## What exactly is Bell's Theorem?

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

### What exactly is Bell's Theorem?

The Wikipedia entry for Bell's Theorem gives,

"If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local."

Which is from the book "Speakable and Unspeakable in Quantum Mechanics" page 65 which actually says,

"But if his extension is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local. This is what the theorem says."

However, in the paragraph before this statement, Bell gives another description of the so-called theorem.

"$A({\bf a}, \lambda)$
$B({\bf b}, \lambda)$ (3)

With these local forms, it is not possible to find functions A and B and a probability distribution $\rho$ which give the correlation (1). This is the theorem."

Correlation (1) is of course the quantum mechanical prediction of -a.b. Now..., someone with proper definitions could possibly make this into a rigorous mathematical theorem. But there not much point in that since Joy has already found local A and B functions that do give the QM correlation. This should really be the end of the debate.
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FrediFizzx
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### Re: What exactly is Bell's Theorem?

FrediFizzx wrote:
Correlation (1) is of course the quantum mechanical prediction of -a.b. Now..., someone with proper definitions could possibly make this into a rigorous mathematical theorem. But there is not much point in that since Joy has already found local A and B functions that do give the QM correlation. This should really be the end of the debate.

I of course agree with your conclusion. But just to be the devil's advocate (because the devils seem to have fled the battleground), what the Bell-believers would say is that by considering four EPR-Bohm type experiments and assuming only locality and realism (aka "counterfactual definiteness") they are able to derive the bounds of +/-2 on the CHSH correlator. The theorem, they claim, is then that no local functions of the form $A({\bf a}, \lambda)=\pm1$ and $B({\bf b}, \lambda)=\pm1$ can be used to exceed those bounds.

But their claim is wrong on at least two counts. To begin with, their claim is based on the same mistaken assumption von Neumann's theorem is based on (as I show in my latest paper). And secondly, it is wrong because of what you say above (which is explicitly shown, for example, in my IEEE Access paper).

***
Joy Christian
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### Re: What exactly is Bell's Theorem?

FrediFizzx wrote:The Wikipedia entry for Bell's Theorem gives,

"If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local."

Which is from the book "Speakable and Unspeakable in Quantum Mechanics" page 65 which actually says,

"But if his extension is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local. This is what the theorem says."

However, in the paragraph before this statement, Bell gives another description of the so-called theorem.

"$A({\bf a}, \lambda)$
$B({\bf b}, \lambda)$ (3)

With these local forms, it is not possible to find functions A and B and a probability distribution $\rho$ which give the correlation (1). This is the theorem."

Correlation (1) is of course the quantum mechanical prediction of -a.b. Now..., someone with proper definitions could possibly make this into a rigorous mathematical theorem. But there not much point in that since Joy has already found local A and B functions that do give the QM correlation. This should really be the end of the debate.
.

When Bell himself talked about his *theorem* he usually meant his *inequality*. As a mathematician, I would rather call the inequality a *lemma*. It’s an easy to prove, elementary mathematical inequality. There are mathematical theorems concerning local realism and quantum mechanics. See for instance Boris Tsirelson’s (RIP) writings on Citizendium, or a not so old paper by Landsman and Cator. In order to qualify as mathematical theorem one has to give careful mathematical definitions of all concepts involved. My *executive summary* of such a theorem is: the formal mathematical structure called “quantum mechanics” is incompatible with mathematical structures possessing the conjunction of three, not two, properties: “locality”, “realism”, and “no-conspiracy”. Of course it is a matter for physicists, or for philosophers of science, to argue about whether or not certain formal mathematical definitions have any “real world” interest.

I have been very busy with Corona virus infection statistics, epidemiology, and hydroxychloroquine treatment. It is a good time for statisticians. Physicists also have very important insights.

Regarding “no-conspiracy: Bell himself of course knew very well that one had to rule out super-determinism or to say it in a positive sense, to assume the existence of something which some people like to call “free will”, but I would say is just a statistical assumption about the possibility to make effectively random choices when setting up experimental parameters of a real world lab experimental.

At this moment, Sabine Hossenfelder and Tim Palmer are two high-profile proponents of super-determinism. Earlier, Gerard ‘t Hooft has been an influential but somewhat lonely proponent of that point of view.
gill1109
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### Re: What exactly is Bell's Theorem?

gill1109 wrote:
When Bell himself talked about his *theorem* he usually meant his *inequality*.

Usually, but not always. Fred's observation is correct. For Bell his "inequality" was incidental --- a means to an end. For him, the "theorem" was about the impossibility of reproducing the correlations -a.b by averaging over the product A(a, h)*B(b, h) of the local functions A(a, h) and B(b, h). On page 65 of his book quoted by Fred, Bell explicitly says "This is the theorem."

However, his theorem, whichever form it takes, is simply wrong. But many things are wrong in this world and people still believe in them. That is the characteristic of a belief system.

gill1109 wrote:
Bell himself knew that one had to rule out super-determinism or to say it in a positive sense, to assume the existence of something which some people like to call “free will” ...

Here I agree with you. Super-determinism is a nonstarter. It dictates not only the answers but also questions that we can ask Nature. That is equivalent to giving up on scientific enterprise.

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Joy Christian
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### Re: What exactly is Bell's Theorem?

Gentlemen, perhaps this video will be of interest to you.

local

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### Re: What exactly is Bell's Theorem?

Joy Christian wrote:
FrediFizzx wrote:
Correlation (1) is of course the quantum mechanical prediction of -a.b. Now..., someone with proper definitions could possibly make this into a rigorous mathematical theorem. But there is not much point in that since Joy has already found local A and B functions that do give the QM correlation. This should really be the end of the debate.

I of course agree with your conclusion. But just to be the devil's advocate (because the devils seem to have fled the battleground), what the Bell-believers would say is that by considering four EPR-Bohm type experiments and assuming only locality and realism (aka "counterfactual definiteness") they are able to derive the bounds of +/-2 on the CHSH correlator. The theorem, they claim, is then that no local functions of the form $A({\bf a}, \lambda)=\pm1$ and $B({\bf b}, \lambda)=\pm1$ can be used to exceed those bounds.
...
***

It's trivial that if you have local A and B functions that produce the QM correlation of -a.b, you will exceed the bounds of CHSH also. So, all it takes is Bell's original "theorem" that I have shown above. With that disproven, all of the rest of it also falls into junk physics theory territory.
.
FrediFizzx
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### Re: What exactly is Bell's Theorem?

FrediFizzx wrote:
Joy Christian wrote:
FrediFizzx wrote:
Correlation (1) is of course the quantum mechanical prediction of -a.b. Now..., someone with proper definitions could possibly make this into a rigorous mathematical theorem. But there is not much point in that since Joy has already found local A and B functions that do give the QM correlation. This should really be the end of the debate.

I of course agree with your conclusion. But just to be the devil's advocate (because the devils seem to have fled the battleground), what the Bell-believers would say is that by considering four EPR-Bohm type experiments and assuming only locality and realism (aka "counterfactual definiteness") they are able to derive the bounds of +/-2 on the CHSH correlator. The theorem, they claim, is then that no local functions of the form $A({\bf a}, \lambda)=\pm1$ and $B({\bf b}, \lambda)=\pm1$ can be used to exceed those bounds.
...
***

It's trivial that if you have local A and B functions that produce the QM correlation of -a.b, you will exceed the bounds of CHSH also. So, all it takes is Bell's original "theorem" that I have shown above. With that disproven, all of the rest of it also falls into junk physics theory territory.

Sure. But to recognize that elementary fact requires honesty, scientific integrity, and competence in physics. That is asking too much from those irrationally committed to Bell orthodoxy.

***
Joy Christian
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### Re: What exactly is Bell's Theorem?

Joy Christian wrote:
FrediFizzx wrote:
Joy Christian wrote:
FrediFizzx wrote:
Correlation (1) is of course the quantum mechanical prediction of -a.b. Now..., someone with proper definitions could possibly make this into a rigorous mathematical theorem. But there is not much point in that since Joy has already found local A and B functions that do give the QM correlation. This should really be the end of the debate.

I of course agree with your conclusion. But just to be the devil's advocate (because the devils seem to have fled the battleground), what the Bell-believers would say is that by considering four EPR-Bohm type experiments and assuming only locality and realism (aka "counterfactual definiteness") they are able to derive the bounds of +/-2 on the CHSH correlator. The theorem, they claim, is then that no local functions of the form $A({\bf a}, \lambda)=\pm1$ and $B({\bf b}, \lambda)=\pm1$ can be used to exceed those bounds.
...
***

It's trivial that if you have local A and B functions that produce the QM correlation of -a.b, you will exceed the bounds of CHSH also. So, all it takes is Bell's original "theorem" that I have shown above. With that disproven, all of the rest of it also falls into junk physics theory territory.

Sure. But to recognize that elementary fact requires honesty, scientific integrity, and competence in physics. That is asking too much from those irrationally committed to Bell orthodoxy.

***

Competence in physics involves having certain broad mathematical skills (and even statistical skills).
gill1109
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### Re: What exactly is Bell's Theorem?

gill1109 wrote:
Joy Christian wrote:
Sure. But to recognize that elementary fact requires honesty, scientific integrity, and competence in physics. That is asking too much from those irrationally committed to Bell orthodoxy.

Competence in physics involves having certain broad mathematical skills (and even statistical skills).

Absolutely. And we must give the benefit of the doubt to those irrationally committed to Bell-orthodoxy that they do possess broad mathematical skills and even statistical skills. Indeed, Bell himself certainly possessed those skills (as evidenced by his excellent work in theoretical particle physics and phenomenological accelerator physics); and so did his followers such as Wigner (a Nobel Laureate) and Shimony (a Lakatos Laureate). And yet, each one of them failed, as did the mighty von Neumann, in applying those skills to the question of hidden variables underlying quantum mechanics, which is of course a profoundly physical question. von Neumann misapplied his superhuman mathematical ability to this physical question, and so did Bell, who ended up making exactly the same mistake as the one von Neumann had made after ridiculing him for making that mistake. Oh ... the irony!

Ref: https://arxiv.org/abs/1704.02876 (2020).

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Joy Christian
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### Re: What exactly is Bell's Theorem?

Sabine Hossenfelder has a recent post about nonlocality and Bell's theorem on her blog:

http://backreaction.blogspot.com/2020/0 ... 3-non.html
Heinera

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### Re: What exactly is Bell's Theorem?

Heinera wrote:
Sabine Hossenfelder has a recent post about nonlocality and Bell's theorem on her blog:

http://backreaction.blogspot.com/2020/0 ... 3-non.html

With all due respect to Sabine, I prefer my own explanation of quantum correlations. My explanation is far more lucid, compelling, and non-mystical: https://arxiv.org/abs/1911.11578.

***
Joy Christian
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### Re: What exactly is Bell's Theorem?

FrediFizzx wrote:The Wikipedia entry for Bell's Theorem gives,

"If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local."

Which is from the book "Speakable and Unspeakable in Quantum Mechanics" page 65 which actually says,

"But if his extension is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local. This is what the theorem says."

However, in the paragraph before this statement, Bell gives another description of the so-called theorem.

"$A({\bf a}, \lambda)$
$B({\bf b}, \lambda)$ (3)

With these local forms, it is not possible to find functions A and B and a probability distribution $\rho$ which give the correlation (1). This is the theorem."

Correlation (1) is of course the quantum mechanical prediction of -a.b. Now..., someone with proper definitions could possibly make this into a rigorous mathematical theorem. But there not much point in that since Joy has already found local A and B functions that do give the QM correlation. This should really be the end of the debate.
.

For some unknown reason, the Bell fans don't want to even accept Bell's own definition of the "theorem". So, now we have Gill's "theorem" which is basically that a local model can't simulate Nature and the experiments. But quantum mechanics can't predict correct individual outcome events for A and B either. So, what is the point? Now what is more, Jay Yablon has successfully demonstrated that quantum mechanics is local for the EBR-Bohm scenario so locality is no longer an issue. QM and... Nature must be local so non-local simulations are completely out.
.
FrediFizzx
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