A physicist who in 2016 threatened to sue Elsevier after the publisher retracted one of his papers has lost another article over concerns about the “integrity of the mathematics” in the paper.
The article, “Eight-dimensional octonion-like but associative normed division algebra,” by Joy Christian, of the Einstein Centre for Local-Realistic Physics in Oxford, UK, appeared in Communications in Algebra in July 2020. According to the notice:
Since publication, concerns have been raised about the integrity of the mathematics in the article. The main error is visible not only in the title, but in the abstract which claims “We present an eight-dimensional even sub-algebra of the 24 = 16-dimensional associative Clifford algebra Cl4,0 and show that its eight-dimensional elements denoted as X and Y respect the norm relation
∥XY∥=∥X∥∥Y∥, ‖XY‖=‖X‖‖Y‖, thus forming an octonion-like but associative normed division algebra ⋯.” In short, the author claims to have found an 8-dimensional normed division algebra over R,R, but there is no such thing. Hurwitz’s theorem 1 says all 8-dimensional normed division algebras over the reals are nonassociative and isomorphic to the octonions. This famous result, published in 1923, has been confirmed with a number of well-established proofs. Thus, based on these factors the Editors assess the author’s main result to be false.
The Editors have discussed this with the author, and sought additional review of these points of concern. As the main result cannot be relied upon, the Editor and Publisher are retracting the article. The author has been notified of the retraction and does not agree with this decision.
Christian has been active on PubPeer, defending his work and taking shots at his most vocal critic, Richard Gill, an emeritus professor at Leiden University in the Netherlands. Gill has pushed for the retraction of several of Christian’s papers, and the pair have waged a public-ish battle on message boards for their field.
In a post this week on PubPeer, Christian wrote:
I strongly disagree with the retraction of my paper by Communications in Algebra because the journal has not provided any evidence of a mathematical mistake in my calculations.
Misled by Richard D. Gill (see my previous post), John C. Baez put tremendous pressure on the journal via his blog, by wrongly claiming that my “main result must be false” without providing any evidence for his claim. In fact, my main result is rather trivially correct and does not conflict with any known theorem. Moreover, to date, no one has found a mistake in the calculation of my main result …
And yet, badly misled by Richard D. Gill, John Baez has claimed that my “main result must be false”, and the journal has accepted his knee-jerk reaction without providing any evidence of a mistake in my calculations …
Christian accused the journal of having “simply buckled under political pressure” and that the retraction was a “travesty of justice” — which he addresses in this document.
And, he added:
I intend to publish the paper again in a journal of higher standing and greater scientific integrity.
Christian also has this 18-month-old expression of concern for a 2018 article in Royal Society Open Science:
Following the publication of ‘Quantum correlations are weaved by the spinors of the Euclidean primitives’ R. Soc. open sci.5, 180526 published 30 May 2018 (http://dx.doi.org/10.1098/rsos.180526), it has been brought to the attention of the Editorial team that there are concerns regarding this study.
A number of readers have written privately via email and also in extensive public correspondence, which is available via the ‘Comments’ section of the original article, raising a number of concerns regarding fundamental aspects of the work.
An investigation into these aspects is under way, and the journal is therefore issuing an expression of concern and will notify readers as to the results of our investigation as soon as possible.
Nine months ago, a representative for the journal posted a comment that the editors were “nearing a resolution” on what to do about the article.
To which one reader — perhaps a mathematician — quipped: “Asymptotically?”
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I always find retractions in mathematics journals fascinating. First, because it should be possible to completely evaluate the paper during peer review (there can’t be any photoshopped figures or experiments left out), so it is a real benchmark of the effectiveness of peer review. Second, it undermines the (somewhat deserved, somewhat undeserved) reputation of mathematics as “objective” and “rigorous”–again because of deficiencies in peer review.
I think that the main issue with peer review in mathematics, and many other fields, is the pressure to publish. People publish more and more and have less and less time for peer review.
Anyhow, even with good peer review, sometimes the error in a math paper is something very subtle which is not easy to spot, especially when it is one line in a 85 pages paper. And in many cases, mathematics research has became so specialized that it is very hard to find a reviewer which is as close to the topics of the paper as the author. And in that case, if the author is missing something very subtle, it is likely that the reviewer will also miss it.
I do agree that the peer review is becoming more and more deficient (and not only in math) which is doing a lot of damage.
It’s true that in mathematics errors are sometimes very hard to spot, but in this case the retracted article claims boldly in its title something that contradicts a rather well-known theorem (something that, for example, a grad student specializing in algebra should know). So in this case, the referee(s) didn’t do their job very well.
there can’t be any photoshopped figures
Hold my beer.
https://eusa-riddled.blogspot.com/2020/06/operators-and-things.html
In my personal experience attempting to correct far, far simpler mathematics errors (undergraduate level stuff) in mathematical papers has also yielded extreme resistance (from authors, editors, and online trolls). Somehow, a poorly reviewed paper (or sometimes fraudulently reviewed — gaming the review process, as discussed on RW) becomes the immutable word of a deity once published by a mediocre journal. Then, when one attempts to point out a clear error (backing it up with rigorous evidence and so on) becomes “sus” (to use a modern colloquialism). It’s unbelievable what lengths, and how much energy, some people will go and expend on erroneous work. In mathematics, there is no gray areas, you either made the logical deduction from the underlying axioms or you didn’t!
Andrew Gelman calls this the Research Incumbency Rule: “Once an article is published in some approved venue, it is taken as truth. Criticisms which would absolutely derail a submission in pre-publication review can be brushed aside if they are presented after publication.”
https://statmodeling.stat.columbia.edu/2009/05/24/handy_statistic/
It may be worth pointing to John Baez’s blog post, where he notes that Christian’s “result” contradicts a well-known theorem and says (in the comments) that the journal had been contacted before he was.
https://golem.ph.utexas.edu/category/2020/09/new_normed_division_algebra_fo.html
During that discussion, we did get a bit into specific places where Christian’s “mathematics” appears to become wishful thinking. Also in the comments there is a pointer to an open letter that carries on the drama elsewhere:
https://www.iqoqi-vienna.at/blog/dishonesty-in-academia-the-deafening-silence-of-the-royal-society-open-science-journal-on-an-accepted-paper-that-failed-the-peer-review-process
My result does not contradict any theorem, well-known or otherwise. John Baez does not know what he is talking about. There is no “wishful thinking” in my mathematics. The calculations are all straightforward and perfectly compatible with Hurwitz’s theorem. Since the retraction, I have been reassured of that fact by a very prominent expert in normed division algebra. So Baez, his online gang, and the editors of the journal are all simply wrong. And the “open letter” you have linked is false propaganda. The author of that blog post claims that he was a referee of my RSOS paper. He wasn’t. He isn’t even qualified to be a referee of any of my papers.
Baez is not the only one who sees it. You really have contradicted Hurwitz Theorem, so the burden of proof is on you: show us where Hurwitz got it wrong.
There is no contradiction between my result and Hurwitz theorem, or any other known theorem for that matter. The burden of proof is indeed on me, and I have provided that proof in my paper, which is freely available from the arXiv:
https://arxiv.org/pdf/1908.06172.pdf
The only thing is that one has to actually read my paper.
I just read it. Your algebra is isomorphic to a direct sum of two copies of the quaternions, projected out by (1 ± I₃ e∞)/2. This algebra has zero divisors, and multiplication doesn’t respect the X·X† norm, as you seem to realize. You try to fix that with the condition (A2) which is equivalent to the two quaternions (not qr and qd but the projections) having the same norm. Unfortunately this subspace (aside from being of dimension 7 and not 8) isn’t closed under addition.
You don’t seem to understand that when you decide standard definitions aren’t Clifford-algebraic enough and change them, you aren’t doing the same thing any more, and you can’t truthfully claim in the abstract to have found an instance of the standard thing. You need to give it a new name, like “Clifford-normed division algebra” or something. But you haven’t found that either, as far as I can tell, since your K^λ doesn’t have division without (A2) and doesn’t have addition with it.
The other paper of yours I read had a similar problem. You argued Bell’s experiment wasn’t Clifford-algebraic enough and changed it so the measurement outcomes were Clifford objects (vectors I think) instead of discrete up/down. The problem is Bell proved one specific experiment incompatible with local hidden variables, and it wasn’t yours. Many experiments are compatible with local hidden variables, and even if everything in the paper was correct, all you did was add another one to that list.
Your observations are not correct. My algebra is a tensor product of quaternions and split complex numbers: H x C’. For this algebra, the composition law holds:
|| X Y ||^2 = ||X ||^2 || Y ||^2,
as I have proved in Eq. (B.14) of the arXiv version of the paper. Because the coefficient algebra involves split complex numbers, the algebra does not conflict with Hurwitz’s theorem, which holds only for the real coefficient algebra.
As for Bell’s theorem, it is a non-starter. But that is another and a very long story:
https://arxiv.org/pdf/1704.02876.pdf
C’ ≅ R + R so H × C’ ≅ H + H.
Joy says that Florin Moldoveanu, author of a blog post on the related RSOS paper, wasn’t a referee of that paper, and wouldn’t be qualified to referee any of Joy’s papers. Florin did referee that paper. I was asked to, but turned down the opportunity (though I did point out the PubPeer debate on the arXiv version). I suggested that the editors should ask Florin, who was perhaps the only serious mathematician at that moment who had gone into Joy’s crazy arguments and posted a careful analysis on arXiv. Since Joy’s early preprints on this topic were never published in journals (they became chapters of a book by him), most early critics’ arXiv refutations were never published.
If you think this is something, do a bit of research on the purported proof of the ABC Conjecture. That controversy has been going on for years, with winners of the Fields Medal involved. Consensus is now that the proof is wrong, but the proposer and a number of his supporters do not accept that.
For the record, the following facts about this retraction should be noted:
Just one individual, Richard D. Gill, who has no expertise in Algebra or General Relativity on which my papers are based, is responsible for trying to have five of my papers retracted.
1) In 2016 he succeeded in having one of my papers retracted from Annals of Physics. That paper is now re-published by IEEE Access:
https://ieeexplore.ieee.org/document/8836453
2) Gill again tried to have the above paper retracted, from IEEE Access, this time by claiming that I have plagiarised one of his computer programs. So far, even after a year, the journal has not contacted me about Gill’s complaint or retracted my paper.
3) In 2018 Gill succeeded in having another paper of mine, published by the Royal Society journal Open Science, earn an Expression of Concern. After two years of investigation, the journal has just published a Final Comment on their investigation and decided not to retract my paper:
https://royalsocietypublishing.org/doi/10.1098/rsos.201777
4) The current paper, published in Communication in Algebra, is the second paper of mine Gill has succeeded in having retracted. I have submitted the retracted article to another journal of higher standing. In the meantime, it can be downloaded from the arXiv:
https://arxiv.org/pdf/1908.06172.pdf
5) Gill has threatened to have two more of my forthcoming articles retracted, so watch this space!
More details can be found on some ten threads started by Gill on PubPeer about my papers. I admire the amount of time (some ten years) and energy Gill has devoted to my papers.
I wonder if Joy Christian would care to explain his position at the University of Oxford as described on his LinkedIn, FaceBook page, Blog page and Researchgate page c.f.
https://uk.linkedin.com/in/joy-christian-oxford
Quote “I am a theoretical physicist based at the University of Oxford (UK), working on the conceptual foundations of quantum and gravitational physics.”
https://www.facebook.com/joyjch
Quote “University of Oxford, Research Physicist · 2014 to present · Oxford, Oxfordshire United Kingdom
http://libertesphilosophica.info/blog/lpmain/
Quote “I am a research physicist based at University of Oxford (UK), working on the foundations of quantum and gravitational physics.”
https://www.researchgate.net/profile/Joy_Christian
Which seems to imply that the “Einstein Centre for Local-Realistic Physics, Oxford, University of Oxford”
The University of Oxford contact directory has no evidence of a Joy Christian being a member of the University
https://www.ox.ac.uk/search or here
https://staff.admin.ox.ac.uk/
Nor does the Department of Physics
https://www2.physics.ox.ac.uk/contacts
Wolfson College doesn’t list a Joy Christian either
https://www.wolfson.ox.ac.uk/search/results
Dear “Watcher”, I would be quite happy “to explain [my] position at the University of Oxford” to you or anyone else. Just drop me an email or send me a regular post at my publicly available addresses and I will be happy to send you all the “explanation” you want in the return mail.
Why not explain here in the comments so everyone can benefit from the knowledge?
Joy, I think many of us will start to question the credibility of the work of any researcher who is perceived as possibly misleading the public about his/her purported institutional affiliations. In the present context, there should be no need for lengthy explanations. Are you _currently based_ at the University of Oxford, yes or no? If no, were you ever _formally affiliated_ with that institution? If yes to either question, a very brief explanation of the nature of your affiliation would be really helpful to your cause.
The answers to both of your questions are: Yes, and Yes. I have been affiliated with the University of Oxford, in particular with Wolfson College and a number of Departments of the University, in various capacities, including (in the past) as a member of its Governing Body (called “the Congregation” in Oxford), for the past 29 years, starting from the 1st of October 1991. If anyone has any doubt about this (and I am fully aware of who has been spreading false rumors about me), then go ahead and ask the Nobel Laureate Prof. Sir Roger Penrose, OM, FRS, whether what I have written above is true or not. Would that be enough? Or do I also need to provide President Obama’s birth certificate to convince people?
What is the nature of your current affiliation?
“The Congregation” has 5,500 members.
Joy, that first Yes, is interesting and you were not clear by carefully using the words “been” and “in the past”.
So what department or college are you currently a member of at the University of Oxford. That was my question. I can see that you were a member, but your social media profiles imply you still are a member of the University of Oxford.
The University contact search does not find you listed in any capacity and a google crawl of the University website, only mentions a last connection back in 2011 .
Actually, Joy Christian’s name does not appear on the Register of members of the Congregation in at least:
2012 (https://gazette.web.ox.ac.uk/files/revisedregisterofcongregation1tono4976pdf)
2015 (https://gazette.web.ox.ac.uk/files/revisedregisterofcongregation-1tono5084pdf)
2016 (https://gazette.web.ox.ac.uk/files/revisedregisterofcongregation-1tono5120pdf)
2019 (https://gazette.web.ox.ac.uk/files/revisedregisterofcongregation-1tono5228pdf) I did not have the patience to find the list of members for other years to confirm he has indeed appeared there in any of the intervening years
The absence of evidence is not evidence of absence.
Maybe you should ask Richard D. Gill why is he so devoted to investigate one and only one mathematician for the last 10 years. What’s going on between these two people? I’m not a mathematician but I can smell rotten relationships from miles.
I’m happy to explain, Robert. I first met Joy Christian at a workshop in Berlin in (I think) 2010. It was so sad to see this very earnest young person (well, ten years younger than me) giving a talk which was so obviously total nonsense. Most participants of the workshop (run by the “Young Academy” of the German Academy of Sciences) proceeded to ignore Christian for the rest of the conference. I felt sorry for him. How he had been invited is also a mystery to me. Some influential German professor must have been taken in. JC assumed that the outcomes of a Bell-type experiment were unit length pure quaternions instead of the numbers +/-1, and so his own new “correlation” was pretty meaningless, and his whole mathematical exercise was totally irrelevant to Bell’s theorem and to Bell-type experiments.
A short while later, I was visiting the Oxford University physics department, and I thought I would try to meet him, to get some more clarification. I was interested in the possibility that Geometric Algebra could give new insights into QM, even if these particular papers were garbage. I let JC into the physics department on a Saturday morning – he didn’t have a key. A bit strange, for someone who claimed to be a member of the department (he had no explanation). But he gave me a delightful tutorial on Geometric Algebra. We parted on good terms. On the way home I read his “one-page paper” which had just come out, and immediately found a sign error. So not only was the whole enterprise scientifically irrelevant, but it was also mathematically flawed. I put out an analysis of his short paper on arXiv, and sent it to him too, of course. He got furious. My analysis was useful because till then, it seems that no-one had noticed that the newest version of his model definition was completely crazy (it changes every few years). The measurement outcomes were now always equal and opposite, equal to +/-1, whatever the measurement settings, so the correlation was always -1, and not the negative cosine which is predicted by quantum mechanics. I was however impressed by his style, and he normally is a rather nice guy, and very erudite too. Lucien Hardy (a well-known person in quantum information and a close friend of JC) told me that JC often has amazingly deep physical insights. I found it really sad that such a talented and interesting person should totally destroy their own academic career by embarking on this fool’s quest.
I already was deeply interested in the psychology of “Bell deniers”. How can quite a few, so intelligent and knowledgable, well-established scientists, convince themselves that such a simple theorem is false? How is it that top physics people in theoretical electronics, or top people in quantum chemistry, or whatever, as well as some famous mathematicians, can convince themselves that this simple maths is wrong and put their academic reputation on the line, in order to come up with a counter-example to the theorem and a whole alternative theory? So Bell’s theorem exerts an unholy fascination on a lot of people. They go into denial and become evangelists on a mission: destroy Bell’s theorem.
And there is another question: how can pretty serious journals be tricked by this charlatan into publishing huge papers which, if correct, would be a bombshell under modern physics? Most of the professors of quantum physics would have to be fired and all the textbooks would have to be rewritten. You can forget the quantum computer, and the massive investments in it made by Microsoft, Google and others.
I too can explain, Robert. But my explanation will not add anything to what must be so obvious to any psychologist; so I won’t bother.
As for Bell’s so-called theorem, it is a non-starter. If anyone is interested in why it is a non-starter, then they can find my answer here:
https://arxiv.org/pdf/1704.02876.pdf
Despite appearances, I actually have no interest in Bell’s “theorem.” Since it is so deeply flawed argument, it is irrelevant for physics. What I am interested in is explaining quantum correlations in such a manner that their explanation becomes compatible with Einstien’s theory of gravity, leading us to a theory of quantum gravity. This is what my work of the past thirteen years is all about, including what is presented in the wrongly retracted algebra paper. Towards this goal, I have been able to publish the following four papers:
1) Macroscopic Observability of Spinorial Sign Changes under 2pi Rotations:
https://link.springer.com/article/10.1007/s10773-014-2412-2
2) Bell’s Theorem Versus Local Realism in a Quaternionic Model of Physical Space:
https://ieeexplore.ieee.org/document/8836453
3) Dr. Bertlmann’s Socks in the Quaternionic World of Ambidextral Reality:
https://ieeexplore.ieee.org/document/9226414
4) Quantum correlations are weaved by the spinors of the Euclidean primitives:
https://royalsocietypublishing.org/doi/10.1098/rsos.180526
This achievement is despite the constant interference and constant efforts to retract my papers by Richard D. Gill for the past ten years. Thus it is quite a success story.
C’ ≅ R + R so H × C’ ≅ H + H.
If the algebra under discussion is indeed A = H (x) C’, where H is the real quaternion algebra, C’ is the split complex algebra R (+) Rj, j^2=1, and the tensor product is taken over R, then various things follow.
1) A is an 8-dimensional real algebra, and so the Hurwitz theorem applies: A cannot be an associative division algebra.
2) A is a tensor product of associative real algebras, and hence A is an associative real algebra.
3) A contains a copy of C’, which is not a division algebra, since 1+j is a non-trivial zero divisor, (1+j)(1-j) = 0, and hence A is not a division algebra.
I should correct (3) slightly:
3) A contains a copy of C’, contains non-trivial zero divisors, since 1+j is a non-trivial zero divisor, (1+j)(1-j) = 0, so A contains non-trivial zero divisors and hence is not a division algebra.
I do not agree with your # 1) and # 2), Richard Pinch. Hurwitz’s theorem does not apply to the algebra presented in my paper, and there are no non-zero zero-divisors. But I was not careful enough to point out here that C’ is not quite split complex numbers. The square of the norm XX* takes the following form:
XX* = (a scalar) + (a scalar) x I,
where I, with I^2 = 1, is what is usually called a pseudoscalar in Geometric Algebra (GA). The important point is that, unlike split complex numbers, the GA conjugate of XX* does not pick up a minus sign. Thus the underlying coefficient algebra resembles that of split complex numbers, but it is not quite the same. The upshot of this is that there are no non-zero zero-divisors in algebra and Hurwitz’s theorem does not apply. The devil is in the details.
Notice how Christian replies. This is canonical JC style. He does not show where Richard Pinch’s argument is wrong. Instead he goes off into a rambling and technical digression which sports impressive looking chit-chat. What the GA conjugate of XX* is, is totally irrelevant. (He’s right about it; but so what?). What he means by “underlying coefficient algebra” is unclear. His algebra is defined “over the reals”. It is exactly the algebra which you and Ben say it is (and he even writes that it is the even sub algebra of Cl(4, 0)(R) hence isomorphic to Cl(0, 3)(R) in his bigger paper in RSOS of which this one is an extract). I guess he dropped that remark when writing for the big audience of Royal Society Open Science.
This rather omits the question: is the underlying algebra we are talking about, call it A for convenience, isomorphic to H (x) C’ as a tensor product of real algebras with C’ = R (+) RI with I^2 = 1 or is it not? I’m not talking about any norm you wish to impose on it at this point, merely the real algebra structure. If it is, then it’s an 8-dimensional real algebra, and as a real algebra it contains an isomorphic copy of C’, which has zero divisors. And if it has zero divisors, then it is not a division algebra.
So the two statements
4) A, as an algebra, is (isomorphic to) H (x) C’ as a tensor product of real algebras with C’ = R (+) RI with I^2 = 1
5) A is a division algebra
cannot both be true. (4) is what was asserted above. If (4) is correct, (5) cannot be correct.
Note here that I’m not discussing any question about norms, just the real algebra structure.
Yes, the underlying algebra A is isomorphic to H (x) C’ as a tensor product of H and C’ = R (+) RI with I^2 = 1.
For this algebra the composition law holds:
|| XY ||^2 = || X ||^2 || Y ||^2,
despite the fact that XX* etc. are not real but resemble split complex numbers. I then reduce the above composition law to the norm relation
|| XY || = || X || || Y ||,
as a special case, where all quantities are real, by imposing a certain normalization condition. That condition removes the zero-divisors from the algebra. Thus the final product is a normed division algebra. That is the construction.
Richard Pinch, you can find the answer to your question in Christian’s RSOS paper, in which he writes out the multiplication table for a basis of 8 elements of Cl(0, 3)(R). His Communications in Algebra paper is just one or two central sections from that paper which can be read “stand-alone”. The underlying algebra we are talking about *is* the even sub algebra of Cl(4, 0)(R). As he himself states explicitly. He even goes on to define the norm as the usual L^2 norm on R^8 and claims it is multiplicative. The proof of that fact has some obvious errors, of course. They were pointed out almost immediately after the arXiv original of the RSOS paper came out. By the way, that same paper had briefly been accepted in a formerly prestigious physics journal (International Journal of Theoretical Physics; most of the editorial board are elderly Nobel prize recipients), but inexplicably vanished due to a clerical error. Joy Christian is persistent, and I suppose he has some influential supporters, too.
The norm (or whatever you want to call it) is irrelevant. There are two statements
(6) the underlying algebra A is isomorphic to H (x) C’ as a tensor product of H and C’ = R (+) RI with I^2 = 1.
(7) That condition “removes” the zero-divisors from the algebra.
(8) What remains is a normed division algebra
(9) of dimension 8
Statement (6) appears to be common ground. Statement (7) makes no sense. If “that” condition, whatever may be, “removes” certain elements, such as 1(x)1 + 1(x)J, which are zero-divisors, then we are no longer talking about A but about some subset of it. The algebra A, as defined by (6) contains zero-divisors. To “remove” them means to replace A by some subset, not made precise, but which I shall call B. So the discussion appears to be about two different things, A, which is a well-understood 8-dimensional real algebra which contains zero-divisors and hence is not a division algebra; and B which is a subset of A, claimed to be an associative division algebra.
Furthermore, B is a proper subset of A, and so has dimension less than that of A, which is 8. So B is not an 8-dimensional algebra.
Well, let’s work out what B is. A contains a copy of C’ = R(+)RI with I^2=1. B contains an identity 1 = 1+0i+0j+0k (x) 1+0I. We are told that B does not contain any zero divisor, so B does not contain any element W = 1+0i+0j+0k (x) x+yI, with y non-zero, because if it did it would contain 1 + (W – x1)/y = 1+0i+0j+0k (x) 1+1I which is a zero-divisor. So all the elements of B lie in H (x) R which is just H.
So B isomorphic to a sub-algebra of H. In summary, the only division algebras consistent with the condition of being a subalgebra of A are in fact subalgebras of H.
In summary, the construction does not yield an
… 8-dimensional real associative division algebra.
This discussion is not helpful. You wish to discuss in terms that are well defined to you but they do not correspond precisely enough to what I have presented in my paper. What I have presented is an 8-dimensional algebra. It is an associative Clifford algebra. In fact, it is an even subalgebra of Cl(4, 0), which happens to be isomorphic to H (x) C’, where C’ resembles split complex numbers. Then, the questions I have answered in my paper are the following:
1) Does the composition law hold for this algebra? The answer is Yes.
2) Does the norm relation hold for this algebra? The answer is Yes, under certain conditions.
3) Do these conditions change the dimensions of the algebra in any way? The answer is No, they do not.
4) Are there non-zero zero-divisors in the algebra thus constructed? The answer is No.
The details can be found here: https://arxiv.org/pdf/1908.06172.pdf
These statements cannot all be correct. You have identified the algebra as isomorphic to H (x) C’. This algebra contains non-trivial zero-divisors, one of which has been exhibited. No discussion about “norm conditions” can change this elementary fact. In that sense, I too would say this this discussion is not helpful.
Richard Pinch, I find the discussion very heartening, and am very grateful for your efforts. You express more eloquently than me problems with this paper which I had already pointed out to the author and posted on PubPeer. I hope your efforts will help the journal “Royal Society Open Science” to understand the mess they have got themselves into by publishing the paper “Quantum correlations are weaved by the spinors of the Euclidean primitives” by Joy Christian (2018) https://royalsocietypublishing.org/doi/10.1098/rsos.180526 . That paper is built on top of the algebra result we are talking about here. RSOS has added an “expression of concern” but seems still to think that by publishing this paper they are contributing to open scientific debate on controversial issues. They also think that five scientists (not including myself) who wrote in to them, arguing that the paper should be retracted, should instead have submitted “comment” papers, which of course would all be responded to and “refuted” by J.C. This is what he does. Thus, yet more noise, on top of noise which they have created themselves. I think they should hold an internal investigation into how their paper got past peer review, and the only discussion about the RSOS paper which is worth having, is about the increasing failure of peer review in modern science publishing. Apparently, fatal mistakes in pure maths in a physics paper are now merely “alternative facts”, and a matter of opinion.
It is of course also amazing that the algebra paper, which is the original subject of this thread, got published in a fairly decent pure maths journal.
RSOS is not as insecure a journal as Annals of Physics and Communications in Algebra where Gill has had some success of spooking the chief editors. Unlike the latter two journals, RSOS did not buckle under the pressure put on them by Gill and his friends. Neither did they deviate from their scientific duty. They rightfully took the complaint made by Gill and his friends seriously and conducted two years of a very thorough investigation into the validity of the results presented in my paper. At the end of their investigation, they have decided not to retract the paper.
My algebra paper is also flawless. Just because someone on the Internet said this or that about my paper does not make the argument presented in my paper invalid. No one has found a mistake in my paper because there is none. Consequently, my RSOS paper is on very secure grounds.
“You wish to discuss in terms that are well defined to you but they do not correspond precisely enough to what I have presented in my paper.”
This comment seems significant. The terms which I am using are those which are completely standard in pure mathematics. If they do not correspond precisely to the terms used in the paper under discussion, then that paper is using standard mathematical terminology in a non-standard way, which is surprising to say the least.
“Just because someone on the Internet said this or that about my paper does not make the argument presented in my paper invalid”
Indeed, it is the content of what was said rather than the standing of the person saying it or the venue in which it was said that one should focus on. Speaking as “someone on the Internet”, albeit someone who is a professional mathematician posting openly under their own name, though, I feel confident that that my comments are sufficient to invalidate the claim that H(x)C’ is an 8-dimensional real algebra without zero divisors. Any argument that purports to establish that claim that must therefore be invalid. When someone sends me a preprint containing long and complex calculations which purport to show that 2+2=5, I do not need to find the exact mistake, because I know that there must be one. Indeed, my experience of reading unorthodox manuscripts in my own specialist field tells me that quite often there is nothing in such arguments to point to as “the” mistake, because there is not an argument as such, merely a sequence of assertions.
You have not done what you claim to have done. You have invented your own algebra and invalidated it. That is nice, but that has little to do with what I have presented in my paper. To date, no one has invalidated the construction I have presented in my paper. On the other hand, at least one authority on normed division algebras has found it worthy enough to comment on it constructively (albeit privately), and that has led to my adding Appendix B in the paper, thereby consolidating the result I have presented in the main text:
https://arxiv.org/pdf/1908.06172.pdf
“You have invented your own algebra”
That is false. I have used nothing but the terminology and results of standard algebra such as would be understood by any student of the subject, familiar with, say Lang’s Algebra. As such, it refutes the claim, explicitly made here (“Are there non-zero zero-divisors in the algebra thus constructed? The answer is No.”) that H (x) C’ is an 8-dimensional real algebra without zero-divisors. I have exhibited a non-zero zero-divisor in this algebra. What could be clearer than that?
As a followup, using the notation of https://arxiv.org/pdf/1908.06172.pdf and following the multiplication table of “TABLE I: Multiplication Table” in that preprint, the product (1 + lambda I_3 e_infinity)*(1 – lambda I_3 e_infinity) is rather clearly equal to zero.
Incidentally, I’m not disputing that the “norm” (actually a C’-valued quadratic form) is multiplicative: it clearly is, using Euler’s four-square identity with coefficients in C’. The false claim is that H(x)C’ is free of zero-divisors.
Why that is not a valid argument is discussed at length in Appendix A.
Ah yes, Appendix A, which is headed “Illustration of How the Definition (b) of the Norm Precludes Zero Divisors”. Formula A5 explicitly presents non-zero zero-divisors, X and Y and immediately following (A6) it is stated that ||X|| = ||Y|| = 1 but ||XY|| = 0. So in (A7) This is using (24) to compute ||X||.
Now, this is obviously a problem for you, since you want ||.|| to be multiplicative, as in equation (10), and yet here is a counter-example. The resolution comes from (23), where it is correctly recognised that the quadratic form on H(x)C’ obtained from the four-squares theorem is a C’-valued form. The norm ||.|| is then arbitrarily defined to be a real number by using the “normalisation condition” which essentially says that the pseudo-scalar part of the quadratic form (as seen in (21)) can be neglected. Definition (b) then states that the norm ||.|| is simply the real part of (21) whether or not the pseudoscalar part is zero.
And this is the mistake. The quadratic form which is multiplicative is C’-valued, not R-valued. Restricting it to a subset of the algebra on which it is R-valued (“normalisation”) means that there is no longer an algebra structure, so you don’t have a positive definite normed algebra, to have a positive definite norm on something which isn’t an algebra. Arbitrarily modifying it by dropping the pseudoscalar part (24) means that it is no longer multiplicative, as (A7) plainly shows. As the first part of Appendix A shows, you can have a definition of a norm which has multiplicativity on all of the algebra but which is not definite, or a different definition of the norm which has positive definiteness on a proper subset of the algebra but which is not itself the whole algebra, indeed not even closed under addition.
You can’t have both.
In fact, Appendix A is simply circular reasoning. You take elements which don’t satisfy the normalisation condition (A6) and derive a contradiction from the assumption of multiplicativity of ||.|| to demonstrate that they don’t satisfy the normalisation condition. Actually, what you should be deducing from that contradiction is that on elements which don’t satisfy the normalisation condition you don’t always have multiplicativity of ||.||.
Definition (b) is not arbitrarily chosen. It comes from using Geometric Algebra correctly rather than taking the scalar part of the geometric product as the norm and ignoring the non-scalar part.
The definition then consistently allows recovery of the norm relation (10) or (49) from the general composition law proved in (B14) that retains non-scalar parts. The resulting structure is a beautiful algebraic structure if one has the eye to see it.
To be sure, it imposes the constraint (53) on the scalar coefficients. That is the price that must be paid for retaining associativity in eight dimensions. This is a very welcome tradeoff from a physics and engineering point of view. Mathematicians will come around eventually.
Finally, the ad hoc coefficients you have arbitrarily chosen in your counterexample apparently showing non-zero zero-divisors are not permitted by the constraint (53). That is why I have stated in my previous posts that there are no non-zero zero-divisors in the construction I have presented in my paper.
“To be sure, it imposes the constraint (53) on the scalar coefficients.” and “not permitted by the constraint (53)”.
These tell us that you are excluding certain elements of the algebra H(x)C’ and restricting yourself to some proper subset. So you don’t have an algebra isomorphic to H(x)C’ after all. In fact, since you have restricted yourself to some subset of it, satisfying this condition (53), you haven’t even proved that it is a real algebra at all, and even if it were to turn out to be one, it would not be 8-dimensional, since a proper subalgebra of a finite dimensional algebra has strictly smaller dimension.
I have pointed out several times in my posts above that the algebra I am talking about is not quite H(x)C’ and C’ is not quite the C’ you have in mind. The algebra is nevertheless eight-dimensional, as I have shown in the paper. Whether it can be called an algebra over the reals is debatable, but that is not a claim you will find in my paper. It is an eight-dimensional associative algebra that satisfies the composition law (B14) and the multiplicative norm relation (49) (as explicitly proven), and that is all that has been claimed in my paper.
You have said “My algebra is a tensor product of quaternions and split complex numbers: H x C’.” When you say that “the algebra I am talking about is not quite H(x)C’” this is a contradiction.
You have said C’ “resembles” split complex numbers. Well, that is again a contradiction to the assertion just quoted, and more to the point your C’ actually is the split complex numbers, being the 2-dimensional real algebra generated by 1 and lambda I_3 e_infinity.
“Whether it can be called an algebra over the reals is debatable”. Well, what you claim to have constructed starts off as an 8-dimensional real vector space with a multiplication, which you further claim is an associative division algebra. That is a type of algebra over the reals, so unless you want to modify your claim, I don’t think there’s any doubt that you’re claiming it to be a very special kind of algebra over the reals. (In passing, being an algebra over C’ does not preclude its being an algebra over the reals — quite the opposite in fact: if it is a C’-algebra then it is necessarily also an R-algebra.)
In fact it now seems that you are selecting a subset of this space which satisfies your “normalisation condition”. This is not an 8-dimensional vector space, and therefore not an 8-dimensional algebra.
So, what you have constructed is a proper subset of an 8-dimensional real algebra H(x)C’, selected by certain conditions, for which there is a “norm” ||.|| which, when it is defined, is multiplicative and definite. You further assert, without proof, that it is an algebra (that would require it to be closed under addition and multiplication), a division algebra and of dimension 8, which it is not.
However, there is in fact a much simpler way of summarising this discussion. The “normalisation condition” is a condition on a quaternion over the reals being zero. That is four real conditions on eight real parameters. So there are four free real parameters in the objects you’re considering. It seems to me, therefore, that you have succeeded in recovering the quaternions by an extremely roundabout method. The quaternions do indeed form a normed real associative division algebra, but of dimension 4, not 8.
Richard, you write “[Joy Christian has] succeeded in recovering the quaternions”. But you earlier mention that J.C. does not actually *prove* that the subset he is talking about is closed under addition and multiplication. *If* that proof could be supplied, *then* it seems he could have recovered the quaternions.. But right now, nothing is proved. Do you agree?
What Richard Pinch has written has little or nothing to do with what I have presented in my paper. Richard Gill, on the other hand, is lost in both worlds. What I have presented in my paper is a thing of beauty that is lost on both Richards. Therefore, further discussion is pointless.
In mathematics, unlike in the arts, things of beauty should also make sense.
In mathematics, making sense IS beauty. What makes sense is beautiful and what is beautiful makes sense.
What an odd thing to say, when I was at pains to quote the paper and give formula reference numbers. But it was an interesting exercise to find out what was going on, even though it was not, of course, an 8-dimensional associative normed division algebra.
Well, I am certainly not claiming that as a certainty. Indeed, I may be falling into the trap of reading more into a fallacious proof that is warranted. The quaternions are certainly present in the structure he describes, and they satisfy all the conditions he wants to hold, except that of course they are a 4-dimensional algebra. In the absence of a completely explicit statement of any restrictions he imposes beyond (A2), it’s not clear. As I said, if there’s an algebra free of zero-divisors in there, it’s a proper subalgebra of the 8-dimensional algebra H(x)C’ and hence as an algebra is of dimension strictly less than 8. If he has succeeded in defining a division algebra, it must be of dimension at most 4.
However, if the only condition that he imposes is the normalisation condition (A2), then his system is not closed under addition. Both 1.1+0.epsilon and 0.1 + 1.epsilon satisfy his condition (A2) but their sum 1.1 + 1.epsilon does not, and indeed is one of the non-zero zero-divisors that give him so much trouble.
I’m reminded of the remarks of Imre Lakatos on “monster-barring” in his stimulating tract “Proofs and Refutations”.
As a followup, let’s consider condition (a2) in a little more detail. It reads qr.qd^dagger + qd.qr^dagger = 0, where qr and qd are real quaternions, and dagger is the quaternion involution. This is a non-linear condition, corresponding to four quadratic equations on eight real variables, defining an affine algebraic variety of dimension 4. This contains the linear varieties qr=0 and qd=0, which correspond to two copies of the quaternions, containing the two points 1.1+0.epsilon and 0.1 + 1.epsilon respectively. But this variety is not itself a linear space, as we have just seen.
I think we need more specialists on this topic entering this debate. Because it is clear that Richard David Gill has some quarrel with Joy Christian. And it is very crazy how people are very prompt to condemn to the stake someone without even thinking reasonable about their actions.
The results of my paper retracted from the journal Communications in Algebra are now republished in Section 2.8 of the journal Royal Society Open Science, vol 9, 220147 (2022):
DOI:https://doi.org/10.1098/rsos.220147
Section 2.7 of this paper also includes refutations of the incorrect claims made by Richard Gill in his posts above and elsewhere, as well as of similar incorrect claims made by others.
Sections 2.7 and 2.8 of my paper also refute the rationale given for the retraction of the original paper by the journal Communications in Algebra.
This was the second paper of mine Richard Gill had managed to have retracted by making false claims, as I explained in one of my posts above. Both papers are now republished in journals of higher standards and greater reputations.