What exactly is Bell's Theorem?

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

Re: What exactly is Bell's Theorem?

Postby gill1109 » Sun Sep 27, 2020 4:50 am

Joy Christian wrote:Here is ... an irrefutable scientific argument: The additivity of expectation values is not a valid or acceptable assumption for any hidden variable theory, regardless of locality or reality. On the other hand, the only way to derive the bounds of +/-2 on the CHSH correlator is by assuming the additivity of expectation values. If the additivity of expectation values is not assumed, then the bounds on the CHSH correlator are +/-4, not +/-2. In the experiments, the bounds of +/-2 are exceeded. Therefore the assumption of the additivity of expectation values is ruled out by the experiments. Locality and realism remain untouched and unscathed, contrary to what Bell and his followers believe.

The argument is easily refutable. A hidden variable theory is a theory in which outcomes which would be observed if various different measurements were done are all defined, or more generally, can all be defined. Moreover, this is done in such a way that the theory predicts the same correlations between jointly observable variables as quantum mechanics does; or at least, it predicts the same correlations up to very close approximation. Additivity of expectation values is not an assumption. Joint existence (in a mathematical sense) is the central assumption of a hidden variables theory. Your own hidden variables theories are of this kind. You define functions A(a, lambda) and B(b, lambda) etc etc etc.

Linearity of expectations is now a corollary.

Itamar Pitowsky tried to escape this by assuming non measurability so that expectation values could not be defined in the way of modern measure-theoretic probability theory (Kolmogorov). Others try to escape this by assuming some inconsistency in the usual ZFC axioms of set theory. My own work shows that by assuming randomness only in the setting choices, Bell’s theorem follows without any probabilistic assumptions on the hidden variables model at all. Only standard discrete (finite, counting) probability is needed.

“Bell followers” are not members of some religious sect. No modern scientist is a “Bell follower”. There are hundreds of proofs of hundreds of variants of Bell’s theorem. The maths and the logic are not difficult. The problem is that nobody can claim to *understand* quantum mechanics, though one can become familiar with the mathematical structure.

I suppose that you are also not the leader of a religious sect, but a scientist.
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Re: What exactly is Bell's Theorem?

Postby Joy Christian » Sun Sep 27, 2020 5:29 am

gill1109 wrote:
Joy Christian wrote:Here is ... an irrefutable scientific argument: The additivity of expectation values is not a valid or acceptable assumption for any hidden variable theory, regardless of locality or reality. On the other hand, the only way to derive the bounds of +/-2 on the CHSH correlator is by assuming the additivity of expectation values. If the additivity of expectation values is not assumed, then the bounds on the CHSH correlator are +/-4, not +/-2. In the experiments, the bounds of +/-2 are exceeded. Therefore the assumption of the additivity of expectation values is ruled out by the experiments. Locality and realism remain untouched and unscathed, contrary to what Bell and his followers believe.

The argument is easily refutable. A hidden variable theory is a theory in which outcomes which would be observed if various different measurements were done are all defined, or more generally, can all be defined. Moreover, this is done in such a way that the theory predicts the same correlations between jointly observable variables as quantum mechanics does; or at least, it predicts the same correlations up to very close approximation. Additivity of expectation values is not an assumption. Joint existence (in a mathematical sense) is the central assumption of a hidden variables theory. Your own hidden variables theories are of this kind. You define functions A(a, lambda) and B(b, lambda) etc etc etc.

Linearity of expectations is now a corollary.

In a hidden variable theory, all observables have definite values and those values must be eigenvalues of the corresponding quantum mechanical operators. The eigenvalue x(r,s,t,u) of the observable that correponds to the quantum mechanical operator R+S+T+U and appears on the right-hand side of the assumption of additivity of expectation values in the derivation of the CHSH inequalities is not a linear combination r+s+t+u of the eigenvalues of R, S, T, and U. But Bell and followers wrongly assume that it is a linear combination r+s+t+u and use that to derive the wrong bounds of +/-2. The use of wrong eigenvalue leads them to wrong bounds. It is a rookie mistake. For the correct derivation of the correct bounds +/-2\/2, see my paper:

https://arxiv.org/abs/1704.02876.

gill1109 wrote:
“Bell followers” are not members of some religious sect.

In the light of the above rookie mistake, it is clear that Bell’s theorem is a politically sustained belief system. Therefore, in my books, the followers of Bell are members of a religious sect.

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Re: What exactly is Bell's Theorem?

Postby gill1109 » Mon Sep 28, 2020 12:55 am

Indeed. But the “observable” R+S+T+U is not observed directly, hence its eigenvalues are irrelevant. This is a quite subtle point. Joy is really the first person who brings it clearly out into the open. We only *deduce* things about that “observable”, by making the working assumption that a local hidden variable model is possible. We reach a contradiction, exactly the contradiction which Joy discusses. Hence our no-go theorem. The working assumption must be rejected. Proof by contradiction. To prove something is impossible, you assume it to be true, and see where that brings you.
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Re: What exactly is Bell's Theorem?

Postby Joy Christian » Mon Sep 28, 2020 3:33 am

Joy Christian wrote:
gill1109 wrote:
Joy Christian wrote:Here is ... an irrefutable scientific argument: The additivity of expectation values is not a valid or acceptable assumption for any hidden variable theory, regardless of locality or reality. On the other hand, the only way to derive the bounds of +/-2 on the CHSH correlator is by assuming the additivity of expectation values. If the additivity of expectation values is not assumed, then the bounds on the CHSH correlator are +/-4, not +/-2. In the experiments, the bounds of +/-2 are exceeded. Therefore the assumption of the additivity of expectation values is ruled out by the experiments. Locality and realism remain untouched and unscathed, contrary to what Bell and his followers believe.

The argument is easily refutable. A hidden variable theory is a theory in which outcomes which would be observed if various different measurements were done are all defined, or more generally, can all be defined. Moreover, this is done in such a way that the theory predicts the same correlations between jointly observable variables as quantum mechanics does; or at least, it predicts the same correlations up to very close approximation. Additivity of expectation values is not an assumption. Joint existence (in a mathematical sense) is the central assumption of a hidden variables theory. Your own hidden variables theories are of this kind. You define functions A(a, lambda) and B(b, lambda) etc etc etc.

Linearity of expectations is now a corollary.

In a hidden variable theory, all observables have definite values and those values must be eigenvalues of the corresponding quantum mechanical operators. The eigenvalue x(r,s,t,u) of the observable that correponds to the quantum mechanical operator R+S+T+U and appears on the right-hand side of the assumption of additivity of expectation values in the derivation of the CHSH inequalities is not a linear combination r+s+t+u of the eigenvalues of R, S, T, and U. But Bell and followers wrongly assume that it is a linear combination r+s+t+u and use that to derive the wrong bounds of +/-2. The use of wrong eigenvalue leads them to wrong bounds. It is a rookie mistake. For the correct derivation of the correct bounds +/-2\/2, see my paper:

https://arxiv.org/abs/1704.02876.

gill1109 wrote:
Indeed. But the “observable” R+S+T+U is not observed directly, hence its eigenvalues are irrelevant. This is a quite subtle point. Joy is really the first person who brings it clearly out into the open. We only *deduce* things about that “observable”, by making the working assumption that a local hidden variable model is possible. We reach a contradiction, exactly the contradiction which Joy discusses. Hence our no-go theorem. The working assumption must be rejected. Proof by contradiction. To prove something is impossible, you assume it to be true, and see where that brings you.

According to the Hilbert space formulation of quantum theory, the correspondence between observables and self-adjoint operators is one-to-one. Now, it is correct that R+S+T+U is never observed in any Bell-test experiments that test singlet correlations. But in the Hilbert space of the singlet state, the sum R+S+T+U is a self-adjoint operator because R, S, T, and U are all self-adjoint operators themselves, and therefore R+S+T+U is observable in principle, at least counterfactually. In other words, a "God" can observe it, at least counterfactually. Therefore, the eigenvalue of the operator R+S+T+U is anything but irrelevant in a local or nonlocal hidden variable theory in which counterfactual possibilities are on par with the actual occurrences.

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Re: What exactly is Bell's Theorem?

Postby gill1109 » Thu Oct 01, 2020 5:14 am

Not all physicists will agree that *every* self adjoint operator must correspond to a physical observable. But OK, if you do take that as axiomatic, then there is indeed an issue here. If you go that way, then your argument is a proof of a version of the Kochen-Specker theorem. It’s nothing to do with locality. Nice! Congratulations.
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Re: What exactly is Bell's Theorem?

Postby Joy Christian » Thu Oct 01, 2020 6:25 am

gill1109 wrote:
Not all physicists will agree that *every* self adjoint operator must correspond to a physical observable. But OK, if you do take that as axiomatic, then there is indeed an issue here. If you go that way, then your argument is a proof of a version of the Kochen-Specker theorem. It’s nothing to do with locality. Nice! Congratulations.

My conclusion, to put it in my words, which appear in the last line of my (yet to be published) arXiv paper, is that "what is ruled out by Bell-test experiments is not local realism, but the additivity of expectation values (21), which does not hold for hidden variable theories to begin with."

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Re: What exactly is Bell's Theorem?

Postby gill1109 » Thu Oct 01, 2020 1:48 pm

Joy Christian wrote:
gill1109 wrote:
Not all physicists will agree that *every* self adjoint operator must correspond to a physical observable. But OK, if you do take that as axiomatic, then there is indeed an issue here. If you go that way, then your argument is a proof of a version of the Kochen-Specker theorem. It’s nothing to do with locality. Nice! Congratulations.

My conclusion, to put it in my words, which appear in the last line of my (yet to be published) arXiv paper, is that "what is ruled out by Bell-test experiments is not local realism, but the additivity of expectation values (21), which does not hold for hidden variable theories to begin with."

***

Exactly. Why should it? That's what I am saying. It depends, of course, on what you mean by "a hidden variable theory".Which depends on what you mean by a whole load of other things.

The maths is clear. The "meaning" you give to it is up to you.The meaning of the maths is socially, culturally, determined. It's relative to a more or less shared world view. That's challenging, that's diversity!
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Re: What exactly is Bell's Theorem?

Postby FrediFizzx » Thu Oct 01, 2020 1:59 pm

gill1109 wrote:
Joy Christian wrote:
gill1109 wrote:
Not all physicists will agree that *every* self adjoint operator must correspond to a physical observable. But OK, if you do take that as axiomatic, then there is indeed an issue here. If you go that way, then your argument is a proof of a version of the Kochen-Specker theorem. It’s nothing to do with locality. Nice! Congratulations.

My conclusion, to put it in my words, which appear in the last line of my (yet to be published) arXiv paper, is that "what is ruled out by Bell-test experiments is not local realism, but the additivity of expectation values (21), which does not hold for hidden variable theories to begin with."

***

Exactly. Why should it? That's what I am saying. It depends, of course, on what you mean by "a hidden variable theory".Which depends on what you mean by a whole load of other things.

The maths is clear. The "meaning" you give to it is up to you.The meaning of the maths is socially, culturally, determined. It's relative to a more or less shared world view. That's challenging, that's diversity!

What a bunch of mumbo jumbo. Is that all you can do? Mumbo jumbo?
.
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Re: What exactly is Bell's Theorem?

Postby gill1109 » Fri Oct 02, 2020 1:23 am

FrediFizzx wrote:
gill1109 wrote:
Joy Christian wrote:
gill1109 wrote:Not all physicists will agree that *every* self-adjoint operator must correspond to a physical observable. But OK, if you do take that as axiomatic, then there is indeed an issue here. If you go that way, then your argument is a proof of a version of the Kochen-Specker theorem. It’s nothing to do with locality. Nice! Congratulations.

My conclusion, to put it in my words, which appear in the last line of my (yet to be published) arXiv paper, is that "what is ruled out by Bell-test experiments is not local realism, but the additivity of expectation values (21), which does not hold for hidden variable theories to begin with."

Exactly. Why should it? That's what I am saying. It depends, of course, on what you mean by "a hidden variable theory". Which depends on what you mean by a whole load of other things. The maths is clear. The "meaning" you give to it is up to you. The meaning of the maths is socially, culturally, determined. It's relative to a more or less shared world view. That's challenging, that's diversity!

What a bunch of mumbo jumbo. Is that all you can do? Mumbo jumbo?

No, it is not all I can do. Participate in our conference and help us re-formulate Bell's theorem.https://gill1109.com/2020/10/02/time-reality-and-bells-theorem/
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Re: What exactly is Bell's Theorem?

Postby Joy Christian » Fri Oct 02, 2020 2:33 am

gill1109 wrote:
No, it is not all I can do. Participate in our conference and help us re-formulate Bell's theorem.https://gill1109.com/2020/10/02/time-reality-and-bells-theorem/

:lol: Fred, if I were you, I wouldn't touch Gill's "conference" with a barge pole. No matter how one tries to "re-formulate" garbage, it remains garbage. A dog born in a barn remains a dog. :)

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Re: What exactly is Bell's Theorem?

Postby gill1109 » Fri Oct 02, 2020 5:15 am

Joy Christian wrote:
gill1109 wrote:No, it is not all I can do. Participate in our conference and help us re-formulate Bell's theorem.https://gill1109.com/2020/10/02/time-reality-and-bells-theorem/

:lol: Fred, if I were you, I wouldn't touch Gill's "conference" with a barge pole. No matter how one tries to "re-formulate" garbage, it remains garbage. A dog born in a barn remains a dog. :)

The conference is organised by Sabine Hossenfelder, Ivette Fuentes, Jay Yablon and Jan-Ake Larsson. I am merely a facilitator. And I plan to give a birthday party. Klaas Landsman is going to give a talk. He said to me "Joy would probably like my lecture, by the way ...". Referring to his prize-winning essay, which he says destroys the whole meeting. "I am finished with Bell", he said to me. https://fqxi.org/community/essay/winners/2020.1. Time to bury the hatchet, I would say! Come and smoke the peace pipe. We have good stuff in the Netherlands, it's even legal... If I'm a dog born in a barn, I'm proud of that.
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Re: What exactly is Bell's Theorem?

Postby gill1109 » Fri Dec 18, 2020 3:01 am

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.

"
(3)

With these local forms, it is not possible to find functions A and B and a probability distribution 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.

That was not the end of the debate, since many people did not agree with Joy's claim. But I must say this particular quotation from Bell's works is very good, and it is part of a really excellent and very short paper "Locality in quantum mechanics: reply to critics" by John Bell, just two pages in the book "speakable and unspeakable" (it's chapter 8 there).
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Re: What exactly is Bell's Theorem?

Postby gill1109 » Fri Dec 18, 2020 5:24 am

This is the complete paper by Bell to which Fred referred. For reasons of internet typography I replaced lambda with mu, rho with pi, an overline with an underline, and dropped Bell's hats on bold a and bold b. This was written by John Bell some time after his paper Bell (1964) appeared, but before the Bell papers using CHSH started coming out. "His theorem" is therefore his 1964 theorem, not his later, stronger, theorems.
=======================================================================================================

The editor has asked me to reply to a paper, by G. Lochak [1], refuting a theorem of mine on hidden variables. If I understand correctly, Lochak finds that I failed somehow to allow for the effect on these variables of the measuring equipment. I will try to explain why I do not agree. The opportunity will also be taken here to comment on another refutation [2], by L. de la Peña, A. M. Cetto and T. A. Brody, and on another [3] by L. de Broglie. Yet another refutation of the same theorem, by J. Bub[4], has already been refuted by S. Freedman and E. P. Wigner [5].

Let us recall a typical context to which the theorem is relevant. A ‘pair of spin 1/2 particles’ is produced in a space-time region 3 and activates counting systems, preceded by Stern–Gerlach magnets, in space-time regions 1 and 2. The system at 1 is such that one of two counters (‘up’ or ‘down’) registers each time the experiment is done; correspondingly we label the result there by A ( = + 1 or – 1). Likewise the system at 2 is such that one of two counters registers each time the experiment is done, giving B ( = + 1 or – 1). We are interested in correlations between the counts in 1 and 2, and define a correlation function AB which is the average of the product of A and B over many repetitions of the experiment.

Now it would certainly be better to give a purely operational, technological, macroscopic, description of the equipment involved. This would avoid completely any use of words like ‘particle’ and ‘spin’, and so avoid the possibility that someone feels obliged to form a personal microscopic picture of what is going on. But it would take quite long to give such a purely technological specification. So, please accept that the words ‘particle’ and ‘spin’ are used here only as part of a conventional shorthand, to invoke without lengthy explicit description the kind of experimental equipment involved, and with no commitment whatever to any picture of what, if anything, really causes the counters to count.

Suppose that part of the specification of the equipment is by two unit vectors a and b (e.g., the directions of certain magnetic fields at 1 and 2). Then according to ordinary quantum mechanics situations exist for which

AB = –a.b (1)

to good accuracy.

Actually it is this last statement which is challenged by de Broglie. Although his paper is called ‘Sur la réfutation du théorème de Bell’, it is not in fact concerned with any reasoning of mine. He is of the opinion that the correlation function (1) simply cannot occur for macroscopic separations, either in nature or in ordinary quantum mechanics: ‘Nous échappons complétement à cette objection puisque, pour nous, les mesures du spin sur des électrons éloignés ne sont pas corrélées’. As regards ordinary quantum mechanics, de Broglie disagrees here with most students of the subject, and I am unable to follow his reasons for doing so. As regards nature, he seems to disagree also with experiment [6].

Now we investigate the hypothesis that the final state of the system, in particular A and B, would be fully determined by the equations of some theory if the initial conditions were fully specified. So to parameters like a and b, subject to experimental manipulation, we add a list of hypothetical ‘hidden’ parameters µ. We can take these µ to be the initial values (say just after the action of the source) of some corresponding dynamical variables. We have no interest in what subsequently happens to these variables except in so far as they enter into the measurement results A and B. But in so far as they do enter into A and B we allow fully for the effect of the measuring equipment by allowing A and B to depend not only on the initial values µ of the hidden parameters but also on the parameters a and b, specifying the measuring devices:

A(a, b, µ), B(a, b, µ). (2)

We have no need to enquire into the precise nature of this dependence on a and b, nor into how it comes about, whether by the effect of the measuring equipment on the hidden variables of which the µ are the initial values, or otherwise.

Can one find some functions (2) and some probability distribution π(µ) which reproduces the correlation (1)? Yes, many, but now we add the hypothesis of locality, that the setting b of a particular instrument has no effect on what happens, A, in a remote region, and likewise that a has no effect on B:
A(a, µ), B(b, µ). (3)

With these local forms, it is not possible to find functions A and B and a probability distribution π which give the correlation (1). This is the theorem. The proof will not be repeated here.

Lochak illustrates the way in which the output of a single instrument A depends on its setting a, as allowed for in (3), in the hidden parameter theory of de Broglie. I think this is very instructive. But more instructive for the present purpose is the case of two instruments and two particles. Then one finds that in de Broglie’s theory the dependence is not of the local form (3) but of the nonlocal form (2). I have made this point on several occasions, in two of the three papers referred to by Lochak and elsewhere[7]. It may be that Lochak has in mind some other extension of de Broglie’s theory, to the more-than-one-particle system, than the straightforward generalization from 3 to 3N dimensions that I considered. 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.

The objection of de la Peña, Cetto, and Brody is based on a misinterpretation of the demonstration of the theorem. In the course of it reference is made to

A(a', µ), B(b', µ)

as well as to

A(a, µ), B(b, µ).

These authors say “Clearly, since A, A', B, B' are all evaluated for the same µ, they must refer to four measurements carried out on the same electron–positron pair. We can suppose, for instance, that A' is obtained after A, and B' after B”. But by no means. We are not at all concerned with sequences of measurements on a given particle, or of pairs of measurements on a given pair of particles. We are concerned with experiments in which for each pair the ‘spin’ of each particle is measured once only. The quantities

A(a', µ), B(b', µ)

are just the same functions

A(a, µ), B(b, µ)

with different arguments.

References

  [1] G. Lochak, Fundamenta Scientiae (Universite de Strasbourg, 1975), No 38, reprinted in Epistemological Letters, p. 41, September 1975.
  [2] L. de la Pena, A. M. Cetto and T. A. Brody, Nuovo Cimento Letters 5, 177 (1972).
  [3] L. de Broglie, CR Acad. Sci. Paris 278, B721 (1974).
  [4] J. Bub, Found. Phys. 3, 29 (1973).
  [5] S. Freedman and E. Wigner, Found. Phys. 3, 457 (1973).
  [6] S. J. Freedman and J. F. Clauser, Phys. Rev. Lett. 28, 938 (1972). A brief account is given by M. Paty, Epistemological Letters, p. 31, September 1975.
  [7] J. S. Bell, On the Hypothesis that the Schrödinger Equation is Exact, CERN Preprint TH. 1424 (1971).
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