The Mistakes by Bell and von Neumann are Identical

[Joy Christian]

***
I have published a new paper on the arXiv. In the paper, I compare the mistakes by John S. Bell and John von Neumann in their respective theorems against hidden variable theories.

Here is the paper: https://arxiv.org/abs/1704.02876. I also demonstrate in the paper how John Bell exposed a mistake in the no-go theorem by von Neumann against hidden variable theories, and then went on to make the same mistake himself in his own no-go theorem against their locally causal counterpart. The cost of Bell's mistake to physics is immeasurable.

Bell's Mistake: The linear sum of eigenvalues is not an eigenvalue of the linear sum of the observables as Bell and his followers assume to derive the bounds of on the CHSH.

Once Bell's mistake is corrected, it is straightforward to derive the correct bounds on the Bell-CHSH correlator, and they work out to be and , not and .

So what dictates that the bounds on Bell-CHSH correlator must be And what dictates that they cannot possibly be A surprising ansswer: Einstein's local-realism.

So, the takeaway: The correct local-realistic bounds on the Bell-CHSH correlator are and , not and .

Here are the title and abstract of my paper (this paper is completely independent of my quaternionic 3-sphere model for the singlet correlations: https://arxiv.org/abs/1911.11578):



Here is the conclusion:



And here is a parody of the famous quote from John Bell's interview by the science-fiction magazine Omni, published in May 1988 (p. 88). I have replaced "von Neumann" with [ Bell ]:

Yet the [ Bell ] proof, if you actually come to grips with it, falls apart in your hands! There is nothing to it. It is not just flawed, it is silly! ...When you translate [ his assumptions ] in terms of physical disposition they’re nonsense. You may quote me on that: The proof of [ Bell ] is not merely false but foolish!

***

[gill1109]

I have published a new paper on the arXiv. In the paper, I compare the mistakes by John S. Bell and John von Neumann in their respective theorems against hidden variable theories. Here is the paper:https://arxiv.org/abs/1704.02876v4. I also demonstrate in the paper how John Bell exposed a mistake in the no-go theorem by von Neumann against hidden variable theories, and then went on to make the same mistake himself in his own no-go theorem against their locally causal counterpart. The cost of Bell's mistake to physics is immeasurable.

I see that the new paper is a version of a paper from 2017, with a new title. And indeed, it is also much improved. As usual, it is beautifully written.

I have explained what I think is wrong with the author's argument in my paper Does Geometric Algebra provide a loophole to Bell’s Theorem?, https://arxiv.org/abs/1203.1504v9, https://www.mdpi.com/1099-4300/22/1/61, published in the journal "Entropy". I write at the beginning of Section 6.5 The RSOS Paper, 2018

He [Christian] claims that the use of expressions involving simultaneously measurement functions evaluated at several different measurement settings cannot possibly have any physical meaning and hence that proofs which depend on the evaluation of such expressions are physically meaningless and hence invalid. But under local realism, the local hidden variable λ is supposed to exist, even if it cannot be observed directly, and various functions thereof are also supposed, by the assumption of local realism, to exist. Hence one can mathematically consider any combinations of those functions one likes, and if that leads to observable consequences, for instance, bounds on correlations which can be observed in experiments, then those bounds must be valid, if local realism is true. Note that when I use the word “exists", I speak as a mathematician, and I am discussing mathematical models. For me, local realism means the existence of a mathematical model which reproduces the predictions of quantum mechanics (or approximately does so). It is up to physicists and to philosophers to thrash out what they mean when they use words like local, real, or exist. The mathematics which they do should “stand alone”, the words used to describe various objects and to describe the relations between them are irrelevant.

[Joy Christian]

I see that the new paper is a version of a paper from 2017, with a new title. And indeed, it is also much improved. As usual, it is beautifully written.

It is a new paper. The arXiv moderators did not allow me to generate a new ID for it. So read the paper before making nonsensical comments about it. Read also what I have written above.

***

[Joy Christian]

The mathematics which they do should “stand alone”, the words used to describe various objects and to describe the relations between them are irrelevant.

Lecture that to John Bell, who thrashed von Neumann for ignoring physics and relying too much on mathematics: Here is what he had to say about von Neumann's reliance on mathematics:

Yet the von Neumann proof, if you actually come to grips with it, falls apart in your hands! There is nothing to it. It is not just flawed, it is silly! ...When you translate [ his assumptions ] in terms of physical disposition they’re nonsense. You may quote me on that: The proof of von Neumann is not merely false but foolish!

This famous quote is from John Bell's interview by the science-fiction magazine Omni, published in May 1988 (p. 88):

***

[Joy Christian]

He [Christian] claims that the use of expressions involving simultaneously measurement functions evaluated at several different measurement settings cannot possibly have any physical meaning and hence that proofs which depend on the evaluation of such expressions are physically meaningless and hence invalid. But under local realism, the local hidden variable λ is supposed to exist, even if it cannot be observed directly, and various functions thereof are also supposed, by the assumption of local realism, to exist. Hence one can mathematically consider any combinations of those functions one likes, and if that leads to observable consequences, for instance, bounds on correlations which can be observed in experiments, then those bounds must be valid, if local realism is true. Note that when I use the word “exists", I speak as a mathematician, and I am discussing mathematical models. For me, local realism means the existence of a mathematical model which reproduces the predictions of quantum mechanics (or approximately does so). It is up to physicists and to philosophers to thrash out what they mean when they use words like local, real, or exist. The mathematics which they do should “stand alone”, the words used to describe various objects and to describe the relations between them are irrelevant.

The above comments by Gill are irrelevant.

It is the very notion of local-realism used by Bell and his followers that invalidate their derivation of the bounds -2 and +2. That is the beauty of the argument presented in my paper.

***

[Heinera]

I don't see anything new in this paper that you and minkwe haven't harped on before.

[Joy Christian]

I don't see anything new in this paper that you and minkwe haven't harped on before.

The world will go around irrespective of what you can see or not see. You are not even Gill.

***

[gill1109]

He [Christian] claims that the use of expressions involving simultaneously measurement functions evaluated at several different measurement settings cannot possibly have any physical meaning and hence that proofs which depend on the evaluation of such expressions are physically meaningless and hence invalid. But under local realism, the local hidden variable λ is supposed to exist, even if it cannot be observed directly, and various functions thereof are also supposed, by the assumption of local realism, to exist. Hence one can mathematically consider any combinations of those functions one likes, and if that leads to observable consequences, for instance, bounds on correlations which can be observed in experiments, then those bounds must be valid, if local realism is true. Note that when I use the word “exists", I speak as a mathematician, and I am discussing mathematical models. For me, local realism means the existence of a mathematical model which reproduces the predictions of quantum mechanics (or approximately does so). It is up to physicists and to philosophers to thrash out what they mean when they use words like local, real, or exist. The mathematics which they do should “stand alone”, the words used to describe various objects and to describe the relations between them are irrelevant.

The above comments by Gill are irrelevant. It is the very notion of local-realism used by Bell and his followers that invalidate their derivation of the bounds -2 and +2. That is the beauty of the argument presented in my paper.

Perhaps you can try to explain what is the notion of local-realism that you ascribe to "Bell and his followers" (as if they were some kind of religious sect). I suspect you see a different notion there than I do.

[Joy Christian]

Perhaps you can try to explain what is the notion of local-realism that you ascribe to "Bell and his followers" (as if they were some kind of religious sect). I suspect you see a different notion there than I do.

You really should read my paper to fully understand the problem with your handwaving conception of realism. But let me make a few comments here. You write:

...under local realism, the local hidden variable λ is supposed to exist, even if it cannot be observed directly, and various functions thereof are also supposed, by the assumption of local realism, to exist. Hence one can mathematically consider any combinations of those functions one likes ...

But by no means! You are not at liberty to invent your own pet notion of realism. The correct notion of realism is the one defined by Einstein, Podolsky, and Rosen, and mathematically formulated by von Neumann. So what is this notion of realism? It amounts to a simultaneous assignment of definite values to all of the observables of a quantum system within a hidden variable theory based on dispersion-free states. What you have been using instead is a handwaving conception of realism invented by you. Why should anyone care about your personal notion of realism?

***

[gill1109]

Perhaps you can try to explain what is the notion of local-realism that you ascribe to "Bell and his followers" (as if they were some kind of religious sect). I suspect you see a different notion there than I do.

You really should read my paper to fully understand the problem with your handwaving conception of realism. But let me make a few comments here. You write:

...under local realism, the local hidden variable λ is supposed to exist, even if it cannot be observed directly, and various functions thereof are also supposed, by the assumption of local realism, to exist. Hence one can mathematically consider any combinations of those functions one likes ...

But by no means! You are not at liberty to invent your own pet notion of realism. The correct notion of realism is the one defined by Einstein, Podolsky, and Rosen, and mathematically formulated by von Neumann. So what is this notion of realism? It amounts to a simultaneous assignment of definite values to all of the observables of a quantum system within a hidden variable theory based on dispersion-free states. What you have been using instead is a handwaving conception of realism invented by you. Why should anyone care about your personal notion of realism?

EPR did not define realism. They only gave a sufficient condition (not a necessary condition) for existence of an "element of reality".
The condition does not assume a quantum system.
It is not equivalent to the von Neumann concept.
I don't care whether or not anyone cares about my personal notions.
Everyone is free to come up with their own conception and offer it for open discussion to the community of interested scientists.
I did not come up with a new conception of realism. I did use the word "exist" a few times. And I suggested that if lambda *exists*, and certain functions of directions and of lambda *exist*, then one may combine those functions mathematically just how one likes and compute whatever you like. If, by doing that, you are able to deduce observable consequences of your assumptions, and if those consequences conflict with experimental data, then your assumptions need to be reviewed.

My argument does not depend on what *you* mean by exist. I think it is an argument which any decent notion of "exist" should support. Especially if, as I said, I am talking about *mathematical models* for physical applications. Philosophers have endlessly discussed what "exists" means in ordinary language. Mathematicians, of course, have their own discussions about what mathematical existence means. The notion is still controversial in the foundations of mathematics. But it seems to me that those discussions are not terribly pertinent here.

[Joy Christian]

Perhaps you can try to explain what is the notion of local-realism that you ascribe to "Bell and his followers" (as if they were some kind of religious sect). I suspect you see a different notion there than I do.

You really should read my paper to fully understand the problem with your handwaving conception of realism. But let me make a few comments here. You write:

...under local realism, the local hidden variable λ is supposed to exist, even if it cannot be observed directly, and various functions thereof are also supposed, by the assumption of local realism, to exist. Hence one can mathematically consider any combinations of those functions one likes ...

But by no means! You are not at liberty to invent your own pet notion of realism. The correct notion of realism is the one defined by Einstein, Podolsky, and Rosen, and mathematically formulated by von Neumann. So what is this notion of realism? It amounts to a simultaneous assignment of definite values to all of the observables of a quantum system within a hidden variable theory based on dispersion-free states. What you have been using instead is a handwaving conception of realism invented by you. Why should anyone care about your personal notion of realism?

EPR did not define realism. They only gave a sufficient condition (not a necessary condition) for existence of an "element of reality".
The condition does not assume a quantum system.
It is not equivalent to the von Neumann concept.
I don't care whether or not anyone cares about my personal notions.
Everyone is free to come up with their own conception and offer it for open discussion to the community of interested scientists.
I did not come up with a new conception of realism. I did use the word "exist" a few times. And I suggested that if lambda *exists*, and certain functions of directions and of lambda *exist*, then one may combine those functions mathematically just how one likes and compute whatever you like. If, by doing that, you are able to deduce observable consequences of your assumptions, and if those consequences conflict with experimental data, then your assumptions need to be reviewed.

My argument does not depend on what *you* mean by exist. I think it is an argument which any decent notion of "exist" should support. Especially if, as I said, I am talking about *mathematical models* for physical applications. Philosophers have endlessly discussed what "exists" means in ordinary language. Mathematicians, of course, have their own discussions about what mathematical existence means. The notion is still controversial in the foundations of mathematics. But it seems to me that those discussions are not terribly pertinent here.

I see that you still haven't read my paper: Oversights in the Respective Theorems of von Neumann and Bell are Homologous. Read the paper to recognize your mistake (and Bell's mistake).

The bottom line is that you assume something that is demonstrably not real. It is pure fiction. By assuming pure fiction that can't even be counterfactually real, you derive the bounds of -2 and +2 on the CHSH correlator. Therefore those bounds are also pure fiction. It is therefore not surprising at all that the bounds of -2 and +2 are not respected in the EPRB experiments.

***

[gill1109]

I see that you still haven't read my paper: Oversights in the Respective Theorems of von Neumann and Bell are Homologous https://arxiv.org/abs/1704.02876v5. Read the paper to recognize your mistake (and Bell's mistake). The bottom line is that you assume something that is demonstrably not real. It is pure fiction. By assuming pure fiction that can't even be counterfactually real, you derive the bounds of -2 and +2 on the CHSH correlator. Therefore those bounds are also pure fiction. It is therefore not surprising at all that the bounds of -2 and +2 are not respected in the EPRB experiments.

I certainly have read your paper. But your arguments do not convince me.

Could we agree on this: suppose we temporarily (provisionally) assume something that may or may not be a fiction. (Einstein firmly believed in it, or at least, dearly wanted to believe in it). It is something called "local realism". We derive the bounds -2 and +2. We observe that these bounds are resoundingly violated in the real world. We conclude that "local realism" definitely is a fiction.

[Joy Christian]

I see that you still haven't read my paper: Oversights in the Respective Theorems of von Neumann and Bell are Homologous https://arxiv.org/abs/1704.02876v5. Read the paper to recognize your mistake (and Bell's mistake). The bottom line is that you assume something that is demonstrably not real. It is pure fiction. By assuming pure fiction that can't even be counterfactually real, you derive the bounds of -2 and +2 on the CHSH correlator. Therefore those bounds are also pure fiction. It is therefore not surprising at all that the bounds of -2 and +2 are not respected in the EPRB experiments.

I certainly have read your paper. But your arguments do not convince me.

Could we agree on this: suppose we temporarily (provisionally) assume something that may or may not be a fiction. (Einstein firmly believed in it, or at least, dearly wanted to believe in it). It is something called "local realism". We derive the bounds -2 and +2. We observe that these bounds are resoundingly violated in the real world. We conclude that "local realism" definitely is a fiction.

Einstein's local realism dictates the bounds of -2\/2 and +2\/2 on the CHSH correlator, not of -2 and +2. And the bounds of -2\/2 and +2\/2 have never been violated in the experiments.

***

[FrediFizzx]

…
Einstein's local realism dictates the bounds of -2\/2 and +2\/2 on the CHSH correlator, not of -2 and +2. And the bounds of -2\/2 and +2\/2 have never been violated in the experiments.
***

Yep, Bell (and followers) derived the wrong inequality.
.

[gill1109]

Einstein's local realism dictates the bounds of -2\/2 and +2\/2 on the CHSH correlator, not of -2 and +2. And the bounds of -2\/2 and +2\/2 have never been violated in the experiments.

Yep, Bell (and followers) derived the wrong inequality.

Yes. Local realism has to be discarded. The good inequality is Tsirelson's ("the bounds of -2\/2 and +2\/2"). Which, by the way, has been violated many times in the past, too. For instance, see
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.97.170408
Experimental Violation of Bell’s Inequality beyond Tsirelson’s Bound
Yu-Ao Chen, Tao Yang, An-Ning Zhang, Zhi Zhao, Adán Cabello, and Jian-Wei Pan
Phys. Rev. Lett. 97, 170408 – Published 27 October 2006

The correlations between two qubits belonging to a three-qubit system can violate the Clauser-Horne-Shimony-Holt-Bell inequality beyond Tsirelson’s bound [A. Cabello, Phys. Rev. Lett. 88, 060403 (2002)]. We experimentally demonstrate such a violation by 7 standard deviations by using a three-photon polarization-entangled Greenberger-Horne-Zeilinger state produced by Type-II spontaneous parametric down-conversion. In addition, using part of our results, we obtain a violation of the Mermin inequality by 39 standard deviations.

We did it before that, in Leiden
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.92.217901
How to Observe High-Dimensional Two-Photon Entanglement with Only Two Detectors
S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman
Phys. Rev. Lett. 92, 217901 – Published 24 May 2004
https://arxiv.org/abs/quant-ph/0401148
Violation of local realism in a high-dimensional two-photon setup with non-integer spiral phase plates
S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, J. P. Woerdman
(Submitted on 23 Jan 2004 (v1), last revised 10 Sep 2004 (this version, v5))

We propose a novel setup to investigate the quantum non-locality of orbital angular momentum states living in a high-dimensional Hilbert space. We incorporate non-integer spiral phase plates in spatial analyzers, enabling us to use only two detectors. The resulting setup is somewhat reminiscent of that used to measure polarization entanglement. However, the two-photon states that are produced, are not confined to a 2X2-dimensional Hilbert space, and the setup allows the probing of correlations in a high-dimensional space. For the special case of half-integer spiral phase plates, we predict a violation of the Clauser-Horne-Shimony-Holt version of the Bell inequality (S<=2), that is even stronger than achievable for two qubits (S=2*(2^1/2)), namely S=16/5

[Joy Christian]

Yes. Local realism has to be discarded.

Poppycock.

***

[JohnDuffield]

Good stuff Joy. Fancy a trip to Poole sometime?

[Joy Christian]

Good stuff Joy. Fancy a trip to Poole sometime?

Thanks, John. Poole is a coastal town, so perhaps in warmer weather.

***

[FrediFizzx]

… The good inequality is Tsirelson's ("the bounds of -2\/2 and +2\/2"). Which, by the way, has been violated many times in the past, too. For instance, see
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.97.170408
Experimental Violation of Bell’s Inequality beyond Tsirelson’s Bound
Yu-Ao Chen, Tao Yang, An-Ning Zhang, Zhi Zhao, Adán Cabello, and Jian-Wei Pan
Phys. Rev. Lett. 97, 170408 – Published 27 October 2006
…

Not surprising. The way the experimenters do it, the bound on the CHSH type inequality is +/- 4. Which is the maximum. Now..., I'd be really surprised if that bound was ever exceeded.
.

[Joy Christian]

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Extreme prejudice exists within the physics community against hidden variable theories of all kinds. This prejudice is totally irrational. No theoretical argument, no-go theorem, or experimental findings have ruled out hidden variable theories of any kind. Since hidden variable theories can completely eliminate interpretational difficulties of quantum mechanics and lead us to a successful theory of quantum gravity, we should embrace hidden variable theories instead of attacking them irrationally.

Because of Bell's no-go "theorem" against local hidden variable theories, most of the said prejudice comes from those who have vested interests in protecting this so-called "theorem."

However, Bell's "theorem" is a fundamentally flawed argument. Let me illustrate here the mistaken step in the proof of the "theorem" as clearly as possible (more details are in this paper).

Let R, S, T, and U be Hermitian operators whose eigenvalues are r = AB, s = AB', t = A'B, and u = A'B', where A, B, A', and B' = +1 or -1 are the measurement results observed by Alice and Bob.

Now, in the proof of Bell's "theorem," it is assumed that the sum of the individual eigenvalues, (r + s + t - u), is an eigenvalue of the sum (R + S + T - U) of the corresponding operators.

Without this assumption, the stringent bounds of -2 and +2 on the CHSH correlator cannot be derived.

But this assumption is simply false. No such correspondence between the Hermitian operator (R + S + T - U) and the sum (r + s + t - u) of individual eigenvalues exists mathematically.

Therefore, this step in the proof of Bell's "theorem" is a major error. In other words, no theorem against local hidden variable theories exists despite the widespread belief to the contrary.

***