## The Mistakes by Bell and von Neumann are Identical

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

### Re: The Mistakes by Bell and von Neumann are Identical

Joy Christian wrote:***
I have worked out the correct eigenvalue of the operator (R + S + T - U) relevant for the Bell's implicit assumption [cf. eq. (16) or (29) of this paper: https://arxiv.org/abs/1704.02876].

The correct eigenvalue of the operator (R + S + T - U) is

(1) $\sqrt{(r + s + t - u)^2 + z\;}$,

where $z = l + m + n - o - p - q$, and l, m, n, o, p, and q are the eigenvalues of the operators L = RS - SR, M = RT - TR, N = TS - ST, O = US - SU, P = UR - RU, and Q = UT - TU, respectively.

Now, implementing what they think is the demand of local realism, Bell and his followers assume that the eigenvalue of the operator (R + S + T - U) is (r + s + t - u). But that is true if and only if the operators R, S, T, and U commute with each other. This is easy to see from the above eq. (1). When R, S, T, and U all commute with each other, then z = 0 and the eigenvalue reduces to (r + s + t - u). But in the Bell-test experiments the operators R, S, T, and U do not commute with each other because they correspond to different detections made at mutually exclusive measurement directions. So Bell and his followers assume a wrong eigenvalue of the operator (R + S + T - U) and thus incorrectly implement Einstein's notion of local realism. It is a simple mathematical mistake. And it invalidates the bounds of -2 and +2 on the CHSH correlator. The correct bounds follow if we use the correct eigenvalue (1) worked out above. The correct bounds work out to be $-2\sqrt{2}$ and $+2\sqrt{2}$, exactly as those predicted by quantum mechanics. Thus there is no incompatibility between quantum mechanics and local realism.

There is a dishonest attempt by a Bell-believer to deflect from the above calculation. Note to other readers: Don't fall for that deflection. Concentrate on what I have presented above.

***
Joy Christian
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### Re: The Mistakes by Bell and von Neumann are Identical

Joy Christian wrote:[...]
There is a dishonest attempt by a Bell-believer to deflect from the above calculation. Note to other readers: Don't fall for that deflection. Concentrate on what I have presented above.

Could you please be more specific why you think this is dishonest? Did you not write the paper I cited? Does it not describe an experiment where the operators R, S, T, and U do indeed commute?
Heinera

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### Re: The Mistakes by Bell and von Neumann are Identical

Heinera wrote:
Joy Christian wrote:[...]
There is a dishonest attempt by a Bell-believer to deflect from the above calculation. Note to other readers: Don't fall for that deflection. Concentrate on what I have presented above.

Could you please be more specific why you think this is dishonest? Did you not write the paper I cited? Does it not describe an experiment where the operators R, S, T, and U do indeed commute?

I do not see any self-adjoint operators on a complex Hilbert space in the description of the experiment proposed in my paper, let alone a specific linear combination like (R + S + T - U) of self-adjoint operators. Therefore, the question of their commutation or non-commutation does not even arise. As you always do, you are simply trying to deflect from the major mistake in Bell's argument I have identified. It is a pathetic attempt to deflect. Moreover, the discussion of my experiment is off-topic for this thread. It has been discussed elsewhere in this forum.

***
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### Re: The Mistakes by Bell and von Neumann are Identical

Joy Christian wrote:I do not see any self-adjoint operators on a complex Hilbert space in the description of the experiment proposed in my paper, let alone a specific linear combination like (R + S + T - U) of self-adjoint operators. Therefore, the question of their commutation or non-commutation does not even arise.
***

Do you see any self-adjoint operators on a complex Hilbert space in the proof of Bell's theorem, as laid out in this paper, let alone a specific linear combination like (R + S + T - U) of self-adjoint operators?

https://journals.aps.org/ppf/pdf/10.110 ... zika.1.195
Heinera

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### Re: The Mistakes by Bell and von Neumann are Identical

Heinera wrote:
Joy Christian wrote:I do not see any self-adjoint operators on a complex Hilbert space in the description of the experiment proposed in my paper, let alone a specific linear combination like (R + S + T - U) of self-adjoint operators. Therefore, the question of their commutation or non-commutation does not even arise.
***

Do you see any self-adjoint operators on a complex Hilbert space in the proof of Bell's theorem, as laid out in this paper, let alone a specific linear combination like (R + S + T - U) of self-adjoint operators?

https://journals.aps.org/ppf/pdf/10.110 ... zika.1.195

Yes, I do.

The fact that you don't see any says a lot about your knowledge and understanding of Bell's theorem. I bet you have never read Bell's 1966 paper to understand his 1964 paper. Sad, really.

PS: Bell's 1966 paper was written before his 1964 paper. The dates have to do with an editorial mistake. His 1964 paper is effectively just an appendix to his 1966 paper (see last footnote).

***
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### Re: The Mistakes by Bell and von Neumann are Identical

Heinera wrote:
Joy Christian wrote:I do not see any self-adjoint operators on a complex Hilbert space in the description of the experiment proposed in my paper, let alone a specific linear combination like (R + S + T - U) of self-adjoint operators. Therefore, the question of their commutation or non-commutation does not even arise.
***

Do you see any self-adjoint operators on a complex Hilbert space in the proof of Bell's theorem, as laid out in this paper, let alone a specific linear combination like (R + S + T - U) of self-adjoint operators?

https://journals.aps.org/ppf/pdf/10.110 ... zika.1.195

Oh for heavens sake, there is no proof of Bell's junk physics theory. The inequalities prove themselves.
.
FrediFizzx
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### Re: The Mistakes by Bell and von Neumann are Identical

FrediFizzx wrote:
Heinera wrote:
Joy Christian wrote:I do not see any self-adjoint operators on a complex Hilbert space in the description of the experiment proposed in my paper, let alone a specific linear combination like (R + S + T - U) of self-adjoint operators. Therefore, the question of their commutation or non-commutation does not even arise.
***

Do you see any self-adjoint operators on a complex Hilbert space in the proof of Bell's theorem, as laid out in this paper, let alone a specific linear combination like (R + S + T - U) of self-adjoint operators?

https://journals.aps.org/ppf/pdf/10.110 ... zika.1.195

Oh for heavens sake, there is no proof of Bell's junk physics theory. The inequalities prove themselves.

And even the inequalities were proven by George Boole 100 years before Bell.

***
Joy Christian
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### Re: The Mistakes by Bell and von Neumann are Identical

Joy Christian wrote:And even the inequalities were proven by George Boole 100 years before Bell.

They are utterly elementary. He did not prove them in his book. He gave them in his book as a simple exercise for schoolboys.

Of course, Bell's theorem (as opposed to Bell's inequality) - QM is incompatible with locality+realism+freedom - is really just a simple variation of EPR. You could say, it's obtained by a rotation of Bob's lab 45 degrees in the Q-P plane. Together with the switch to EPR-B ... due to Bohm and Aharonov.
gill1109
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### Re: The Mistakes by Bell and von Neumann are Identical

gill1109 wrote:
Of course, Bell's theorem (as opposed to Bell's inequality) - QM is incompatible with locality+realism+freedom - is really just a simple variation of EPR. You could say, it's obtained by a rotating Bob's lab 45 degrees in the P-Q plane.

This is just plain wrong. You keep repeating this mantra because you have no understanding of what Einstein's (or EPR's) notion of local realism actually is. As I have shown in my paper, quantum mechanics is perfectly compatible with loclity+realism+freedom (see below). Einstein's notion of local realism has to do with the simultaneous contextual assignment of specific eigenvalues of the respective operators to all observables of a given physical system. But Bell and his followers have botched the implementation of this idea in the proof of Bell's so-called theorem. And when their mistake is pointed out to them, they turn their "theorem" into an article of faith, and repeat the mantra that "QM is incompatible with locality+realism+freedom."

Joy Christian wrote:
I have worked out the correct eigenvalue of the operator (R + S + T - U) relevant for the Bell's implicit assumption [cf. eq. (16) or (29) of this paper: https://arxiv.org/abs/1704.02876].

The correct eigenvalue of the operator (R + S + T - U) is

(1) $\sqrt{(r + s + t - u)^2 + z\;}$,

where $z = l + m + n - o - p - q$, and l, m, n, o, p, and q are the eigenvalues of the operators L = RS - SR, M = RT - TR, N = TS - ST, O = US - SU, P = UR - RU, and Q = UT - TU, respectively.

Now, implementing what they think is the demand of local realism, Bell and his followers assume that the eigenvalue of the operator (R + S + T - U) is (r + s + t - u). But that is true if and only if the operators R, S, T, and U commute with each other. This is easy to see from the above eq. (1). When R, S, T, and U all commute with each other, then z = 0 and the eigenvalue reduces to (r + s + t - u). But in the Bell-test experiments the operators R, S, T, and U do not commute with each other because they correspond to different detections made at mutually exclusive measurement directions. So Bell and his followers assume a wrong eigenvalue of the operator (R + S + T - U) and thus incorrectly implement Einstein's notion of local realism. It is a simple mathematical mistake. And it invalidates the bounds of -2 and +2 on the CHSH correlator. The correct bounds follow if we use the correct eigenvalue (1) worked out above. The correct bounds work out to be $-2\sqrt{2}$ and $+2\sqrt{2}$, exactly as those predicted by quantum mechanics. Thus there is no incompatibility between quantum mechanics and local realism.

***
Joy Christian
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### Re: The Mistakes by Bell and von Neumann are Identical

Joy Christian wrote:Einstein's notion of local realism has to do with the simultaneous contextual assignment of specific eigenvalues of the respective operators to all observables of a given physical system.

Well, I could agree with one sentence in your post. The two named topics certainly do have quite a lot to do with one another. John Conway and Simon Kochen, Eric Cator and Klaas Landsman, and many other notable mathematicians would also agree and have proved very interesting theorems on the relationship.
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### Re: The Mistakes by Bell and von Neumann are Identical

gill1109 wrote:
Joy Christian wrote:
Einstein's notion of local realism has to do with the simultaneous contextual assignment of specific eigenvalues of the respective operators to all observables of a given physical system.

Well, I could agree with one sentence in your post.

If you agree with the definition of Einstein's notion of local realism I have stated above in my sentence, then the conclusion of my paper is inescapable. The abstract of my paper reads:

***
Joy Christian
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### Re: The Mistakes by Bell and von Neumann are Identical

Joy Christian wrote:If you agree with the definition of Einstein's notion of local realism I have stated above in my sentence ...

What you stated was not a definition and anyway, nothing like Einstein's notion. BTW, EPR carefully avoided *defining* local realism. They just gave a sufficient condition for something to be considered an element of reality. Then, in your paper, you write
within quantum theory, using normalized state vectors | ψi >, his [Bell's] assumption is: Given the observables R, S, T, and U, there exists an observable
a R + b S + c T + d U such that...

But Bell was at this point in his paper explicitly *not* working within quantum theory. So to say that this is Bell's assumption is grossly unfair to the person you seem to be accusing of having perverted the course of science.
gill1109
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### Re: The Mistakes by Bell and von Neumann are Identical

gill1109 wrote:
But Bell was at this point in his paper explicitly *not* working within quantum theory.

No, he was not. But he was certainly working within von Neumann's framework of hidden variable theories. His 1964 paper is an "appendix" to his 1966 paper on hidden variable theories. See the last footnote of Bell's 1966 paper. Thus, he was very much concerned about reproducing quantum mechanical expectation values as an average of specific eigenvalues of quantum mechanical operators. But he blundered in doing so, as I explain in my paper: https://arxiv.org/abs/1704.02876.

***
Joy Christian
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### Re: The Mistakes by Bell and von Neumann are Identical

gill1109 wrote:
Joy Christian wrote:
Einstein's notion of local realism has to do with the simultaneous contextual assignment of specific eigenvalues of the respective operators to all observables of a given physical system.

Well, I could agree with one sentence in your post.

Good. So you agree with my sentence. You agree with the fact that "Einstein's notion of local realism has to do with the simultaneous contextual assignment of specific eigenvalues of the respective operators to all observables of a given physical system." It is then inescapable that Bell's assumption, eq. (16) of my paper, is wrong. Thus the rest of Bell's argument is wrong too.

***
Joy Christian
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### Re: The Mistakes by Bell and von Neumann are Identical

***

John S. Bell was singularly well-equipped to spot the mistake in his theorem but failed to do so. That is the vital lesson one can learn from my paper: https://arxiv.org/abs/1704.02876.

***
Joy Christian
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### Re: The Mistakes by Bell and von Neumann are Identical

Joy Christian wrote:
gill1109 wrote:
Joy Christian wrote:
Einstein's notion of local realism has to do with the simultaneous contextual assignment of specific eigenvalues of the respective operators to all observables of a given physical system.

Well, I could agree with one sentence in your post.

Good. So you agree with my sentence. You agree with the fact that "Einstein's notion of local realism has to do with the simultaneous contextual assignment of specific eigenvalues of the respective operators to all observables of a given physical system." It is then inescapable that Bell's assumption, eq. (16) of my paper, is wrong. Thus the rest of Bell's argument is wrong too.

***

I disagree with your deduction. You say “it is then inescapable...”. I think that your physical intuition is brilliant. I cannot follow your “logic”. But then I’m merely a statistician

By the way: to follow the lead of Jarek Duda in another internet discussion group: is anyone interested in a group video meeting? Of course, time zones are a big issue to truly international internet communities. And who knows how long internet will keep working.
gill1109
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### Re: The Mistakes by Bell and von Neumann are Identical

gill1109 wrote:
Joy Christian wrote:
gill1109 wrote:
Joy Christian wrote:
Einstein's notion of local realism has to do with the simultaneous contextual assignment of specific eigenvalues of the respective operators to all observables of a given physical system.

Well, I could agree with one sentence in your post.

Good. So you agree with my sentence. You agree with the fact that "Einstein's notion of local realism has to do with the simultaneous contextual assignment of specific eigenvalues of the respective operators to all observables of a given physical system." It is then inescapable that Bell's assumption, eq. (16) of my paper, is wrong. Thus the rest of Bell's argument is wrong too.

***

I disagree with your deduction. You say “it is then inescapable...”. I think that your physical intuition is brilliant. I cannot follow your “logic”. But then I’m merely a statistician

Ok. I am happy to spell out my logic. Or better still, spell out the logical fallacy in Bell's argument. It is not even a deep fallacy. Simply put, Bell's argument is circular. He assumes what he wants to prove. In his proof, Bell wants to conclude that the bounds on the CHSH correlator are -2 and +2. But in doing so he has smuggled-in that conclusion in one of his unacknowledged assumptions, namely, in his assumption I have written as eq. (16) of my paper. Even though mathematically my eq. (16) is a trivial identity, physically it is an unjustifiable assumption, as Bell himself has pointed out in the context of von Neumann's theorem against hidden variable theories. Thus Bell has fallen for the same circular argument he had ridiculed von Neumann for using. In other words, in the "proof" of his "theorem", he has assumed what he wants to prove. As I said, it is not even a deep fallacy.

***
Joy Christian
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