## Physics journal retracts paper without alerting author

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Christian received a response from Marc Chahin, an executive publisher at Elsevier, saying that his paper had been withdrawn from the journal, adding: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

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 yablon@alum.mit.edu.

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.”