## Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

Foundations of physics and/or philosophy of physics, and in particular, posts on unresolved or controversial issues

### Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

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In another thread, the issue of perfect anti-correlation in the EPR-Bohm setup and Dr. Bertlmann's socks came up. If you are unfamiliar with Dr. Bertlmann's socks, then I recommend reading chapter 16 of Bell's book. The following is a variant of Bell's amusing example of Dr. Bertlmann's socks, which I recently posted in another thread:

Joy Christian wrote:
"Imagine you are walking from your home in winter and midway to your destination you reach out in your pockets for hand-gloves, but you find only one of them. At that moment you instantly know that you forgot the other one at home. Not only do you know that, but you also instantly know that the one you forgot at home is a lefthand glove if the one you pulled out from your pocket happens to be a righthand glove. Note that this is true even if your home happens to be in the far corner of the Universe. You have instant information about the handedness of the glove you forgot at home by simply looking at the glove you just pulled out from your pocket. That is perfect anti-correlation and it has nothing to do with nonlocality of any sort. It is just simple classical physics. The same is true for the anti-correlation of spins in the EPR-Bohm setup."

Here I want to stress the importance of this example in my 3-sphere (or quaternionic) model for the singlet correlations. The mathematical details of this model can be found in this paper. But the point I want to make here is that ALL correlations are Dr. Bertlmann's socks type classical correlations within my 3-sphere model, not just the perfect anti-correlations. You may wonder: How is that possible? Well, recall that the perfect anti-correlations are ensured by the condition A(a)B(b) = -1, where A(a) and B(b) are results observed about the measurement directions a and b. On the other hand, the singlet correlations predicted by quantum mechanics are of the sinusoidal form: << A(a)B(b) >> = -a.b. That requires sign flips from A(a)B(b) = -1 to A(a)B(b) = +1 for at least some of the directions a and b. How is that possible for Dr. Bertlmann's socks type correlations? Well, it is indeed not possible if we assume that we live in a flat Euclidean space R^3. But it is possible to have such sign flips naturally within a quaternionic 3-sphere. The sign flips from A(a)B(b) = -1 to A(a)B(b) = +1 are induced by the twists in the U(1) bundle over S^2 constituting a quaternionic 3-sphere. In my view, the strong correlations we observe in Nature are proof that we live in such a quaternionic 3-sphere. Let me reproduced an essential page from an influential paper. It discusses the twists in the U(1) bundle over S^2. It is from page 272 of T. Eguchi, P. B. Gilkey, and A. J. Hanson, Physics Reports, Volume 66, No. 6, pp 213-393 (1980):

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### Re: Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

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Admittedly, quaternionic 3-sphere, or S^3, is not easy to intuit or visualize, even though it is just a three-dimensional space. But we can use a 2D toy model to visualize what I am saying about the sign flips from A(a)B(b) = -1 to A(a)B(b) = +1, which are essential to arrive at the correlations << A(a)B(b) >> = -a.b. In Appendix 1, page 20, of this paper I have illustrated how such sign flips come about in a toy model of a Mobius world of 2D Alice and Bob. The twists in the Hopf bundle of S^3 mentioned in the previous post are just more complicated versions of the twist in a Mobius strip. This may seem like poetry to some, but it is more beautiful poetry than of Bell's theorem.

Needless to say, the toy model is not to be taken too seriously. It is for an intuitive understanding only.

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### Re: Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

Something I learned from Niles Johnson's web site.
https://nilesjohnson.net/hopf.html

A Hopf band. I guess it is like a double Mobius.
.
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### Re: Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

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In the above picture, it does look like a double Mobius --- i.e., with two twists instead of one. The explanation given on Niels Jhonson's webpage is the following: "The collection of fibers over an arc form an annulus whose boundary circles are linked. This is known as a Hopf band; it is a Seifert surface for the Hopf link." Fascinating!

As for the strong EPR-Bohm correlations, the key observation is the following: The perfect anti-correlation between the observations of Alice and Bob demands that if A(a) = +1, then B(b) = -1 for a = b, and vice versa. But for Dr. Bertlmann's socks type correlations this is true even for a =/= b. If I find a righthand glove in my pocket then the one I forgot at home must be a lefthand glove regardless of the choice of a and b. Thus the correlations between Dr. Bertlmann's socks are always perfect anti-correlation regardless of the choice of a and b, implying that the product A(a)B(b) = -1, always. But the strong correlations demand that we must have the product A(a)B(b) = +1 at least for some a and b in order to get << A(a)B(b) >> = -a.b. And this is where the Mobius-like twists in the Hopf bundle of S^3 come to rescue. As counterintuitive as it may seem, within S^3 it is possible to find a righthand glove in one's pocket and be certain that the one forgotten at home must also be a righthand glove at least for some of the a =/= b. And this is not some empty talk but has been explicitly proven by rigorous mathematics. Consequently, we end up getting << A(a)B(b) >> = -a.b, as observed.

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### Re: Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

Joy Christian wrote:***
In the above picture, it does look like a double Mobius --- i.e., with two twists instead of one. The explanation given on Niels Jhonson's webpage is the following: "The collection of fibers over an arc form an annulus whose boundary circles are linked. This is known as a Hopf band; it is a Seifert surface for the Hopf link." Fascinating!

As for the strong EPR-Bohm correlations, the key observation is the following: The perfect anti-correlation between the observations of Alice and Bob demands that if A(a) = +1, then B(b) = -1 for a = b, and vice versa. But for Dr. Bertlmann's socks type correlations this is true even for a =/= b. If I find a righthand glove in my pocket then the one I forgot at home must be a lefthand glove regardless of the choice of a and b. Thus the correlations between Dr. Bertlmann's socks are always perfect anti-correlation regardless of the choice of a and b, implying that the product A(a)B(b) = -1, always. But the strong correlations demand that we must have the product A(a)B(b) = +1 at least for some a and b in order to get << A(a)B(b) >> = -a.b. And this is where the Mobius-like twists in the Hopf bundle of S^3 come to rescue. As counterintuitive as it may seem, within S^3 it is possible to find a righthand glove in one's pocket and be certain that the one forgotten at home must also be a righthand glove at least for some of the a =/= b. And this is not some empty talk but has been explicitly proven by rigorous mathematics. Consequently, we end up getting << A(a)B(b) >> = -a.b, as observed.

***

Very good, you put your finger on what I think is the sore point. In my opinion, and with all respect, this has not been explicitly proven by rigorous mathematics. I think it is quite simply not true. People can take your word for this, or mine, or judge for themselves.

I will further draw attention to the sore point at the symposium. If people don't trust their own facilities, I hope they would trust John Baez.
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### Re: Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

gill1109 wrote:
Very good, you put your finger on what I think is the sore point. In my opinion, and with all respect, this has not been explicitly proven by rigorous mathematics. I think it is quite simply not true. People can take your word for this, or mine, or judge for themselves.

I will further draw attention to the sore point at the symposium. If people don't trust their own facilities, I hope they would trust John Baez.

Your opinions about my Bell-work have been rejected by three international journals, whose editors and referees believed that I have proved what I claim to have proved. The editors and referees of these journals were fully aware of your online and in-print criticisms of my work, but they have rejected your claims by publishing my papers:

2) https://royalsocietypublishing.org/doi/ ... sos.180526

3) https://ieeexplore.ieee.org/document/8836453

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### Re: Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

gill1109 wrote: If people don't trust their own facilities, I hope they would trust John Baez.

No cult of personality here! We work with logic and truth, not appeal to authority. And for the record, we all trust our "own facilities".
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### Re: Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

local wrote:
gill1109 wrote: If people don't trust their own facilities, I hope they would trust John Baez.

No cult of personality here! We work with logic and truth, not appeal to authority. And for the record, we all trust our "own facilities".

No, indeed, no cult of personality here! And we must each ultimately trust our own faculties. I think I got the spelling wrong, before.

Joy Christian wrote:
gill1109 wrote:Very good, you put your finger on what I think is the sore point. In my opinion, and with all respect, this has not been explicitly proven by rigorous mathematics. I think it is quite simply not true. People can take your word for this, or mine, or judge for themselves.

I will further draw attention to the sore point at the symposium. If people don't trust their own facilities, I hope they would trust John Baez.

Your opinions about my Bell-work have been rejected by three international journals, whose editors and referees believed that I have proved what I claim to have proved. The editors and referees of these journals were fully aware of your online and in-print criticisms of my work, but they have rejected your claims by publishing my papers:

2) https://royalsocietypublishing.org/doi/ ... sos.180526

3) https://ieeexplore.ieee.org/document/8836453

*Some* editors and some referees of those journals believed you were right. I'm glad the papers got published. The editors and referees of the first journal also published a critique by me of your paper. They did not, as far as I know, publish your rebuttal of my critique. The editor involved in the second two journals was the same person, Derek Abbott. He has elsewhere published a paper in his journal IEEE Access arguing that one should not bother with rigour in mathematics, but just use intuition and sound engineering sense. He promotes Geometric Algebra and he uses the Joy Christian papers as an example of a success story thereof. He asked me to referee the RSOS paper but I declined, because it is unfair that I should repeatedly referee work of the same person.
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### Re: Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

gill1109 wrote:
Joy Christian wrote:
Your opinions about my Bell-work have been rejected by three international journals, whose editors and referees believed that I have proved what I claim to have proved. The editors and referees of these journals were fully aware of your online and in-print criticisms of my work, but they have rejected your claims by publishing my papers:

2) https://royalsocietypublishing.org/doi/ ... sos.180526

3) https://ieeexplore.ieee.org/document/8836453

*Some* editors and some referees of those journals believed you were right. I'm glad the papers got published. The editors and referees of the first journal also published a critique by me of your paper. They did not, as far as I know, publish your rebuttal of my critique. The editor involved in the second two journals was the same person, Derek Abbott. He has elsewhere published a paper in his journal IEEE Access arguing that one should not bother with rigour in mathematics, but just use intuition and sound engineering sense. He promotes Geometric Algebra and he uses the Joy Christian papers as an example of a success story thereof. He asked me to referee the RSOS paper but I declined because it is unfair that I should repeatedly referee the work of the same person.

You are not getting the message. You are flatly wrong about my work. You haven't got the message for the past ten years, so you are unlikely to get it now. But do you know why you are flatly wrong? Well, I have explained that very clearly in my response to you on PubPeer: https://pubpeer.com/publications/A60DFD ... 3184125D#2

In this thread, you have an opportunity to learn about your mistakes and correct them. But instead, you have chosen to attack my work once again. That is pathetic.

Even more pathetically, you have now started to attack Prof. Derek Abbott just because his journal IEEE Access published my paper. That is pathetic beyond measure.

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### Re: Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

Joy Christian wrote:Even more pathetically, you have now started to attack Prof. Derek Abbott just because his journal IEEE Access published my paper. That is pathetic beyond measure.

I am not attacking Derek Abbott. He's a good guy. He has done wonderful work in Forensic Science and in Game Theory. i admire his popularization of Geometric algebra. His paper in his journal IEEE Access https://www.researchgate.net/publication/256838918_The_Reasonable_Ineffectiveness_of_Mathematics_Point_of_View "The Reasonable Ineffectiveness of Mathematics" won a "best writings of the year on Mathematics" prize, MIT Press, 2016. He makes some excellent points. But I do disagree with one of the opinions he presents there.

For references, see Abbott's slides of his 2018 Brazil talk https://mat-web.upc.edu/people/sebastia.xambo/A18/Abbott-0727.pdf, and info about the conference and its eminent participants https://www.ime.unicamp.br/~agacse2018/guests. David Hestenes, Anthony and Joan Lasenby, Leo Dorst from Amsterdam.

Obviously, Christian and I disagree about the correctness of his work on Bell's theorem and on Geometric Algebra; and we disagree about the impressiveness of his publication record. I'll try not to bring it up again.

Abbott writes: "Platonism is a viral form of philosophical reductionism that breaks apart holistic concepts into imaginary dualisms. I argue that lifting the veil of mathematical Platonism will accelerate progress. In summation, Platonic ideals do not exist; however, ad hoc elegant simplifications do exist and are of utility provided we remain aware of their limitations". I think that *within* mathematics, Platonism is necessary; just as in Western society, we pretend that our social systems embody ideals of truth, justice, democracy. People who want to apply mathematics to real problems (like engineers who need to build bridges using as little iron and concrete as possible) will obviously not be too bothered with fundamental doubts as to whether the real numbers actually exist, or not. Physicists who want to explain physical reality, and not just calculate , should worry about mathematical truth, I think.

Christian believes that the 8-dimensional real Clifford algebra Cl(3, 0), which is the even subalgebra of Cl(3, 1), and moreover isomorphic to Cl(1, 2) (see https://en.wikipedia.org/wiki/Clifford_algebra#Grading) is a division algebra. I believe that it is not. I think that this question can be decided objectively.
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### Re: Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

gill1109 wrote:
Christian believes that the 8-dimensional real Clifford algebra Cl(3, 0), which is the even subalgebra of Cl(3, 1), and moreover isomorphic to Cl(1, 2) (see https://en.wikipedia.org/wiki/Clifford_algebra#Grading) is a division algebra. I believe that it is not. I think that this question can be decided objectively.

Some things can indeed be decided objectively. One of those things is the fact that you haven't got a clue what you are talking about. What you have written above is evidently and objectively gobbledygook. I do not believe what you claim I believe. Readers can decide for themselves what I believe: https://arxiv.org/abs/1908.06172.

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### Re: Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

Sorry, got your Clifford algebras mixed up. I should have written: Christian believes that the 8-dimensional real Clifford algebra Cl(0, 3), which is the even subalgebra of Cl(4, 0) (see https://en.wikipedia.org/wiki/Clifford_algebra#Grading) is a division algebra. I believe that it is not. I think that this question can be decided objectively. See Christian's RSOS paper or his "pure mathematics paper" https://arxiv.org/abs/1908.06172

One can take as basis for the 8 dimensional real vector space Cl(0, 3) the scalar 1, three unit vectors, three unit bivectors, and the pseudo-scalar. Take any unit vector $u$. it satisfies by definition $u^2 = 1$ hence $u^2 - 1 = (u - 1)(u + 1) = 0$. If the space could be given a norm such that the norm of a product is the product of the norms, we would have $\|u - 1\|. \|u + 1\| = 0$ hence either $\|u - 1\| = 0$ or $\|u + 1\| = 0$ (or both), implying that u = 1 or u = -1.

Recall that a normed division algebra is an algebra that is also a normed vector space and such that the norm of a product is the product of the norms; a division algebra is an algebra such that if a product of two elements equals zero, then at least one of the two elements concerned must be zero.
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### Re: Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

gill1109 wrote:Sorry, got your Clifford algebras mixed up. I should have written: Christian believes that the 8-dimensional real Clifford algebra Cl(0, 3), which is the even subalgebra of Cl(4, 0) (see https://en.wikipedia.org/wiki/Clifford_algebra#Grading) is a division algebra. I believe that it is not. I think that this question can be decided objectively. See Christian's RSOS paper or his "pure mathematics paper" https://arxiv.org/abs/1908.06172

One can take as basis for the 8 dimensional real vector space Cl(0, 3) the scalar 1, three unit vectors, three unit bivectors, and the pseudo-scalar. Take any unit vector $u$. it satisfies by definition $u^2 = 1$ hence $u^2 - 1 = (u - 1)(u + 1) = 0$. If the space could be given a norm such that the norm of a product is the product of the norms, we would have $\|u - 1\|. \|u + 1\| = 0$ hence either $\|u - 1\| = 0$ or $\|u + 1\| = 0$ (or both), implying that u = 1 or u = -1.

Recall that a normed division algebra is an algebra that is also a normed vector space and such that the norm of a product is the product of the norms; a division algebra is an algebra such that if a product of two elements equals zero, then at least one of the two elements concerned must be zero.

I have already refuted this waffle elsewhere: https://royalsocietypublishing.org/doi/ ... sos.180526 (see the comments on the Disqus thread just below the paper).

But I do admire the tenacity of Richard D. Gill. He has amazing capacity to keep repeating his refuted arguments week after week, month after month, year after year.

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### Re: Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

Dear Joy,
We two form a little mutual admiration society! Keep on fighting!
Richard
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### Re: Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

Let's try again.

I hereby revise my previous claim. I made a mistake, sorry!

I should have written the following:

Christian argues that the 8-dimensional real and associative Clifford algebra Cl(0, 3), which is the even subalgebra of Cl(4, 0) (see https://en.wikipedia.org/wiki/Clifford_algebra#Grading) is a division algebra. I believe that it is not. I think that this question can be decided objectively. See Christian's RSOS paper or his "pure mathematics paper" https://arxiv.org/abs/1908.06172

One can take as vector space basis for the 8 dimensional real vector space Cl(0, 3) the scalar 1, three vectors, three bivectors, and the pseudo-scalar. According to the definition of Clifford algebras, the three vectors square to -1. Take any unit bivector $v$. It satisfies $v^2 = 1$ hence $v^2 - 1 = (v - 1)(v + 1) = 0$. If the space could be given a norm such that the norm of a product is the product of the norms, we would have $\|v - 1\|. \| v + 1\| = 0$ hence either $\|v - 1\| = 0$ or $\|v + 1\| = 0$ (or both), $v - 1= 0$ or $v + 1 = 0$ (or both), implying that v = 1 or v = -1.

Recall that a normed division algebra is an algebra that is also a normed vector space and such that the norm of a product is the product of the norms; a division algebra is an algebra such that if a product of two elements equals zero, then at least one of the two elements concerned must be zero.

Perhaps Joy can tell us exactly what he disagrees with here. For instance, let's take it step by step:

1. Over the real numbers, is the even subalgebra of Cl(4, 0) isomorphic to Cl(0, 3)?

2. In Cl(0, 3), the basis vectors e_1, e_2 and e_3 anti-commute and square to -1. The algebra is associative. I can define the pseudoscalar M = e_1 e_2 e_3. It follows that M^2 = -1. I can define three bivectors v_1, v_2 and v_3 by v_i = M e_i. They anticommute and square to +1.

3. As a real vector space one can take the basis of Cl(0, 3) to be 1 (the scalar), the three basis vectors e_i, the three basis bivectors v_i, and the pseudo-scalar M.
Last edited by gill1109 on Wed Oct 23, 2019 8:15 pm, edited 1 time in total.
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### Re: Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

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### Re: Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

Joy Christian wrote:***

***

Well, I am so sorry. I have re-posted the comment (and corrected a typo!) on the thread about the symposium since I table it as item for debate (with you or with one of your supporters) at the symposium. Of course, if you would simply agree with my analysis, then there is no need to discuss it further.

It has consequences for your paper on your 3-sphere model, which everyone including myself would like to understand more deeply.

My agenda is to shed light on your work, which means that misunderstandings about it need to be cleared up.
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### Re: Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

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Getting back to the main topic of this thread, here is something from the Wikipedia page for Seifert surfaces mentioned on Niles Jhonson's page linked above by Fred:

So, unlike the Mobius band, Seifert surfaces are orientable. That is no good for the argument for the AB = -1 to AB = +1 transition I have discussed above. That argument works for Mobius bands because Mobius bands are not orientable. So Seifert surfaces do not seem to be playing a significant role in the 3-sphere model. The twists in the Hopf bundle of S^3 are what is responsible for the AB = -1 to AB = +1 transition, and therefore for the strong correlations -a.b.

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### Re: Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

I have deleted this post and moved to an appropriate thread
Last edited by gill1109 on Sun Oct 27, 2019 2:23 am, edited 1 time in total.
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### Re: Dr Bertlmann's socks and the 3-sphere model of EPR-Bohm

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Again, your post is off-topic. Moreover, I am entirely unexcited by the abstract. My work has nothing to do with wavefunction and all the unpalatable baggage it brings in.

Don't disrupt the flow of this thread. If you have something to contribute about the fibration of S^3, then let us know. Otherwise, keep the wavefunctions, etc., out of it.

***
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