right and left handedness is a complete red herring. Anybody who continues to insist on this “sign error” is just plain wrong. I now intend to move on and discuss the real issues as to the merits of Christian’s broader claims about strong correlations and Bell’s theorem.
Further, while we here have no way of knowing for sure what the thinking was of the AOP editorial board, to the extent that their retraction was based on their being told and concluding that Christian made a sign error, the retraction was manifestly in error.
From here, at RW, we should start to consider the rest of the legitimate bases upon which the retraction might have been made, and cease chasing our tails over the illegitimate basis of a sign error. The sign error was made only by Christian’s critics who insist and in some cases continue to insist that “left hand = right hand,” with precise specifics as laid out in the above.
Jay
That fallacious and persistent claim by the critics of my model has hurt physics for at least six years, ignoring its detrimental personal, financial and academic effects on me. I am of course expendable, but physics and science are not. And we have no evidence that Annals of Physics peer-reviewed the reports of an error in my paper, or checked the background of those who reported an error. They simply removed my paper without telling me for months.
Your have 2 disjoint expressions in alpha yet use both in your “proof”. The equation you should have inserted -beta for alpha in was (1.4) not (1.5). If you did this, you would have come up with the proper other hand beta rules, instead of the incorrect (1.6), which essentially says the right and left algebras have the same rules, which is certainly incorrect. Using then the wrong beta rules, you are not entitled to do a beta=-alpha substitution when you see fit, and come up with anything meaningful.
That is exactly your mathematical mistake, leading to manifestly incorrect physics!
There is nothing I can do if you do not understand how parity inversion works despite repeated explanations. But the problem is that you refuse to understand, and you insist that it is others who do not understand when you are the one who does not understand. You are wrong, and this is the last time I will spend by time by explaining why your are wrong. After this I am moving on with the discussion which the folks at RW have been kind enough to host for so long, and can only hope that other people understand even if you do not.
Alpha are the bi-vectors against which one represents parity inversion. When you have a right-handed function R(x), parity inversion comes from taking L(x) = R(-x). There is nothing “disjoint” about this way of using coordinates any more than there is something disjoint about the use of -alpha in (1.5). Look at any textbook about parity inversion. You transform x -> -x, y -> -y, z -> -z for the object being inverted. Nobody insistently says that these coordinates are disjoint and you must rename everything to a = -x, b = -y, c = -z. Sorry, this is your own fiction; it has nothing to do with physics. What IS disjoint are the functions L and R. That is why those have different names, and are related by L(x) = R(-x). The proper approach is using L(x) = R(-x), NOT defining some a=-x. And I have no suitable adjective for your repeated claims, in essence, that these physics relations are irrelevant, and that all you need is algebra. Such claims are tantamount to insisting we can do physics while ignoring actual empirical physics, not unlike the pre-Gallilain belief that heavier objects fall faster than lighter ones.
No, we do NOT insert -beta for alpha into (1.4). This is still part of your fiction. We use physics, not fiction, to tell us how parity inverts, and that is in (1.2) applied in (1.5).
Finally, (1.6) does NOT say the right and left algebras have the same rules. It says L(alpha)=R(beta), but beta already has the minus sign embedded in its definition. You do not get to use your minus sign more than once. The only reason (1.6) is even in my proof is to humor your insistence that we define beta = -alpha event though this is entirely superfluous and subject to being horribly misapplied as has you have repeatedly done here, and has been done for six years with the persistent false claims that Christian made a sign error when he did no such thing. We do not invert parity by setting a = -x, b = -y, c = -z. We transform x -> -x, y -> -y, z -> -z inside of whatever function is being inverted.
There are other legitimate matters to be discussed about Christian’s theory. This red herring is not one of them. I have nothing more to say to you about this. I cannot make somebody understand something that they refuse to understand, and after days of trying patiently to do so, I am done trying.
Joy, you have my sympathies, nobody should be subjected to any of this. We still have to work through the rest of your model, but my every instinct tells me you are on the right side of history. And I know with certainty that history will show that the allegations you made a sign error were abjectly false, and as a matter of humanity, profoundly unfair. This is the last I will talk about these bogus sign error claims, aside from suggesting that anybody who has leveled this accusation against you in the past owes you a profound apology.
I found it difficult to follow the commonsense argument with RH and LH gloves as you cannot rotate a LH glove into a RH glove. But I have followed the argument better, I think, with a Rubic cube which after all is just a very big particle(!). I took two diagonally opposite corners of the cube as anchor points and rotated the cube about that diagonal. I had the blue, yellow and orange faces nearest to my eyes. Looking at the orange face, the little orange sub-cube touching my finger does not sweep through much space but the diagonally opposite orange face sweeps through most space. During the rotation there is a RH torsion in that orange face? The blue and yellow faces have the same handedness torsion when they rotate. And they all have the opposite torsion when they rotate the other way. I feel sure that I am correct about these torsions but when I look closely and slowly by eye the torsions just disappear. I don’t feel sure enough of myself to say that the faces definitely gain torsions just by the rotation of the Rubic cube, though I am reasonably confident that this is what is happening. What do you think? I was hopelessly lost when trying to think about this with gloves, despite it being an apparently obvious case. For example, for some types of glove it is only really the thumbs that determine handedness so are the rest of the gloves the same handedness for both gloves? Etc etc with more difficulties in visualisation.
Hi Lurker B,
I am amazed to meet someone who finds a Rubic cube to be easier to work with than a glove! 🙂
To avoid having to visualize, just go get a pair of gloves. Then turn the right-handed glove inside out. Now it will fit your left hand just like the left-handed glove. We then write this result as:
Right (inside out) = Left (outside out).
Everything else is just mathematical dressing up, and generalizing from gloves to any configuration of matter that looks (and fits) differently inside out than it does outside out.
Jay
Jay,
Parity has very little to do with this, and provides absolutely no guidance for how to combine results from two separate orientations. This is one way you miss that you on on the wrong tack. This is strictly a matter for the rules of algebra, and you have not correctly applied them in your clearly incorrect “proof “.
When you make statements like “(1.6) does NOT say the right and left algebras have the same rules” you are demonstrating you have much to learn about algebra, for (1.6) is -by definition- the definition of your beta algebra. An algebra is defined by the basis element product rules, which is precisely what (1.6) represents. (1.6) is one for one with your stated alpha rules in (1.4) used later, so your (1.4) and (1.6) describe the very same algebra, and not separate orientations. The basis names alpha and beta being different, and any consideration external to these two equations will not change the fact that the two definitions you carry forward are the same algebra. As such, your substitution of beta = – alpha is invalid. And since your “proof” requires this invalid substitution it also is invalid. This is fact, not my opinion.
I would have thought a visual picture person like you would be able to see immediately you would be able to mimic the definitions for (1.4) and (1.6) with the same hand. You get the need to switch hands when there is a sign change in front of the epsilon. (1.4) and (1.6) indicate the same sign.
It is not me doing a double negation but you. Once was going to (1.5) and another going to (1.6). If you want to stay with (1.6) then you must move forward with (1.5), from where (1.6) came, for alpha instead of the alpha definition (1.4). Then you will get the necessary sign change in front of epsilon.
If you made the alpha and beta rules correctly indicate the epsilon sign change, you would actually prove Christian DID make a sign error.
To further refute Christian’s claim what I am saying has nothing to do with his model, and give the clearest picture of his sign error, I refer you and everyone else to Christian’s “Refutation of Richard Gill’s Argument Against my Disproof of Bell’s Theorem” arXiv:1203.2529v1[quant-ph] equations (15) and (16)
(15) (+I.a)(+I.b) = -a.b – (+I).(a x b)
(16) (-I.a)(-I.b) = -a.b – (-I).(a x b)
I would hope everyone could glance at this and see Christian’s sign error. The two left hand sides are equal because the minus signs in (16) cancel each other out, they are not some spooky algebra at a distance. The right hand sides differ by his sign error.
Of course Christian needs the signs to be incorrect such that he can add (15) and (16), subtracting out the cross products, divide by two and get his desired -a.b. Unfortunately that is not what GA says is correct.
It is unfortunate that even after such a clear explanation by Jay, Lockyer is still unable to recognize his mathematical and conceptual mistakes, and continues to accuse me for making them. The two equations he has quoted from my paper https://arxiv.org/abs/1203.2529 , namely equations (15) and (16), should have settled this specious issue back in 2011. They are exactly the two equations for the right- and left-handed physical systems discussed by Jay, namely Jay’s equations (1.3) and (1.5), respectively. Needless to say, there is no error of any kind in these two equations. They are exactly the correct equations for the right- and left-handed physical systems, such as a pair of right- and left-handed gloves. You can find these equations in any good textbook on Geometric Algebra, such as the one by Doran and Lasenby. Here are the relevant pages I posted from this standard textbook in my earlier post:
http://libertesphilosophica.info/blog/wp-content/uploads/2017/01/IMG.pdf .
Thus my equations (15) and (16) are textbook equations. And yet Lockyer has been claiming that there is a “sign error” in them. The right-handed equation, namely equation (15), is traditionally used in a right-hand-based geometric algebra, while the left-handed equation, namely equation (16), can be found in the pioneering works of Sir William Rowan Hamilton as well as in any modern-day textbook on classical or quantum mechanics. See, for example, in the following well known and highly cited paper by Hartung, at the bottom of the page 908:
http://libertesphilosophica.info/blog/wp-content/uploads/2017/02/Hartung.pdf .
Thus Lockyer has been telling us for the past six years that all of the textbooks on classical and quantum mechanics of the past 150 years have “sign errors” in them. I rest my case.
I told Jay his (1R) and (1L) both in alpha were acceptable, which parallels your claim about finding similar statements in texts. The problem is your model requires adding the results from two separate algebras. I pointed out the acceptability is ONLY when the math involved ONE algebra. I don’t think you will find any examples in text books where results from both orientations are added together, and certainly not in a way that results in such an obvious error. The reason why is the authors understand the map between orientation representations is an automorphism, and what that means.
Your claim of finding such representations in texts does not help you. Your (15) and (16) are clearly inconsistent and clearly demonstrate your sign error.
I disagree. Jay has already explained to you clearly what is wrong with your argument. Many others, including myself, have also explained your mathematical and conceptual mistakes to you for the past six years. Nevertheless, I invite you again to publish your argument on the arXiv where many of my papers are published, and at least one of which is also published in a peer-reviewed physics journal, namely in the International Journal of Theoretical Physics:
http://link.springer.com/article/10.1007%2Fs10773-014-2412-2 .
Let me remind the readers of RW that my model is based on at least three subjects:
(1) Bell’s theorem, which requires understanding what is meant by a hidden variable by Bell.
(2) General Relativity, which requires understanding what is meant by an orientation of a manifold, such a the vector space Cl(3, 0) [ cf. the Definition V.1 in the above IJTP paper ].
(3) Geometric Algebra, which requires understanding of the equations used on my papers.
In one of your comments above you have stated that even a published paper, like my IJTP paper, can be “garbage.” If so, then it would be useful if you answered my earlier question:
I think this debate can be ended if either side picks some numbers to write down a specific example of Jay’s “proof of no sign error” equation (1.10) without any letters or greek symbols. The equality will either hold or not, and if it does there will be an extra step of showing the example is not a special case where it works but others do not.
(And all of the basis vectors should be written out with numbers too…)
Indeed, it is easy to check that (1.10) is false.
Jay’s “proof of no sign error” is nonsense. Equation (1.3) is not a “right handed function”. It is not a “distribution of matter”. His relation (1.2) is irrelevant. There are no functions R and L here.
Jay Yablon’s https://jayryablon.files.wordpress.com/2017/01/proof-of-no-sign-error-by-christian.pdf derives contradictory assertions (1.3) and (1.5), using an alleged physical fact (1.2).
alpha_i alpha_j = delta_ij – epsilon_ijk alpha_k (1.3)
alpha_i alpha_j = delta_ij + epsilon_ijk alpha_k (1.5)
I conclude that (1.2) is wrong.
Equation (1.3) is a system of 9 equations. It is not a “distribution of matter in space-time”. There are no functions R and L defined in (1.3). Relation (1.2) is irrelevant.
Hi Richard, good to hear from you again.
First, (1.3) and (1.5) in https://jayryablon.files.wordpress.com/2017/01/proof-of-no-sign-error-by-christian.pdf are not contradictory assertions. They are oppositely-handed relations among bi-vectors. My right hand may be different from my left hand, but my right hand does not contradict my left hand. And a right-handed eigenstate of the operator matrix isigma.a (which matrix is isomorphic to alpha.a, per my post about an hour ago) is likewise different from but in no way contradicts a left-handed eigenstate of that same operator matrix.
But before we even get to the above, of which handedness is implicitly a part, let’s get to the parity / handedness relation (1.2) which is at the root of our present impasse. First you say “I conclude that (1.2) is wrong.” Then in the next breath you say “Relation (1.2) is irrelevant.” Those are two different assertions. Is (1.2) wrong? Or is it (in your view) irrelevant? That is my general question. My more specific questions are these:
Are you really saying that handedness is irrelevant to any discussion of (1.3) and (1.5)? And if you are, then referring now to equation numbers in my earlier https://jayryablon.files.wordpress.com/2017/01/geometric-algebra-tutorial-4.pdf (which I will try not to confuse with numbers in https://jayryablon.files.wordpress.com/2017/01/proof-of-no-sign-error-by-christian.pdf), are you saying that we may not use alpha_1 alpha_2 alpha_3 = +1 in (1.5) which is deduced directly from (1.4) [the same as (1.3) above and in https://jayryablon.files.wordpress.com/2017/01/proof-of-no-sign-error-by-christian.pdf%5D which has the indexes in an xyz order, to ascribe “right-handedness” to (1.3) above and in https://jayryablon.files.wordpress.com/2017/01/proof-of-no-sign-error-by-christian.pdf? And are you saying that we may not use alpha_3 alpha_2 alpha_1 = +1 which can be derived directly from (1.5) above and in https://jayryablon.files.wordpress.com/2017/01/proof-of-no-sign-error-by-christian.pdf, and which has a zyx index order, to ascribe “left-handedness” to (1.5) above and in https://jayryablon.files.wordpress.com/2017/01/proof-of-no-sign-error-by-christian.pdf?
Thanks, Jay
The links in the last paragraph above may be more confusing than the confusion I wanted to avoid between the two documents, so let me try it this way: I shall call the document https://jayryablon.files.wordpress.com/2017/01/proof-of-no-sign-error-by-christian.pdf, D1. I shall call the document https://jayryablon.files.wordpress.com/2017/01/geometric-algebra-tutorial-4.pdf, D2. With these, the final paragraph above which should be a much easier read, is as follows:
“Are you really saying that handedness is irrelevant to any discussion of (1.3) and (1.5)? And if you are, then referring now to equation numbers in D2, are you saying that we may not use alpha_1 alpha_2 alpha_3 = +1 in (1.5) of D2 which is deduced directly from (1.4) of D2 [which (1.4) of D2 is the same as (1.3) above and in D1] which has the indexes in an xyz order, to ascribe “right-handedness” to (1.3) above and in D1? And are you saying that we may not use alpha_3 alpha_2 alpha_1 = +1 which can be derived directly from (1.5) above and in D1, and which has a zyx index order, to ascribe “left-handedness” to (1.5) above and in D1?”
I am pleased to announce that the winner of the early spring break contest is . . . Joy Christian! (I actually was unsure if Joy would see this, but he knows this material too well to not have seen it.)
Equation (1.11) and the paragraph in which this is embedded is indeed the most important part of the entire document at https://jayryablon.files.wordpress.com/2017/01/geometric-algebra-tutorial-4.pdf. The silver medal equation (all numbers below also refer to this document) is
alpha_i = I e_i = I . e_i (1.23),
because it provides the mathematical connection which allows us to include:
I=e_1e_2e_3 (1.6)
in this isomorphism when needed or desired.
Let me reproduce below, the main point made in that paragraph housing (1.11):
“Additionally, because of this isomorphism, it is to be anticipated that the strong correlations
E [ (isigma.a) (isigma.b) ] = – a.b
said to be predicted by quantum mechanics will and must find an isomorphic representation through geometric algebra, in the form of the expectation value:
E [ (alpha.a) (alpha.b) ] = – a.b. (1.11)
That is because geometric algebra and spinor geometry are simply two different tools used to represent the same underlying geometry. So irrespective of any viewpoint that someone may hold about the origins of strong correlations, it is to be expected and indeed must be a requirement, that the geometry represented by each of these two mathematical toolsets will have the same objective characteristics. Conversely, the failure of these two tools to represent the same physics would indicate that one or the other or both tools are being misapplied.”
I will add to this the observation that Einstein would roll over in his grave at the suggestion that the physics occurring within the spacetime geometry or even the space geometry, could ever depend upon the particular mathematical language that we call upon to describe that physics. In one language we use the Pauli matrices isigma to represent the commutativity proprieties of space; in the other we isomorphically use bivectors alpha to do the exact same thing. The physics within that space cannot be affected one iota. So it is really the principle of physics invariance with respect to mathematical language, that is the north star of these discussions, and should provide a common point of universal agreement.
I also want to make clear that I have not said said nor have I implied anything at all about Joy Christian’s model; I am merely talking about geometric algebra (GA) on its own as a mathematical descriptor of the geometry of physical space, entirely independently of how someone might try to apply GA to understand the strong correlations.
In my opinion, one of the mistakes Joy has made — which is a presentation mistake not a mathematical or physics mistake — is that the has not sufficiently segregated the underlying structure of GA as an independent mathematical tool, from his application of that tool to the strong correlations puzzle growing from the EPR paradox. That segregation must be clearly made, utilizing the very central role of the isomorphism laid out in the (1.11) paragraph, and doing so can overcome and settle much of the confusion and consequent disagreement which has attended to Joy’s model.
I am planning to prepare a detailed mathematical document that will show the numerous ways in which the isomorphism highlighted in the (1.11) paragraph is of fundamental importance and is indeed the north star Polaris with respect to settling the discussions here. My time over the next 5-6 weeks will be in flux, however, because I am traveling for a week on business beginning Wednesday, then my daughter-in-law is due later in the month which will take me off the field for another two week period which of course is unknowable in advance. So while am am promising to prepare and post this document as soon as I have the time, I can make no promises as to what time will be available to me in these next 5-6 weeks.
Jay
I will let Jay defend his equations, but his overall proof of “no sign error” is impeccably true. In particular his final result (1.10) is trivially true, and so is his starting physical relation (1.2).
As I have already pointed out, Jay’s final result can be very simply proved in my notation, as I have done in my extensive critique of Gill’s earlier claims: https://arxiv.org/abs/1501.03393 . We begin with equations (49) and (56) of this paper: https://arxiv.org/abs/1405.2355 . Equation (49) defines a pair of “hidden equations”, which can be written as “Heads or Tails”:
L(a, +) L(b, +) = -a.b – L(a x b, +)
or
L(a, -) L(b, -) = -a.b – L(a x b, -).
Using equation (56) this pair of equations can be easily rewritten in terms of D(a x b) as
L(a, +) L(b, +) = -a.b – D(a x b) = D(a) D(b)
or
L(a, -) L(b, -) = -a.b + D(a x b) = -b.a – D(b x a) = D(b) D(a).
Thus we have the following identities used in deriving the result (79) of the above paper:
L(a, +) L(b, +) = D(a) D(b)
and
L(a, -) L(b, -) = D(b) D(a).
Since these identities are a part of the definition of the local hidden variable lambda = +/-1 in my model, the so-called “sign error” is a complete non-issue. Substituting the above defining identities into the simplified version of the eq. (72) of the above paper, i.e., into
1/2 { L(a, lambda = +1) L(b, lambda = +1) + L(a, lambda = -1) L(b, lambda = -1) } ,
then immediately leads to the strong EPR-Bohm correlation derived in eq. (79) of the paper:
E(a, b) = 1/2 { D(a) D(b) + D(b) D(a) } = -1/2 { a b + b a } = -a.b . This disproves Bell’s claim.
I have prepared an uploaded a more formal reply to the above, at https://jayryablon.files.wordpress.com/2017/02/proof-of-no-sign-error-by-christian-2.pdf. This does not rely in any way on (1.2), and is even simpler than my earlier proof. Once again: Christian has made no sign error. Jay
Thank you, Jay. For the past six years Gill has been claiming that the left-handed subalgebra
alpha_i alpha_j = delta_ij + epsilon_ijk alpha_k
has a “sign error” in it on the RHS, despite the fact that it has been a textbook equation in both classical and quantum mechanics for at least one hundred years, and despite the fact that I have pointed out his mistake to him long ago: https://arxiv.org/abs/1203.2529 .
PS: Please also note that in his post above Gill has made a different sign mistake in both equations, one of which I have reproduced from him verbatim as
alpha_i alpha_j = delta_ij + epsilon_ijk alpha_k ,
but the correct equation for the left-handed subalgebra is actually
alpha_i alpha_j = – delta_ij + epsilon_ijk alpha_k .
There are several such mistakes by Gill, which I have carefully brought out in my formal replies to his “critiques” in https://arxiv.org/abs/1501.03393 , and in the one linked above.
Sure: if alpha_i satisfy the right-handed relations (1.1a) then beta_i = – alpha_i satisfy the left-handed relations (1.1b).
So both equations are true, but they concern different objects, not the same objects.
Suppose the same alpha_i satisfy both relations. Then adding them, and dividing by two, we obtain alpha_i alpha_j = – delta_ij
Using the same symbols to stand for two different things is a recipe for disaster.
If you want to work with both left-handed and right-handed bases at the same time then you had better distinguish them in your notation. For instance, with a superscript “R” or “L”. There is no problem with defining alpha_i^L = – alpha_i^R. If the alpha_i^R satisfy (1.1a) then the alpha_i^L will satisfy (1.1b), and vice versa.
The left-handed basis and right-handed basis have always been carefully distinguished in my work from day one (cf. my very first paper, or any of my papers for that matter). They are also carefully distinguished in my derivation below. Unfortunately the same cannot be said about some of the critics of my work who have never published a peer-reviewed paper on Clifford algebra or general relativity (i.e., tensor algebra) on which my papers are based.
Unfortunately, (49) and (56) contradict one another.
From (56) it follows that L(a, lambda) L(b, lambda) = lambda^2 D(a) D(b) = D(a) D(b) which is independent of lambda.
That is incorrect. Equations (49) and (56) do not contradict one another. From equation (56) it does not follow that L(a, lambda) L(b, lambda) = D(a) D(b) independent of lambda. The correct calculations are done in my post above. The correct results within geometric algebra are
L(a, lambda = +1) L(b, lambda = +1) = D(a) D(b)
and
L(a, lambda = -1) L(b, lambda = -1) = D(b) D(a) .
Your first two equations here are a restatement of your sign error, so your D(b)D(a) commutation that follows from it is also erroneous. This is common to all of your attempts to justify you have not made an error, you repeat the error axiomatically, perhaps dressed up differently, and claim the remainder is “proof”.
Lambda is a real number. Your definition is L(a, lambda) = lambda (I.a).
It is completely obvious that
L(a, -1)L(b, -1) = (-1)L(a,+1)(-1)L(b,+1) = (-1)^2 L(a,+1)L(b,+1) = L(a,+1)L(b,+1)
So in this rendition of your sign error, the two left hand sides in the first equation set here are equal and the left hand sides are not, differing by your patently obvious sign error.
You have done yourself a great disservice not understanding, or perhaps not admitting, for over 6 years now, that lambda is a real number, and how that fact kills your fair coin orientation choice model. There is no conspiracy out to suppress a contrary opinion, or out to damage you personally. If you have been damaged, you have yourself to blame, not those who have tried to help you by pointing out your errors.
Two right hand sides are not, …. sorry
I have revised my paper slightly to emphasize the simple argument above. The new version is now online. The only difference between the previous version and the new version are the two lines added in equations (77) and (78) for clarity: https://arxiv.org/abs/1405.2355 .
I have been watching this thread for some time, and got interested in all the fuss over GA. I decided to work through Jay’s paper “Geometric Algebra from the Ground Up, Right and Left”. To do this I used Mathematica and set up a representation of GA using Pauli matrices for the e1, e2 and e3 vectors. Section 1 worked out fine. Everything seems to be correct. Then I got to section 2, on the left hand basis (according to Joy Christian’s critics). I set up three new vectors, f1 = -e1, f2 = -e2 and f3 = -e3 to represent the space conjugate coordinate system using these matrices and built a left hand system. Equation (2.1) does not agree with what the matrix representation gives since a minus sign has been added, and everything after that is also incorrect. How many of Joy’s critics find Jay’s analysis correct?
I suggest you divide everything by zero, as you did elsewhere, and it will all workout fine.
Speaking of working out fine, that detection loophole in your simulations is also doing a good job for you.
Yeah, all I have to do is divide all the loopholes by infinity, and the simulation starts running.
Email me your Mathematica notebook file and I will check it out. It is perhaps due to Mathematica being in a fixed right hand basis just like GAViewer.
It’s very easy to do. Just set e1:=PauliMatrix[1], e2:=PauliMatrix[2], and e3:=PauliMatrix[3]
then go through Jay’s paper step by step. The alpha bivectors are formed by matrix multiplication of e1, e2 and e3.
When you get to section 2, set f1=-e1, f2=-e2 and f3=-e3 and see what you get when you form beta1, beta2 and beta3 in this left hand system by matrix multiplication. The extra minus sign shown in (2.1) isn’t needed. In fact if you insert it, it will these will no longer be correct left hand bivectors. If you look at the bivectors without the minus sign they will be in the correct left hand orientation. If the minus sign is added, you are back in the right hand orientation which is wrong.
Fred, Jay’s equation (2.1) has nothing to do with the flips f1=-e1, f2=-e2 and f3=-e3. The mistake here is jReed’s.
The mathematical mistake in the above claim by Lockyer is the same mathematical mistake he has been making for the past six years. He is assuming what he wishes to prove; namely
L(a, -1)L(b, -1) = L(a,+1)L(b,+1) .
Not only is this equality physically fictitious as Jay has painstakingly explained, it is also mathematically meaningless. Because Lockyer has derived it by presupposing that the order of the product L(a, -1)L(b, -1) is the same as the order of the product L(a, +1)L(b, +1). Thus, not surprisingly, he ends up proving the order of the product L(a, -1)L(b, -1) to be the same as the order of the product L(a, +1)L(b, +1). Lockyer proves proposition X by presuming proposition X. No deep understanding of logic is needed to understand what I am saying.
To be sure, his “proof” is impeccable, and it is produced using just one line of equalities:
L(a, -1)L(b, -1) = (-1)L(a,+1)(-1)L(b,+1) = (-1)^2 L(a,+1)L(b,+1) = L(a,+1)L(b,+1) .
But note that my definition L(a, lambda) = lambda (I.a) he has used for his proof says nothing about how it is to be applied to a product like L(a, -1)L(b, -1). We are not at liberty to apply the definition (56) haphazardly as Lockyer has done. We are supposed to follow the specific rules of geometric algebra to derive the relationship between the product L(a, -1)L(b, -1) and the product L(a, +1)L(b, +1) = D(a)D(b). These rules are specified by the bivector subalgebra, which is equivalent to the identity (49) of my paper: https://arxiv.org/abs/1405.2355 .
We must therefore begin without assuming anything about how the order of the product L(a, -1) L(b, -1) is related to the order of the product L(a, +1) L(b, +1). We must therefore proceed with equation (49) of my paper defining a pair of the “hidden equations”
L(a, +1) L(b, +1) = -a.b – L(a x b, +1)
or
L(a, -1) L(b, -1) = -a.b – L(a x b, -1).
Using definition (56) this pair of equations can then be rewritten in terms of D(a x b) as
L(a, +1) L(b, +1) = -a.b – D(a x b) = D(a) D(b)
or
L(a, -1) L(b, -1) = -a.b + D(a x b) = -b.a – D(b x a) = D(b) D(a).
Thus we arrive at the following identities used in deriving the result (79) of the above paper:
L(a, +1) L(b, +1) = D(a) D(b)
and
L(a, -1) L(b, -1) = D(b) D(a).
This is the correct calculation and the correct result, both physically and mathematically. Unlike Lockyer’s “proof”, it does not presuppose the order of the product L(a, -1) L(b, -1) to be equal to D(a) D(b), but derives it to be equal to the product D(b) D(a) using GA.
What this shows is quite simply that (49) and (56) are incompatible. Christian has written down a collection of assumptions which contradict one another.
How can (49) and (56) be incompatible with one another, or contradict one another when they are defining equations of a quaternionic 3-sphere, well established for over 150 years?
Direct substitution from assumed correct relationships is ALWAYS acceptable. I have no issues with your definition L(a, lambda) = lambda (I.a), and since this is an equality I am certainly free to replace L(a, lamba) with lambda (I.a) any time I please, and since GA is not a commutative algebra I am PRECLUDED from changing the order given in L(a, lambda)L(b, lambda) when doing the substitution. Therefore
L(a, lambda) L(b, lambda) = (lambda)^2 (I.a)(I.b) = (I.a)(I.b) independent of lambda since it is a choice between the real numbers +1 and -1.
This is exactly the same as
L(a,-1) L(b,-1) = L(a,+1) L(b,+1)
Or
(-I.a)(-I.b) = (+I.a)(+I.b)
You have been invited to show the previous equality is NOT true by explicitly doing both sides in their respective GA bases using nothing more than their GA multiplication rules but you have refused to do so. Instead you put out invalid equations involving (a x b) where you simply drop the right and left bivector bases attached. Since the map between them is a negation you get the sign error you need.
Doing what I asked, you will get the following
Define real number cross product like coefficients, summation convention assumed
CP_k = a_i b_j epsilon_ijk
E.g. CP_1 = (a_2 b_3 – a_3 b_2), a real number NOT a bivector
With this and taking the right and left orientation bivector bases as previously defined alpha and beta (alpha uses + epsilon and beta uses – epsilon) you will get, sum (i)
(+I.a)(+I.b) = -a.b – CP_i alpha_i
(-I.a)(-I.b) = -a.b + CP_i beta_i
Since alpha_i = – beta_i these two equations are equivalent, in perfect alignment with the obvious mutual cancellation of the 2 minus signs in (-I.a)(-I.b).
You on the other hand toss alpha and beta, violating the rules of GA, claiming the sum of the right and left orientation bivector products is -2 a.b.
This has nothing to do with material objects. A right glove added to a left glove does not result in no gloves
Changing topic from the “sign mistake” debate (I hope both sides can agee on a key equality, and its interpretation, and we can just plug in numbers to unambiguously check)… Is this the central idea of Joy’s theory:
“Something about the initial hidden state is a relative quantity, and it is relative to both Alice and Bob’s later detector setting choices.”
If so, are there any similar widely accepted theories in the literature? I am aware that the length of say a rocketship is a quantity relative to how fast it is moving from the standpoint of the observer in textbook special relativity theory. Here Alice, Bob, and the source of the particles are all at rest relative to one another. And the speed of the particles flying to Alice and Bob, and how fast they may be spinning as they do so, I think is irrelevant, and furthermore, can be very small compared to the speed of light, as in Joy’s macroscopic exploding balls experiment. And there are no supermassive objects bringing in relativistic gravitational effects. Where in the mainstream literature is there a similar relativistic effect? I could then read up on that, to aid in forming an opinion on the plausibility of Joy’s theory (issues of a sign error or not aside).
Richard, I cannot disagree with anything you said above.
And I cannot disagree with any of this either. To address your concern about notation, I have prepared a new document at https://jayryablon.files.wordpress.com/2017/02/ga-parity-inversion-1-0.pdf. Please take a look.
I hope I do not eat my words, but I believe that Christian and Gill would both agree on everything that is contained in this new document. 🙂 If that is so, then we may finally have a common core of mathematical agreement from which to discuss Christian’s model.
So, to both Joy and to Richard: Do you each agree with the content of https://jayryablon.files.wordpress.com/2017/02/ga-parity-inversion-1-0.pdf? And Joy, please don’t start talking about your model. And Richard, please do not start talking about Joy’s model. I want to see whether for at least one brief single moment* you can at least both agree over the left- and right-handed sub-algebras available in geometric algebra, before we dip even a single toe into questions about how these two bases may or may not be applied in Joy’s model. Joy, agree? Richard, agree?
Jay
* “Don’t let it be forgot, that once there was a spot, for one brief shining moment, that was known as Camelot.”
Your document looks fine. I agree with your equation (2.7), which is as error-free as mine.
Section 1 is mainly OK, section 2 is mainly nonsense. Formula (2.1) is full of nonsense. You make several sign errors.
It is clear anyway that (2.7) must be wrong: the left hand side doesn’t depend on lambda, since one may delete a factor lambda^2 = 1.
I disagree with Gill. Jay’s entire document is perfectly fine. There are no sign errors. Most importantly, Jay’s equation (2.7) is trivially correct. It simply combines textbook equations for the right- and left- handed bivector subalgebras in a convenient and compact notation. Mere change of notation cannot make wrong mathematical equations that are well established for over 100 years. Jay’s equation (2.7) is identical to the pair of correct equations in his (2.3).
Jay, I get your point: sometimes lambda is a “handedness indicator” not just a scalar +1 or -1. Can you pick an equation in the latest version of Joy’s paper (I think that is v8) where objects of different handedness are added or multiplied together, and plug in specific numbers? And show all the details of a calculation checking the equality?
Equation (2.6) is actually two sets of equations: one for a right-handed basis, one for a left-handed basis. On the top line, the alpha_i stand for a right-handed basis. On the bottom line, the alpha_i stand for a left-handed basis.
Equation (2.7) is similarly two sets of equations. When lambda = +1 it is the set of equations which characterise a right-handed basis. When lambda = -1 it is the set of equations which characterise a left-handed basis.
So equation (2.7) is deeply misleading. If the alpha_i satisfy (2.7) when lambda = +1 they do not satisfy (2.7) when lambda = -1.
Jay Yablon has successfully created the situation in which Joy Christian makes his sign mistake, using the same technique: sloppy notation: using the same symbols to stand for two different things at the same time. It would have been wise to add a superscript (lambda) to the alpha_i to indicate that they must change according to which value lambda is taking.
The above claims by Gill of “sign mistake” and “sloppy notation” are false. I have debunked both of his claims years ago in my formal reply to him: https://arxiv.org/abs/1203.2529 .
There is still a lot to be worked out with the notation. You write alpha’_i = -alpha_i, but if you substitute -alpha_i for alpha’_i in (2.1) the result is seen to be inconsistent.
You have proclaimed “nonsense” and “sign errors” without being specific at all. Please be specific. To help you, I will lay out (2.1) of https://jayryablon.files.wordpress.com/2017/02/ga-parity-inversion-1-0.pdf right here:
The transformation is
alpha_i -> alpha’_i = – alpha_i
and this is used in (2.1) to write the six lines:
R (alpha_i)
= alpha_i alpha_j = -delta_ij -epsilon _ijk alpha_k
-> R(alpha’_i) = alpha’_i alpha’_i = -delta_ij -epsilon _ijk alpha’_k (2.1)
= ( -alpha’_i) ( -alpha’_i) = -delta_ij -epsilon _ijk ( -alpha’_k)
= alpha_i alpha_j = -delta_ij + epsilon _ijk alpha_k
= L (alpha_i)
Is there something wrong with the transformation alpha_i -> alpha’_i = – alpha_i? If so, what? This is the exact calculation Christian’s critics have been urging, except that instead of using the name beta_i, I call the transformed bi-vector alpha’_i. Am I prohibited from using a different name for this?
In lines 1 and 2 I have written an equation you have previously agreed with. Have you changed your mind?
In line 3 I have applied the transformation you have previously agreed with, except that your beta_i is named my alpha’_i. And I have obtained a sub-algebra equation which Christian’s critics have insisted all along is the correct sub-algebra equation, other than the name alpha’_i in lieu of beta_i. Are you now changing your mind about the correctness of the equation in line 3?
In line 4 I simply apply math, since alpha’_i = – alpha_i is part of the transformation. This is just another name for beta_i = – alpha_i which Christian’s critics have always insisted be used. Are you changing your mind now that I am using a different name for beta_i? Am I not allowed to do so? What I obtain in line 4 is merely the (1.2) that you have previously agreed with for the left-handed sub-algebra.
Line (5) is also math. And line 6 is the name (1.12) for the left hand sub-algebra. Anything wrong here?
Proclaiming “nonsense” and “sign error” is insufficient. If there is an error somewhere, you should be able to point out exactly where and why. I have fully laid out (2.1) in six lines above to help you to do so. Which line in (2.1) above goes amiss?
I just wrote up a very direct derivation of (2.7), posted at https://jayryablon.files.wordpress.com/2017/02/gill-reply-2-8-17.pdf. If there is an error, please be specific exactly whether the error occurs and what the error is.
I agree that (2.7) is two sets of equations. And I agree that “Equation (2.6) is actually two sets of equations: one for a right-handed basis, one for a left-handed basis. On the top line, the alpha_i stand for a right-handed basis. On the bottom line, the alpha_i stand for a left-handed basis.” I would also agree that for any single “event,” you cannot simultaneously have lambda = +1 and lambda = –1.
If you think that combining these two equation sets into one in (2.7) is “sloppy notation” or “deeply misleading,” then although I disagree, I am happy to only use (2.6) as two separate equations until I provide justification as to the circumstances under which the two equation sets (2.6) can be utilized together (for example, added or averaged).
Can we agree that there is nothing wrong with (2.6) so long as we are clear that these are “actually two sets of equations: one for a right-handed basis, one for a left-handed basis”? If so, then I will only use each of the two equation sets in (2.6) separately, unless and until I can justify using them together. Deal?
Jay
HR, Do you see that “Equation (2.6) is actually two sets of equations: one for a right-handed basis, one for a left-handed basis,” as Richard has put it? And are you comfortable with the idea that your right hand is different from your left hand but they can both exist at the same time in a common physical space? And are you comfortable that we can fill a room with multiple people all of whom have left hands and right hands? If so, then you should be able to get comfortable that there is no inconsistency. The right and left hands can and do both exist in a common invariant space, but they cannot turn into one another.
If your answer is, OK, but how does this get used in Christian’s model?, then that is a very fair question which I hope we will soon discuss once the present confusions have hopefully abated.
Jay
My answer is that this must somehow be clear through the notation used, and so far it is not. In particular we should never write down an equality if the two sides cannot be substituted in subsequent expressions.
Mr. Yablon, I hope I’m not being repetitive (I haven’t read all the comments up to here), but equation (2.7) seems to be simply mistaken. If I set i=1,j=2, and k=3 then for lambda=1 it states
a1a2 = -a3
and for lambda = -1 it states
a1a2 = a3
Which is a contradiction, unless you are claiming that a3 = 0.
What you, FE, should have actually written is
a1a2 = -a3 for lambda = +1
or
a1a2 = +a3 for lambda = -1
This is the correct reading of the pair of equations contained in Jay’s equation (2.7).
What you are saying, instead, amounts to saying that Heads = Tails is a contradiction. But of course it is. But do Heads and Tails occur at the same time for a given event lambda? No.
Your model requires adding left and right results. If both use the same basis names for different basis product definitions the result will be assured to be nonsense. But it certainly will lead to -a.b as you want.
Jay,
(1) alpha_i alpha_j = -delta_ij – epsilon_ijk alpha_k
where alpha_1 = e2^e3, alpha_2 = e3^e1, alpha_3 = e1^e2
(2) beta_i beta_j = -delta_ij + epsilon_ijk beta_k
where beta_1 = e3^e2, beta_2 = e1^e3, beta_3 = e2^e1
Do (1) and (2) accurately represent left and right orientation GA bivector basis products?
Is the following representative of the GA bivector product (+I.a)(+I.b)?
(3) (a1 alpha_1 + a2 alpha_2 + a3 alpha_3) (b1 alpha_1 + b2 alpha_2 + b3 alpha_3)
Is the following representative of the GA bivector product (-I.a)(-I.b)?
(4) (a1 beta_1 + a2 beta_2 + a3 beta_3) (b1 beta_1 + b2 beta_2 + b3 beta_3)
Are the following acceptable representations of right handed (a x b) scalar coefficients?
cp_1 = (a2 b3 – a3 b2)
cp_2 = (a3 b1 – a1 b3)
cp_3 = (a1 b2 – a2 b1)
If you use just the basis product rules (1) on the bivector product (3), is the following the correct GA algebra result assuming sum over index i?
(5) -a.b – cp_i alpha_i = (3)
If you use just the basis product rules (2) on the bivector product (4), is the following the correct GA algebra result assuming sum over index i?
(6) -a.b + cp_i beta_i = (4)
By the definition of GA and the definitions for alpha and beta, do you agree the map between is alpha_i = -beta_i?
If we replace – alpha_i in (5) with + beta_i does (5) now equal (6)?
If we replace + beta_i in (6) with – alpha_i does (6) now equal (5)?
From this can we state uncategorically, exclusively using the rules of GA, that (5) = (6) = (3) = (4)?
Is the GA rote conclusion (3) = (4) consistent with the obvious claim the two minus signs in (-I.a)(-I.b) cancel making it = (+I.a)(+I.b)?
Does this last question tell us there is nothing in GA that requires special treatment of (-I.a)(-I.b) because of the minus signs that might lead to a commutation of product order that is not indicated for (+I.a)(+I.b)?
Since (+I.a)(+I.b) = (-I.a)(-I.b), and L(m, lambda) = lambda (I.m) for any GA vector m, is the following a true statement?
(7) 1/2 { L(a,+1) L(b,+1) + L(a,-1) L(b,-1) } = (+I.a)(+I.b)
Is all of the above fully consistent with the rules of general math and geometric algebra?
Should you agree there were no equations slipped in as definitions, everything was derived from first principles using nothing but the definition and rules of GA?
If you think the answer to any of the above questions is no, please explain in detail.
If you can’t refute anything above, you proved Christian’s sign errors.
I would hope an end to your quote: “unending confusion about the sign difference….” could finally be achieved by Christian, you and the other Christian supporters by taking an honest and objective look at what is above. Christian’s errors are real, and it is long past time that should have been recognized.
Let me remind everyone that, as mentioned by him earlier, Jay is out of town until next week, so he may not be able respond to any comments until then. And I am not inclined to address issues that I have already addressed repeatedly over the past six years, and again in this very thread (see, for example, this paper: https://arxiv.org/abs/1203.2529 ). But I will be happy, of course, to address any new questions or comments about my model, or relevant papers.
It should be very easy to refute any criticism of the paper “Local Causality in a Friedmann-Robertson-Walker Spacetime”.
In addition to the math, one has only to insert a simple graph into the paper:
A plot of the predicted detector responses A(alpha; s, lambda) = +/–1 and B(beta; s, lambda) = +/–1 as a function of the angles between the detector settings alpha and beta and the spin orientation s.
For a physicist, nothing more is needed in order to evaluate correlations such as
E(alpha, beta) = 1/N * Sum[k:1..N]{ A(alpha; s_k, lambda_k)*B(beta; s_k, lambda_k) } … (1)
where s_k is the spin orientation for each run k of the experiment considered.
The responses A(alpha; s, lambda) and B(beta; s, lambda) may be obtained by individual “projections” which formally map some tailored functions to values +/-1 such as
A(alpha; s, lambda) = +/-1 = lim[s → alpha]{ functionA(s, alpha, lambda) } …………….. (2)
B(beta; s, lambda) = +/-1 = lim[s → beta]{ functionB(s, beta, lambda) } ………………….. (3)
and which fulfill the local-realistic constraint B(gamma; s, lambda) = -1* A(gamma; s, lambda) for all gamma.
At this point, everything is done from a physical point of view: The predicted measurement outcomes are of interest, not the functions which are devised to do the mapping.
To say it quite simple:
a) Randomly select a spin orientation s_k
b) Carry out both “mappings” in order to get the expected detector responses A = +/–1 and B = +/-1
c) Determine E(alpha, beta) by averaging over the resulting products (+/–1)*( +/–1).
In case the model presented in “Local Causality in a Friedmann-Robertson-Walker Spacetime” is capable of producing such a simple graph, the model could be classified as a physically valid, “realistic local-hidden variable model”.
What you have suggested is done already in the paper itself, both theoretically as well as by means of several numerical simulations, produced independently by different authors (see the references). One such simulation is based on Geometric Algebra (see the code and a plot at this link: http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=296#p7322 ).
What I have said above applies to numerical simulations, too. In case the simulation is capable of producing a consistent – regarding the local-realistic constraint – list for the detector responses {spin orientation s_k, A(a; s_k) = +/-1, B(b; s_k) = +/-1}, the numerical simulation could be classified as a physically valid, “manifestly local-realistic simulation”.
PM, “lists” are irrelevant because according to Joy’s theory, some rows are impossible due to spacetime geometry for specific detector setting choices. See my posts above. I look forward to Jay’s response to Rick’s latest post. But I fear there will still be quibbling over the meaning of notation. If either side plugs specific numbers into some of Joy’s equations which mix left and right handed quantities, I think the sign mistake debate will quickly be resolved one way or the other. In response to Joy’s request for “other types of questions”, see my post above where I ask for references to literature about why we might expect relativity to kick in for Joy’s macroscopic exploding balls experiment, which at first glance appears to be modelable with just Newtonian physics.
I didn’t mean “relativity” as in Einstein’s relativity theories involving the speed of light etc. I simply meant that orientation (or “parity”, or handedness) of any manifold, such as S^3, or Cl(3, 0), is a “relative concept.” But to have some intuitive understanding of what is going on in my 3-sphere model you may want to look up the toy model I discuss involving a fictitious “Mobius world” in the first appendix of this paper: https://arxiv.org/abs/1201.0775 .
What does it mean that “some rows are impossible due to spacetime geometry for specific detector setting choices“? We have an electron with a certain spin orientation entering a Stern-Gerlach apparatus. Putting aside experimental limitations: Does it mean that some electrons will never ever leave traces on a fluorescent screen or so?
No. That is not what the model says. The model says that, because of the specific geometry and topology of the 3-sphere, “… a measurement event cannot occur if there does not exist a state which can bring about that event.” This has been explained in the paragraph just after equation (40) and before equation (41).
According to equations (54) and (55) in “Local Causality in a Friedmann-Robertson-Walker Spacetime”, all “local-realistic” spin states with s1 = s2 = s result in detector responses +/-1. What states cannot bring about an event? A simple graph would be of help.
Yes, that is correct. According to equations (54) and (55) of that paper all local-realistic spin states with s1 = s2 = s result in detector responses +/-1. That pair of equations define measurement functions within a Clifford-algebraic representation of the 3-sphere. The questions you are asking are about the tensor representation of the 3-sphere, and for that the measurement functions are defined by equations (26) and (27). But even in that case your question “What states cannot bring about an event?” is not appropriate, because all states defined by the pair (e, s) bring about the events A and B defined in equations (26) and (27). There is one-to-one correspondence between the initial states (e, s) and the measurement results (A, B). Therefore this is not a detection loophole model. In fact there are no loopholes of any kind in my model. A graph of what is happening can be found, for example, in this simulation of the tensor representation of the 3-sphere: http://rpubs.com/jjc/233477 .
PM: Joy says it is NOT a detection loophole. Somehow the LATER detector setting choice determines if a trial happened or not. Yet he says it is NOT backwards causation. So I thought it must be that the initial state must be a relative quantity, like length when speeds approach the speed of light. But it isn’t exactly like that either. He says the relative quantity is the orientation of a manifold. I’d be interested in reading about any similar widely accepted theories if anyone can provide citations.
This simulation represents the detection loophole. The expression:
f = -1 +2/Sqrt(1 + (3s)/Pi)
which is used to select the states which are included in the simulation came from Richard Gill’s paper arXiv:1505.04431v2, “Pearle’s Hidden Variable Model Revisited” the R code he used in that paper is similar to what is used in your simulation. Your simulation is just another example of the detection loophole that you have relied on for many of your simulations.
I disagree, both with the technical contents of your comments about my simulation, as well as with the priority of my simulations you have wrongly attributed to Richard Gill.
To begin with, my model has nothing to do with any loopholes, let alone with the detection loophole, because of the reasons I have already explained. These reasons are pretty obvious from the theoretical contents of my paper as well as from what I have explained in the preambles of most of my simulations (including the one I have posted here recently).
Secondly, Gill’s paper you have mentioned appeared more than a year after my paper was published on the arXiv in 2014. Unfortunately Gill does refer to my paper or any of my simulations. In fact, Gill’s R-code was simply a reformulation of Michel Fodje’s original simulation called “EPR simple”, which was published long before Gill even got involved in the discussion about my model. Gill made only a very minute improvement on Michel’s original choice of the distribution function which you have wrongly attributed to Gill. That function was first officially published in my arXiv paper in 2014 ( https://arxiv.org/abs/1405.2355 ).
Moreover, Michel Fodje has actually objected online at various places to Gill’s interpretation of Michel’s original contribution. All of the priorities of my work and Michel Fodje’s work can be easily proven, because everything I am saying here is on the record on the Internet (I would be happy to provide the evidence citing dozens of links if needed). To be fair to Gill, it is true that during our discussion on Fred’s forum he did dig out the improved distribution function from Pearle’s 1970 paper, but not without help from me and several other people who were involved in the discussion for years on various blogs and online forums. And I do give generous credit to Gill for his part in the early discussions and his contributions to my simulations in the preamble of some of my early simulations that relied on his contributions.
But the most important technical point here is that neither my model nor any of my simulations have anything to do with the detection loophole, or any other loopholes for that matter, because of the reasons I have already explained: There is one-to-one correspondence in my model between the initial state (e, s) and the results (A, B). Therefore there is no question of detection loophole. All states are detected. That is the whole point.
In reading your paper, “Local Causality in a Friedmann-Robertson-Walker Spacetime”, I need some more information on how equation (17) was arrived at. Can you explain where this equation for the normalization came from in greater detail?
As it says just before eqn (17), it is a chosen initial or complete state of the physical system.
I think Joy is arguing that the orientation of a manifold used to describe the initial state is relative to the later detector setting choices, and this rules out the same combos that would be ruled out by Pearle’s models (or maybe there is just a large overlap in the combos ruled out). So aside from the sign mistake debate, we need to consider how plausable it is that the orientation of a manifold exists, can be relative to detector setting choices, and the relativity effect can be big enough to give the strong correlations Joy says can occur in say his macroscopic exploding balls experiment. The simulation alone could be seen to connect to the idea. It would be more convincing if the idea could connect to the math early on, and an error free derivation leads to his conclusions. The sign mistake debate is trying to assess part of that. What do we need to check in the other parts of his derivation?
I am still traveling for another 48 hours, and I’m not easily situated to reply. I will post about the sign issue soon after I return.
But as to Earl of Snowden’s good question about what we need to check in other parts of Joy’s derivation, here is a simple outline: 1) How Joy represents the singlet state at EPR t0 but prior to detection, and how that is represented. 3) How the detection then gets represented.
Jay
Something got messed up in my prior post. I will try again: 1) The representation of the singlet state at EPR tIme less than zero. 2) The representation of the doublet state at t greater than zero but prior to detection . 3) The representation of the detection.
Let me add number four below:
4) doing all of the foregoing such that each detection is a binary valued correlation which is either +1 or -1. That is the most difficult trick of all.
The “sign issue” is a non-issue in my opinion, but I look forward to Jay’s new post about it.
Meanwhile, looking ahead, let me address the four questions raised above by Jay:
(1) At t ≤ 0 the singlet states in my model are represented by the numbers 1 (yes, just 1). In other words, at times t ≤ 0, a series of 1 emerges from the source: 1, 1, 1, 1, 1, … We can index these numbers with k if we like: 1_k, where k runs from 1 to n (which can be infinity).
(2) Then, at t > 0, and prior to detections, the singlet state represented by 1 splits up into two unit bivectors representing the spins, one moving towards Alice, which is named
-L(s1, λ),
and the other moving towards Bob, which is named
+L(s2, λ),
with the crucial condition s1 = s2 = s arising from the conservation of the zero spin angular momentum (or equivalently from the twist in the Hopf bundle of the 3-sphere). Note that, since all unit bivectors square to -1 (like the imaginary number), we have the product
-L(s1, λ) L(s2, λ) = +1 .
The number 1 therefore continues to represent the singlet state even at times t > 0.
(3) The detection processes of the spin bivectors -L(s1, λ) and +L(s2, λ) by Alice and Bob are then represented by the measurement functions A(a, λ) and B(b, λ), respectively, which are defined by the equations (54) and (55) in my paper: https://arxiv.org/abs/1405.2355 .
(4) Finally, note that we have A(a, λ) = +1 or -1, and B(b, λ) = +1 or -1, as required by Bell.
Dr. Christian, could you please tell how do you represent states other than the singlet state in your model? For example, how does this work for the Hardy state \ket{\psi} = \frac1{\sqrt3}(\ket{00} + \ket{01} + \ket{10})?
Good question. Hardy state is of course a lot more complicated state than the singlet state, so its representation within my model is also a lot more complicated. You will find its complete representation on pages 14 to 17 of this paper: https://arxiv.org/abs/0904.4259 . As you can see, it is not easy to reproduce here (by the way, Lucien Hardy happens to be a friend of mine).
For more general states than the Hardy state, the 3-sphere (or the bivector subalgebra) by itself is not enough. We have to switch to the full 8-dimensional basis of the Clifford algebra, or to the octonionic 7-sphere. The basic mechanism of my local-realistic model remains the same, however. But it all gets much more complicated and impractical for the more general quantum correlations. My goal, however (or Einstein’s for that matter), has never been to replace quantum mechanics, but to merely uncover the underlying local-realistic structure.
I have written up a formal response to the claims made by Lockyer in several posts above:
http://einstein-physics.org/wp-content/uploads/2017/02/refutation-lockyer.pdf .
I very much hope that this will be my last post on the supposed “sign errors” in my work.
I was happy to read in your “refutation-lockyer.pdf” above that you finally agree “the geometric product between bivectors remains invariant under orientation changes”. Since your model is summing successive bivector product results from fair coin choices of orientation now agreed to be invariant == the same thing, you have actually proved you did commit an error and can’t dispatch the bivector portions since they have the same sign when they actually can be added.
The remainder of your “refutation-lockyer.pdf” does not make sense mathematically, and leads to the impossible conclusion that (+I.a)(+I.b) and (-I.a)(-I.b) are not equal in your (19) and (20). Once again you got off track when you introduced (a x b) into a geometric algebra expression without giving it a clear geometric algebra definition, that being explicitly indicating it’s GA basis elements. If you would take my long standing advice to always explicitly show the GA bases in your math, I am sure you will find the error(s) between the correct quote above and the erroneous (19) and (20) combination.
In my questions to Jay you quote, I am quite sure I properly indicated how (a x b) fits in to geometric algebra for both orientations in my (5) and (6). Your claim you have not made any math errors is only possibly true if you can refute one of my affirmative claims there. I addressed it to him, but it really was an open challenge. You could bring a rapid conclusion to the impasse by telling us which one of my claims is false, and explicitly stating what is correct and why.
I have finally had a chance to put together a document in reply to the two-week backlog of the points made and questions raised by the various individuals above. You may read this at https://jayryablon.files.wordpress.com/2017/02/ga-right-and-left.pdf.
Jay
Mr. Yablon, you are arguing against basic mathematics in your attached file. If the \alpha_i in equation (3.11) are indeed the same, as you insist in your point 10, then you get immediately the contradiction \alpha_1\alpha_2 = \alpha_3 = -\alpha_3. You cannot write down a pair of equations and call them “non-simultaneous” to forbid them from being considered simultaneously. The only way to avoid the contradiction is to make the \alpha_i different, by introducing for example an extra subscript _R and _L, which you said is “very misleading”.
Sure you can. Consider the following pair of equations:
Coin toss result = Heads = +1
Coin toss result = Tails = -1.
These are a pair of equations for a toss of a fair coin. No one would want to consider them simultaneously and claim that Heads = Tails implies +1 = -1, and that is a contradiction.
Nonsense!
Before tossing a coin, you have to write
{ ( Coin toss result = Heads = +1 ) OR ( Coin toss result = Tails = -1 ) }
Although the above criticism of my model by Gill is both unjustified and inconsequential, I have nonetheless updated my 2012 arXiv reply to him in response. I have added a 3-page appendix in which I use different notations for the right- and left-handed bivectors to again derive the same strong correlations E(a, b) = -a.b: https://arxiv.org/abs/1203.2529 .
In this new appendix I have also tried to explain as clearly as possible what leads some to make the mistaken claim of “sign error” and end up with a result other than E(a, b) = -a.b.
When considering the physically relevant local-realistic constraint s1 = s2 = s, the whole math ends up in E(a, b) = -1!
From a physical point of view, nothing more than the rhs of equations (54), (55) and (67) in “Local Causality in a Friedmann-Robertson-Walker Spacetime” are needed.
While I am of the opinion that using an extra subscript _R and _L is indeed very misleading, it can be done as long as it is done very carefully. So in deference to Mr. FE, I have added a new point 10a to yesterday’s document, and posted it to https://jayryablon.files.wordpress.com/2017/02/ga-right-and-left-1-1.pdf. If someone else has different aesthetic taste than I do, and feels more comfortable with using the subscripts, then point 10a shows how this can be done without introducing a sign error. The overall results are exactly the same, however.
Sorry,
but all the laborious math is completely irrelevant. When considering the physically relevant local-realistic constraint s1 = s2 = s, one ends up in nothing else than
E(a, b) = -1 !!
From a physical point of view, nothing more than the rhs of equations (54), (55) and (67) in “Local Causality in a Friedmann-Robertson-Walker Spacetime” are needed.
While I thank you for the politeness, Mr. Yablon, I don’t see the what is the point in what you did. You just rewrote the two equations with \alpha_{iR} and \alpha_{iL} together with another equation saying that \alpha_{iR} = \alpha_{iL}. With this I just immediately substitute \alpha_{iR} with \alpha_{iL} and get the same contradiction again.
For a right-handed algebra:
\alpha_1\alpha_2 = -\alpha_3 (part of 1.1)
For a left-handed algebra:
\alpha_1\alpha_2 = \alpha_3 (part of 2.2)
Those are two independent, non-simultaneous algebras. See (1.1) and (2.2) in https://jayryablon.files.wordpress.com/2017/02/ga-right-and-left-1-1.pdf. Or are you saying (1.1) and (2.2) are contradictory, and we are not allowed to talk about left-handed algebras once we talk about right-handed algebras?
I made clear in my point 1 that you cannot equate these, but in point 2 and thereafter that you can average those. You only run into the “contradiction” if you insist on equating those. Look at my point 4. You are insisting that we can set L=R and find that -2 = -8 and this is a contradiction. But you cannot set L=R. But you can add them and average them.
Think about it this way: you cannot equate a left glove to a right glove. But you can randomly place a number of right and left gloves into a room, count how many of each are there, and if we designate each right glove by +1 and each left glove by -1, we can obtain an average number from -1 to 1 which tells me about the statistical balance between right and left gloves. If that average number is 0, this tells us that we have an equal number of each type of glove.
Of course you are allowed to talk about both of them, as long as you use different symbols for the different algebras. Using the same symbol for different objects is just a contradiction, and as you know from a contradiction anything follows.
Your substitution of real numbers for alpha and beta in your 4), 5) and 6) make no sense whatsoever.
The bivector bases cannot be represented by real numbers. The remainder of your text contains misrepresentations of GA you have already been told about.
That was an example requested by some of the readers who wanted numbers, and I stated it was an illustration. But insofar as the sign error which you are introducing, it is perfectly on point. Either the sign stays with the bi-vector and is hidden from the GA basis equation, my (3.10) in https://jayryablon.files.wordpress.com/2017/02/ga-right-and-left-1-1.pdf, or it is separated from the bi-vector and appears explicitly in the GA basis equation, my (3.9). You insist on having it both ways, and as a result you are the one who is making a sign error. Your approach leads to a “no left hand allowed” rule, which along with your interpreting a carbon copy of a right hand as a left hand, is nonsense.
“You have already been told about” the sign error that you are introducing, then misrepresenting by attributing your error to Christian. I have spend about four to five days in total preparing multiple documents to make clear how the signs work, mostly to patiently reply to you. I am done trying to convince you. I hope that other readers with open minds who are looking for the truth will come to understand that your six year old claim of a sign error is just plain wrong.
I found where your equation (20) in “Local Causality in a Friedmann-Walker Spacetime” came from. It is from the paper by Pearle:
Hidden-Variable Example Based upon Data Rejection Phys Rev D, 2, pp 1418-1425.
Pearle showed that the quantum mechanical results for the detectors could be obtained from a local realistic model by using data rejection (the detection loophole) with a certain probability distribution that he found for the detections. This involves rejecting events (the detection loophole) that are detected with less absolute amplitudes than the value of f which is given in your R program for the simulation
f <- -1 + (2/sqrt(1 + ((3 * eta)/pi)))
Where eta is a random variable uniformly distributed in the range {0, pi}.
This can be found in your first version of the simulation where it is stated in the comments of the R program:
“Later Richard Gill improved his 3D version by employing the exact probability distribution derived by Phillip Pearle in his classic 1970 paper”. You go on to state that this has nothing to do with data rejection or the detection loophole. The program doesn’t agree with this.
Looking at the R program, it is clear that this is exactly what is being done. First f is computed using the above equation. Then the following statement is used in a do loop over angles from 0 to 360 degrees:
good f & abs(ub) > f
This statement says that the amplitudes of events ua and ub are good if their absolute values are greater than f, in which case they are saved and used to compute the final values. If they are less than f, they are not saved and rejected. This is the detection loophole.
As explained in my paper, neither my model, nor any of its simulations have anything to do with the detection loophole, or any other loopholes for that matter. There is a strict one-to-one correspondence in the model between the initial states (e, s) and the detection results (A, B). Therefore there is no question of detection loophole. All initial states (e, s) are detected without exception. Not a single initial state goes undetected: http://rpubs.com/jjc/233477 .
However, the set of initial states depends on the (future) detector settings.
Besides, the same simulation code also simulates the detection loophole.
You are wrong on both counts. I have already pointed out your mistakes to you in my previous replies. The correct interpretation of my simulations is provided in my paper.
For a change, in this reconciliatory post let me assume the role of a mediator between Jay and those who have been claiming a “sign error” in my work. From the outset, let me acknowledge that what Gill and Lockyer have been arguing for years is mathematically correct. But it also happens to be physically incorrect and irrelevant for my model for the EPR-Bohm correlations. In fact, their argument was put to me years before they got involved, by none other than Prof. David Hestenes, the father of the modern geometric interpretation of Clifford algebra. In 2007, soon after my first paper on the disproof of Bell’s theorem appeared online, my former PhD mentor, Prof. Abner Shimony, contacted David to have a look at my daring claim of having disproved Bell’s theorem. David — despite his deep sympathy for Einstein’s point of view — told me that I was making a “mistake” in my interpretation of the duality relation between the vectors and bivectors in GA. I protested, of course, but David had just gone through a major heart surgery, and so out of sympathy (as well as deep respect) for him I did not push my point too vigorously. Four years later I had a much better opportunity to explain my point to him at a FQXi conference, on a cruise ship sailing from Bergen, Norway to Copenhagen, Denmark.
The point David had raised was exactly the same as the one raised much later by Gill, Lockyer and others. And my physical point David had missed back in 2007 is also exactly the same as the one both Gill and Lockyer have been missing. So let me first explain the mathematically correct part of their argument, in my notation. We begin with a set of bivector basis, { +alpha_i }, and a set of vector basis, { +e_i }, both right-handed, as is customary. In other words, we begin with a pair { +alpha_i , +e_i }. Now the duality between vectors and bivectors in geometric algebra states that
+alpha_i = I . ( +e_i ) , where I is the standard trivector: I = e_x e_y e_z .
So far so good. But now consider a left-handed set of bivectors, { -alpha_i }, and a left-handed set of vectors, { -e_i }. That is, consider a left-handed pair { -alpha_i , -e_i }. What is then the duality relation between this pair? Well, it is quite easy to verify that it is
-alpha_i = I . ( -e_i ) , where again I is the standard trivector: I = e_x e_y e_z .
This is indisputable. And no one is disputing it. But given these two duality relations, it is easy to verify that the corresponding geometric identities for arbitrary vectors a and b are
(I.a) (I.b) = -a.b – I.(a x b) ………. for right hand
and
(I.a) (I.b) = -a.b – I.(a x b). ……….. for left hand
Yes, they are identical, just as Lockyer has been saying. So Gill and Lockyer are right, and the average of these identities does not yield -a.b, but yields (I.a) (I.b), which is no good.
But why should the above two identities not be identical? After all, they have been derived from the fact that right-handed bivectors +alpha_i are dual to the right-handed vectors +e_i, and left-handed bivectors -alpha_i are dual to the left-handed vectors -e_i, and therefore the two minus signs cancel out, just as one would expect from the duality relations. So there is nothing wrong, at least mathematically, with what Gill and Lockyer have been arguing.
The problem with their argument, however, is that in any actual physical experiment the left-handed vector basis { -e } are never used. Therefore, we must translate everything in terms of the right-handed basis { +e }, which Gill and Lockyer fail to do. What is more, there is a perfectly consistent, mathematically rigorous, and physically transparent/meaningful way of translating the second geometric identity above to reconcile it with the actual practice in physics of using only a fixed vector basis { +e_i } to conduct all experiments. This is especially essential for my purposes of unambiguously distinguishing the “up” spin from the “down” spin. But once this translation from { -e_i } to { +e_i } is carefully done, the second geometric identity above transforms itself into
(I.a) (I.b) = -a.b + I.(a x b). ……….. left hand
We can now unambiguously compare this second identity with the first geometric identity, and then take the average of the two, as done in my model. The result is E(a, b) = -a.b.
This is the point I have tried to bring out in the appendix of my updated arXiv reply to Gill: https://arxiv.org/abs/1203.2529 . See especially the discussion between (A10) and (A18).
I have posted a more careful and detailed explanation of the above argument at the following link: http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=301&p=7613#p7613 .
When you substituted I.(-e_i) into beta rules, you mapped them to the alpha rules, clearly since the basis product rules for both were made identical as you note. You were already on the same side of your “mirror” when you did your true double negation, and quite improperly on just one equation in e_i and not the other that will be involved with separate equations you would like to add later. Faulty math proves nothing. Valid math will never show you have not made a sign error.
Some basic questions from someone who has not studied geometric algebra:
Suppose I am writing a computer program which can draw volume elements. And to avoid the complication of trying to represent 3 dimensional objects on a 2D screen, suppose I have the hardware to draw 3D holograms. If I want to draw a cube, I could start with the locations of its eight vertices, and do whatever encoding is necessary to get the program to draw it. The center of the cube would be the point equidistant from all 8 vertices. Suppose my program can add two volume elements (in particular, cubes) together using the rules of geometric algebra.
1. If I add two cubes with the same vertices together, do I get a cube twice as wide, twice as tall, and twice as deep, with the same center?
2. Do I need to encode a handedness of the cubes to do 1?
3. What would be the best way (if any) to represent the handedness of a cube in the drawn hologram?
4. If the two cubes added together have right handedness, does the resulting bigger cube also have right handedness? (Answer for left should be analogus…)
5. If the two cubes added together have different handedness, does the resulting drawing look like a cube twice as wide, twice as tall, and twice as deep with the same center?
6. Would the cube in 5 have a handedness? What would it be?
Talking about HoloLens? 🙂
Just to be sure you’re on the right track: ‘I’ in GA is not a cube. It is a volumetric entity (similar to liter) with a boolean handedness property. The volume is called ‘weight’.
An excellent program like GAViewer (http://www.cgl.uwaterloo.ca/smann/GA/gaviewer_download.html ) therefore has 2 representations for I: a cube and a sphere. The book from these authors describes how to use GA for working with 3d objects on a computer.
The above claim by Lockyer is manifestly wrong. Apart from being based on his previous mathematical mistakes I have brought out above, his claim contradicts the actual practice in physics experiments involving spin measurements. Substituting I.(-e_i) for beta in the mirror-world represented by the pair { beta, -e_i } has nothing to do with the laboratory-world of actual experimenters represented by the pair { alpha, +e_i }.
The pairs { beta, -e_i } and { alpha, +e_i } represent two unconnected, independent worlds — one inside the mirror, and the other outside the mirror. In particular, in geometric algebra the pair { alpha, +e_i } describes a counter-clockwise spin about the axis +e_i, whereas the pair { beta, -e_i } describes a clockwise spin about the axis -e_i. Therefore there is no way to distinguish the two spins without first translating the pair { beta, -e_i } to the pair { beta, +e_i }, as I have done in the appendix of this paper: https://arxiv.org/abs/1203.2529 .
On this point Richard Gill has agreed with me in the past, while disagreeing with Lockyer:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=226#p5872 .
Your chosen measure space S^3 has nothing to do with GA vectors. Each point of S^3 is represented by a magnitude 1 GA scalar-bivector or equivalently by a magnitude 1 quaternion algebraic element. The GA scalar- bivector basis forms a subalgebra of GA that stands on its own, independent of the GA vector and trivector bases. Once you have the proper basis multiplication rules for right and left orientation algebras, it really does not matter how you got there, you have the agreed to alpha and beta basis product rules which define the two possible orientations. S^3 is faithfully represented by quaternion algebra, and GA vectors have absolutely nothing to do with this algebra.
Your pairing {alpha, e_i} and {beta, e_i} ANY sign on the vector bases e_i has nothing to do with S^3, only alpha and beta have relevance. When you use the map function between alpha and beta representations alpha_i = -beta_i by direct substitution either direction, it does not make any difference whether you started with beta and ended up with alpha, or you could not care less about beta and only wanted alpha for your basis choice. So when you DID map beta to alpha as I said, ending up with two identical basis product rule equations in the first referenced “more careful and detailed explanation” you can’t legitimately say, well this one came from beta so it’s sign on e_i needs to be changed but the other identical equation does not need the same sign change because it started as alpha. That makes absolutely no sense even if you look the other way on the irrelevance of e_i to S^3.
This lack of adherence to mathematical logic is similar to your other mistake of making a simplification by assigning s1=s2=s to reduce a product to -1, then in a later dependent equation requiring s1 and s2 to tend in the limit to separate values a and b. Math does not work the way you need it to, and valid physics never requires bad math.
Sorry, but the bad math is yours as pointed out here,
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=226&hilit=lockyer%27s+error#p5859
You have double mapped the multiplication table which would lead one to the false notion that a left hand is physically equal to a right hand.
The thread that you started on your “forum” referenced here is certainly something everyone should read. I don’t think many people will conclude the math errors were mine. I guess you missed Christian’s post here where he said Richard Gill and I did not make any math errors over most of what was covered in this thread and stated numerous times as being my errors. I have already addressed his improper insertion of a minus sign in one of two identical equations in his argument after our agreed to math.
As for your odd claim I have “double mapped the multiplication table”, Christian set the measure space as S^3, and his model adds right and left orientation results of the same bivector product. The two orientations are defined by two multiplication tables, not one. What you may appear to be calling “double mapping” is nothing more than the following simplification adding the two orientation results properly indicated in (5) and (6) of my questions to Jay above, take just the _1 bivector as a demonstration
-cp_1 alpha_1 + cp_1 beta_1
= -cp_1 e2^e3 + cp_1 e3^e2
= -cp_1 e2^e3 – cp_1 e2^e3
= -2 cp_1 e2^e3
This uses nothing but simple math and the undeniable GA identity e3^e2 = -e2^e3. I do not see an alternative allowing the two cp_1 coefficients to be combined as Christian requires other than changing the e2^e3 to -e3^e2 instead and ending up with the equivalent simplification +2 cp_1 e3^e2. Certainly don’t end up with a zero result as he claims.
If you mean something different by “double mapping”, please explain.
Once again, Lockyer makes here the same math error that he has been making for the past six years. His error is aptly called “double mapping” by Fred Diether. It amounts to repeatedly proving that left-hand = right-hand. But I have already shown in considerable detail in the appendix of the following paper and the link that left-hand is not equal to right-hand:
https://arxiv.org/abs/1203.2529
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=301&sid=62eae96a7a915603ae6eeb6c32e9637d#p7613
I disagree.
PS: Just to be clear to the readers, I disagree with Lockyer, and fully agree with Fred Diether.
Joy Christian above objects vehemently to the “urn model” introduced as a pedagogical tool to understand local hidden variables theories. I think it would be helpful to ask why he thinks that his own theory is so different from an urn model. As I see it, you can just put a bunch of slips containing values of (e_o,s_o) in the urn, [the number of slips containing each value of (e_o,s_o) determined by some probability distribution], and on each slip you write down the corresponding values of A(a, e_o, s_o) and B(b, e_o, s_o) as calculated by (26) and (27) from his paper. Then, you draw a slip at random, and the measurement outcomes for a and b are read off from the slip. However you want to define “local hidden variables”, the fact is that Bell’s theorem applies to urn models, therefore (unless I am misunderstanding his purported model) Bell’s theorem applies to Christian’s model, therefore his claim to see Bell-violating correlations must presumably be due to a math error as alleged by various people above.
Here are my reasons for objecting to the “urn model”: http://philsci-archive.pitt.edu/12655/
As you can see from this paper that the error is committed by Bell and his followers, not me.
I have posted a more detailed abstract of the above paper elsewhere, which reads as follows:
“Bell inequalities are usually derived by assuming locality and realism, and therefore violations of the Bell-CHSH inequality are usually taken to imply violations of either locality or realism, or both. But in the appendix of this paper I have been able to derive the Bell-CHSH inequality by assuming only that Bob can measure along b and b’ simultaneously while Alice measures along either a or a’, and likewise Alice can measure along a and a’ simultaneously while Bob measures along either b or b’, *without assuming locality*. Violations of the Bell-CHSH inequality therefore only means impossibility of measuring along b and b’ (or along a and a’) simultaneously.”
I was not asking why you object to the “urn model” in general. After reading through all the above discussions (like the masochist that I am), it was abundantly clear that you have a different view from most people about what a local hidden variable theory should entail, and object to Bell’s theorem for that reason. But what I couldn’t see is why you think that your own model, in particular, is different from an urn model.
The fact that the CHSH inequality is predicted to be violated by QM shows that your own “derivation” of it must be incorrect, unless you think that QM is inconsistent. That is even disregarding the abundant experimental evidence for CHSH violation.
What is incorrect is your assertion that there is “abundant experimental evidence for CHSH violation.” It is very strange to think that something can actually violate the CHSH inequality. Nothing can violate the CHSH inequality — neither QM nor experimental results. Nothing whatsoever can violate any mathematical inequality. If you think otherwise, then please demonstrate, using only +1 and -1 numbers, how the CHSH inequality is violated.
My model is a model for a physical experiment, whereas urn model has nothing to do with physics. Moreover, my model is based on the geometry and topology of the physical space, represented by the Clifford algebra of orthogonal directions in physical space, whereas urn model is based on pure fiction.
My model makes the same prediction as QM, but does so using only local functions A(a, h) and B(b, h) defined by Bell.
Er, actually scratch that last paragraph, as I misunderstood the point you were making. The point remains that in an “urn model” CHSH is satisfied, and your theory is clearly an “urn model”. That you can derive CHSH from other assumptions does not mean that the original derivation was not valid. It certainly is correct for an urn model.
DE,
you hit the nail on the head. However, there is no need for an urn, because the model finally boils down to
A(alpha; lambda) = +1 and B(beta; lambda) = –1 for lambda = +1 ……………………….. (1)
A(alpha; lambda) = -1 and B(beta; lambda) = +1 for lambda = -1 ..……………………… (2)
(see equations (54) and (55) in “Local Causality in a Friedmann-Robertson-Walker Spacetime”)
From a physical point of view, the correlation E(alpha, beta) can thus be written as
E(alpha, beta) = 1/N * Sum[k:1..N]{ A(alpha; lambda_k)*B(beta; lambda_k) } ..………….. (3)
because lambda is now the only relevant free variable. At this point, everything is done. With (1) and (2) one gets for all combinations {alpha, beta}
E(alpha, beta) = 1/N * (N/2 * (-1)*(+1) + N/2 * (+1)*(-1)) = -1 .…………………………. (4)
You have done your calculation incorrectly. The correct calculation is done in my paper. See equations (54) through (80) of https://arxiv.org/abs/1405.2355 . The result of the correct calculation is clearly E(a, b) = -a.b. This result has also been independently verified using a geometric algebra based computer program. The code and a plot of which can be found at this link: http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=296#p7322 .
Equation (68) – (75) in “Local Causality in a Friedmann-Robertson-Walker Spacetime” are physically meaningless because s1 and s2 are redundant variables owing to the local-realistic constraint
s1 = s2 = s.
Physically, a term like
lim[s → alpha, s → beta]{ some function of s, alpha, beta and lambda }
makes no sense. It makes a big difference whether physical constraints are considered before doing some math or after doing some math!
The calculation in PM’s post is the same calculation the experimenters perform, and that is the only calculation that matters.
The calculation in PM’s post is simply wrong, and therefore it does not matter. It violates one of the most basic laws of physics, known since the time of Newton. As I have noted above, the correct calculation of the correlations between events A(a, h) = + or -1 and B(b, h) = +1 or -1 is done in my paper. The result of this correct calculation is E(a, b) = -a.b.
The calculation in PM’s post is simply the calculation the experimenters are using when computing the correlation. If they have been using a wrong calculation for the past 40 years or so, you should tell them.
Please provide reference to just one experimental paper which calculates the correlations exactly as done by PM in his post you are defending.
Joy, let’s suppose we take equations 77 and 78 of v8 of the retracted paper as axioms. Suppose we take a hologram film of your exploding balls experiment. How could an artist represent the two “spinning bivector” L’s on the left hand sides of the equation by adding animated drawings to the film? Which of 77 and 78 is “inside the mirror” in your terminology, and which is “outside the mirror”? How could an artist add animations to the film to represent unseen distinctions between the two mirror worlds to the film? (I guess your answer will be like “draw different barber pole swirls on the ball fragments”.) Are the a.b-D and a.b+D terms in the string of equalities results of the detection process? How would an artist draw a and b on the film? a.b is obviously related to the angle between a and b. What exactly are the detectors doing, and why does a.b end up as a term of the result? The D’s are related to the detector settings and a and b. How would an artist add animation to the film to describe the differt signs a.b-D and a.b+D turning up in the result of a measurement? How would you try to convince someone that mirror worlds exist?
Equation (54) and (55) in “Local Causality in a Friedmann-Robertson-Walker Spacetime” – classical “projections” which map some tailored functions to detector responses +/-1 – define exactly what an observer will observe when performing measurements:
In case lambda = +1 he/she would get
A(alpha; lambda) = +1 and B(beta; lambda) = –1 ………………………………………………… (1)
for all alpha and beta settings.
In case lambda = -1 he/she would get
A(alpha; lambda) = -1 and B(beta; lambda) = +1 ………………………………………………… (2)
for all alpha and beta settings.
Thus, one gets for all alpha and beta settings
E(alpha, beta) = -1
As I already mentioned, you are doing your calculation incorrectly. You are violating the conservation of spin angular momentum, and ignoring the twist in the Hopf bundle of S^3.
Then I would recommend that equations (26), (27), (54), (55), (57) – (60) and (61) – (64) are removed from the paper “Local Causality in a Friedmann-Robertson-Walker Spacetime”. Your arguments imply that these equations do not define the detector responses which would be observed in experiments in case your model is valid.
I reject your recommendation. A(a, h) and B(b, h) defined in my paper are detector responses.
According to the local-realistic model in “Local Causality in a Friedmann-Robertson-Walker Spacetime” we then have for the detector responses:
In 50% of the measurement cases
A(a, +1) = +1 and B(b, +1) = –1 ………………………………………………… (1)
for all a and b, and in 50% of the measurement cases
A(a, -1) = -1 and B(b, -1) = +1 ………………………………………………… (2)
for all a and b. Thus, one gets for the correlation
E(a, b) = -1 ……………………………………………………………………………….. (3)
When calculated correctly, the correlations according to the local-realistic model in “Local Causality in a Friedmann-Robertson-Walker Spacetime” work out to give E(a, b) = -a.b:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=296#p7322 .
There has been a lot of references by Christian and his supporters to left and right hands, mirrors, gloves, coin faces and the like. They have often used the differences in these physical objects as a justification of their position, even making rather bizarre claims I don’t know the difference or have proved opposite handed physical objects are the same, heads = tails etc.
Physical objects have limited utility in helping us understand something abstract like the orientation of a space, which is really what we are talking about here. In my experience I have never seen anyone desire combining the results created in one orientation of a space with results created in another orientation of that same space until Christian’s attempts. There is good reason for this, there is no utility doing so.
If we want a parallel to this notion of utility we need look no further than the force a magnetic field puts on a moving charged particle. From the position of the charged particle emitter and the detector we can easily visualize the invariant direction the particle travels, and by turning the magnetic field on and off visualize the invariant physical direction of the force when the field is on.
On top of this physically real system we can superimpose an xyz coordinate system space and have a choice between right and left orientation for the space, just as we have here. When the Physics of what is going on is Mathematically worked out in either orientation of the space, the particle is deflected in the very same physical direction, without needing to be repaired by inserting a minus sign as Christian did in his last offering.
So in a matter of speaking, right DOES equal left. The orientation choices for a space are free choices, and most people realize it is an arbitrary choice because the the results in one orientation are equivalent to the results in another. There truly is no reason to do more than pick one and consistently apply it.
Christian needs to abandon his fair coin choice of orientation model, the math just is not there to support his conclusions. He has been shown this six ways to Sunday, and I truly believe he has the math skills to understand what has been presented. After all, it is rather trivial.
At the end of the day what we are concerned about here is a local hidden variable theory. In particular, my 3-sphere model is a local hidden variable model. Therefore the sign issue mistakenly brought up by Lockyer, Gill and others for the past six years is a non-issue. We can simply define the local hidden variable h = +/-1 of my model by the ordering relation between the spin bivectors L(a, h) and L(b, h) and the detector bivectors D(a) and D(b):
L(a, h = +1) L(b, h = +1) = D(a) D(b) = (I.a) (I.b)
and
L(a, h = -1) L(b, h = -1) = D(b) D(a) = (I.b) (I.a).
Since the above relations are an intrinsic part of the very definition of the local hidden variable h of my model, substituting them into equation (72) of the paper in question, namely https://arxiv.org/abs/1405.2355 — i.e., substituting the above relations into
1/2 { L(a, h = +1) L(b, h = +1) + L(a, h = -1) L(b, h = -1) } ,
immediately leads to the strong correlation derived in (79):
E(a, b) = 1/2 { (I.a) (I.b) + (I.b) (I.a) } = -1/2 { a b + b a } = -a.b. Nothing more is needed.
Joy, that is equivalent to the suggestion I made in my most recent post above to take equations 77 and 78 of v8 of the retracted paper as axioms. In that post I ask questions to assess whether one should accept those axioms. Joy, would you answer thise questions? (For those who haven’t followed all the previous discussions, in Joy’s model some combinations of Alice and Bob’s settings can never occur with certain initial states, whick breaks it out of Bell’s framework. This aspect of the model leads to separate ongoing debates; see above.)
I have no idea how an artist would represent the two spinning bivectors, or whether that would help. L(+) and D are outside the mirror, and L(-) is inside the mirror. But “mirror” is just a metaphor for the left-handed or clockwise spins. It shouldn’t be taken too seriously. There are no shortcuts to learning Clifford algebra and quaternions. That is some 170 years old mathematics, with plenty of online tutorials, so it shouldn’t be too difficult to learn. Detection processes give only +1 or -1 numbers, as defined in the equations (54) and (55).
See Adrian Rossiter’s antiprism at http://www.antiprism.com/album/860_tori/index.html . Compare the spindle sphere with an s-orbital ( http://media-3.web.britannica.com/eb-media/54/3254-004-DAE1AA41.jpg ) but try to think of it as “the eye of the storm”, as it were. The electron is not some ball, it’s a space-filling standing-wave standing-field construct with a spherically-symmetric centre. And see Hans Ohanian’s What is Spin? here: http://people.westminstercollege.edu/faculty/ccline/courses/phys425/AJP_54(6)_p500.pdf . Spin is real.
A simple advice:
Supplement the paper “Local Causality in a Friedmann-Robertson-Walker Spacetime” with a plot of the predicted detector responses A(alpha; s, lambda) = +/–1 and B(beta; s, lambda) = +/–1 as a function of the angles between the detector settings alpha and beta and the spin orientation s. In case the proposed local hidden variable model is not capable of producing such a simple plot, every reviewer will classify the paper “Local Causality in a Friedmann-Robertson-Walker Spacetime” as physically irrelevant.
Those who are concerned about local hidden variable theories will find the essentials on the MathPages, entry “Quantum Entanglement and Bell’s Theorem” ( http://www.mathpages.com/home/kmath521/kmath521.htm ).
In the reference list of my paper you have cited all such plots and their simulation codes are provided. Two examples of such detailed information can be found at the following links:
http://rpubs.com/jjc/84238 (read through the details of this simulation, all the way down)
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=296#p7322 .
From the first paragraph of the link to the mathpages.com link: “Thus an electron manifests one of only two possible spin states, which we may call “spin up” and “spin down”.”
An entity that only exhibits two states, is COMPLETELY describable by a single bit of information. Single bits of information DO NOT HAVE MULTIPLE, OBSERVABLE COMPONENTS. All Bell-type theorems and experiments have mistakenly assumed that they DO HAVE MULTIPLE components, and then attempt to determine their NON-EXISTENT statistics. That is the problem.
All attempts to measure MULTIPLE COMPONENTS, of an entity that only has one, MUST result in “strange” correlations. If you bother to construct classical objects which manifest only a single, recoverable bit of information, it will be observed that these “strange” correlations obey the exact same statistics as the so-called “quantum correlations”.
See: http://vixra.org/pdf/1609.0129v1.pdf
In the light of some recent comments above I have updated my arXiv response to Gill by adding a second appendix: https://arxiv.org/abs/1203.2529 . The new appendix is only one page long. In it I have presented my local model in such a manner that no one will be able to artificially introduce a sign error in it by double-mapping and then blame me for it.
Equations (B10) to (B12) of this new appendix are also used in my local model that has been peer-reviewed and published in the International Journal of Theoretical Physics since 2015:
https://link.springer.com/article/10.1007%2Fs10773-014-2412-2 .
The arXiv version of the above paper can be found here: https://arxiv.org/abs/1211.0784 .
There is no accountability in science for wrongful retraction and the damage it sustains.
I have now published a paper on the physics arXiv that refutes the claim made by the Editors of Annals of Physics on the webpage of my retracted paper that “violation of local realism … has been demonstrated not only theoretically but experimentally in recent experiments…”:
The Paper: https://arxiv.org/abs/1704.02876 .
Its Abstract: “Bell inequalities are usually derived by assuming locality and realism, and therefore violations of the Bell-CHSH inequality are usually taken to imply violations of either locality or realism, or both. But, after reviewing an oversight by Bell, in the Corollary below we derive the Bell-CHSH inequality by assuming only that Bob can measure along vectors b and b’ simultaneously while Alice measures along either a or a’, and likewise Alice can measure along vectors a and a’ simultaneously while Bob measures along either b or b’, without assuming locality. The violations of the Bell-CHSH inequality therefore only mean impossibility of measuring along b and b’ (or along a and a’) simultaneously.”
A comprehensive and much-expanded version of my paper that was retracted by Annals of Physics is now published by the Royal Society journal Open Science (RSOS):
http://rsos.royalsocietypublishing.org/content/5/5/180526