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

Postby Joy Christian » Fri May 22, 2020 3:22 pm

Joy Christian wrote:
gill1109 wrote:
In pure mathematics one can go for absolute truth but on the other hand those absolute truths are mere tautologies.

That is exactly right. A fine example of that is Bell's theorem. It is tautologous. It assumes (in a different guise) what it wants to prove, in order to prove it. Nice mathematics, bad physics.

Actually, Bell's theorem suffers from a "double whammy." It is a circular argument to begin with, because it assumes something that amounts to assuming the bounds of -2 and +2 on the CHSH correlator that it intends to prove. But what is more, what it assumes [i.e., Eq. (16) of my paper] is a false premise. So it assumes a false premise to derive a false conclusion. :)

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Re: The Mistakes by Bell and von Neumann are Identical

Postby Joy Christian » Fri Jun 26, 2020 6:30 pm

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I have revised the abstract of this paper, which I am reproducing below. The revised version of the paper should be online by Monday.

Image

The Bell-worshipers have been cheering Bell for ridiculing von Neumann for making a silly mistake in his theorem. Now the only options left for them are dishonesty and double-standards.

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Re: The Mistakes by Bell and von Neumann are Identical

Postby gill1109 » Fri Jun 26, 2020 9:59 pm

Joy Christian wrote:***
I have revised the abstract of this paper, which I am reproducing below. The revised version of the paper should be online by Monday.

Image

The Bell-worshipers have been cheering Bell for ridiculing von Neumann for making a silly mistake in his theorem. Now the only options left for them are dishonesty and double-standards.

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I think you are wrong, Joy. If you look at Bell's proof carefully you will see that the addition of expectation values of observations of objects which, under quantum mechanics, are modelled through an abstract mathematical framework involving non commuting operators, is carefully justified *under the assumption of local hidden variables*. Remember, that when we work under a hidden variables theory, we may forget *all* the dogmas of quantum theory. We are limited by experimental reality, only, and possibly other more fundamental principles such as locality and no-conspiracy.

Your confusion is typical of the confusion of physicists who have been brought up in a universe of quantum physics. They have too deeply absorbed dogmas from their teachers. They have absorbed "lies for children" which those teachers learnt from their teachers. I think I should write a short response paper to yours.
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Re: The Mistakes by Bell and von Neumann are Identical

Postby Joy Christian » Fri Jun 26, 2020 11:18 pm

gill1109 wrote:
Joy Christian wrote:***
I have revised the abstract of this paper, which I am reproducing below. The revised version of the paper should be online by Monday.

Image

The Bell-worshipers have been cheering Bell for ridiculing von Neumann for making a silly mistake in his theorem. Now the only options left for them are dishonesty and double-standards.

***

I think you are wrong, Joy. If you look at Bell's proof carefully you will see that the addition of expectation values of observations of objects which, under quantum mechanics, are modelled through an abstract mathematical framework involving non commuting operators, is carefully justified *under the assumption of local hidden variables*. Remember, that when we work under a hidden variables theory, we may forget *all* the dogmas of quantum theory. We are limited by experimental reality, only, and possibly other more fundamental principles such as locality and no-conspiracy.

Your confusion is typical of the confusion of physicists who have been brought up in a universe of quantum physics. They have too deeply absorbed dogmas from their teachers. They have absorbed "lies for children" which those teachers learnt from their teachers. I think I should write a short response paper to yours.

You have absolutely no idea what you are talking about. You have no understanding of what a hidden variable theory is. You have not read my paper to understand my argument. You have not even read my Introduction. What you have written above is nothing but dogmatic folklore of Bell-worshipers. I should have added a third "d": dishonesty, double-standards, and denial.

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Re: The Mistakes by Bell and von Neumann are Identical

Postby gill1109 » Sat Jun 27, 2020 7:08 am

I have very carefully read the current version of Joy Christian’s paper. Obviously, I disagree with his evaluation of my motives and of my intellectual capacities. I will proceed to write out a more full analysis (refutation) of his argument. When I have done that, I’ll publish in a suitable medium and inform folks here. His standpoint is interesting and important because, as I wrote this morning, it represents a still prevalent misconception about Bell’s discovery.
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Re: The Mistakes by Bell and von Neumann are Identical

Postby Joy Christian » Sat Jun 27, 2020 7:24 am

gill1109 wrote:
I have very carefully read the current version of Joy Christian’s paper. Obviously, I disagree with his evaluation of my motives and of my intellectual capacities. I will proceed to write out a more full analysis (refutation) of his argument. When I have done that, I’ll publish in a suitable medium and inform folks here. His standpoint is interesting and important because, as I wrote this morning, it represents a still prevalent misconception about Bell’s discovery.

Bell did not discover anything. Bell simply made the same silly mistake that von Neumann had made in his own little theorem. The revised abstract of my paper explains this quite clearly:

Image

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Re: The Mistakes by Bell and von Neumann are Identical

Postby Joy Christian » Sun Jun 28, 2020 5:20 pm

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Here is a published paper by Richard D. Gill: https://link.springer.com/article/10.10 ... 015-2657-4.

It is a critique of one of my papers, published in the same journal. I have criticized Gill's paper before for different reasons in this unpublished preprint: https://arxiv.org/abs/1501.03393.

But here I want to point out the mistake in all Bell-type arguments I have been highlighting in this thread. This mistake is conspicuous in Gill's published paper, which I reproduce below:

Image

It is the second unnumbered equation seen above that is wrong. Gill is referring to me and my published paper in the above paragraph. Mind you, there is nothing mathematically wrong in his equation. Mathematically it is a trivial equation and its LHS is indeed equal to its RHS. But physically the equation is nonsense. It assumes that the sum of expectation values is equal to the expectation value of the sum. That is not valid for hidden variable theories, as I have explained here: https://arxiv.org/abs/1704.02876. That is the same mistake von Neumann had made in his theorem --- the one that Bell ridiculed him for making.

Exactly the same mistake exists in every single proof of Bell's theorem, starting with the proof presented by Bell in his famous 1964 paper. The physics community has been fooled by Bell and his followers for the past 56 years. And I have to report this here because the followers of Bell have been stonewalling my papers exposing Bell's mistakes for the past thirteen years.

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Re: The Mistakes by Bell and von Neumann are Identical

Postby gill1109 » Sun Jul 05, 2020 11:48 pm

Joy Christian wrote:Here is a published paper by Richard D. Gill: https://link.springer.com/article/10.10 ... 015-2657-4. It is a critique of one of my papers, published in the same journal. I have criticized Gill's paper before for different reasons in this unpublished preprint: https://arxiv.org/abs/1501.03393. But here I want to point out the mistake in all Bell-type arguments I have been highlighting in this thread. This mistake is conspicuous in Gill's published paper, which I reproduce below:
Image
It is the second unnumbered equation seen above that is wrong. Gill is referring to me and my published paper in the above paragraph. Mind you, there is nothing mathematically wrong in his equation. Mathematically it is a trivial equation and its LHS is indeed equal to its RHS. But physically the equation is nonsense. It assumes that the sum of expectation values is equal to the expectation value of the sum. That is not valid for hidden variable theories.

Strange that a trivial arithmetical identity, in whose truth there is no doubt whatsoever, as Joy Christian himself agrees, should suddenly fail to be true *for physical reasons* when the numbers involved happen to be the values taken on by *hidden variables*. The hidden variables theory is a physical theory of what is going on behind the scenes which supposes that there are physical properties of physical objects, which take on numerical values in any particular instance but which happen not to be directly observable by human experimenters. The odd thing is that they apparently do not obey ordinary rules of arithmetic. What rules they do follow is never revealed. The physical meaning of the word "hidden" is that only certified physicists can talk about them. I don't think it makes sense to call such a theory a "theory" at all. It sounds more like a mystery religion to me. The high priest makes pronouncements which make no sense to anybody, but since his actions seemed to ensure that Winter gave way to Spring every year (because he performed those ritual incantations every Winter and so far, hey presto, every year, Spring did come again), the faithful continue to make sacrifices and offerings at his temple, enabling the high priest to live a cushy life, and giving him lots of political influence, since the rich and powerful need to consult him and get scientific backing for their policy decisions.

The amusing thing about Christian's proposed experiment involving colourful exploding balls which disintegrate into pairs of contrarywise spinning hemispheres, tracked by state of the art video cameras and video processing software, is that the so-called hidden variables are not hidden at all, but are in fact directly observed.
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Re: The Mistakes by Bell and von Neumann are Identical

Postby Joy Christian » Mon Jul 06, 2020 2:19 am

gill1109 wrote:
Joy Christian wrote:Here is a published paper by Richard D. Gill: https://link.springer.com/article/10.10 ... 015-2657-4. It is a critique of one of my papers, published in the same journal. I have criticized Gill's paper before for different reasons in this unpublished preprint: https://arxiv.org/abs/1501.03393. But here I want to point out the mistake in all Bell-type arguments I have been highlighting in this thread. This mistake is conspicuous in Gill's published paper, which I reproduce below:
Image
It is the second unnumbered equation seen above that is wrong. Gill is referring to me and my published paper in the above paragraph. Mind you, there is nothing mathematically wrong in his equation. Mathematically it is a trivial equation and its LHS is indeed equal to its RHS. But physically the equation is nonsense. It assumes that the sum of expectation values is equal to the expectation value of the sum. That is not valid for hidden variable theories.

Strange that a trivial arithmetical identity, in whose truth there is no doubt whatsoever, as Joy Christian himself agrees, should suddenly fail to be true *for physical reasons* when the numbers involved happen to be the values taken on by *hidden variables*. The hidden variables theory is a physical theory of what is going on behind the scenes which supposes that there are physical properties of physical objects, which take on numerical values in any particular instance but which happen not to be directly observable by human experimenters. The odd thing is that they apparently do not obey ordinary rules of arithmetic. What rules they do follow is never revealed. The physical meaning of the word "hidden" is that only certified physicists can talk about them. I don't think it makes sense to call such a theory a "theory" at all. It sounds more like a mystery religion to me. The high priest makes pronouncements which make no sense to anybody, but since his actions seemed to ensure that Winter gave way to Spring every year (because he performed those ritual incantations every Winter and so far, hey presto, every year, Spring did come again), the faithful continue to make sacrifices and offerings at his temple, enabling the high priest to live a cushy life, and giving him lots of political influence, since the rich and powerful need to consult him and get scientific backing for their policy decisions.

The amusing thing about Christian's proposed experiment involving colourful exploding balls which disintegrate into pairs of contrarywise spinning hemispheres, tracked by state of the art video cameras and video processing software, is that the so-called hidden variables are not hidden at all, but are in fact directly observed.

Your emotional rambling has near-zero scientific content. You say:
Richard D. Gill wrote:
The odd thing is that they apparently do not obey ordinary rules of arithmetic. What rules they do follow is never revealed. The physical meaning of the word "hidden" is that only certified physicists can talk about them.

There is no truth in these statements, except perhaps in the last one. Yes, you have to be a certified physicist, such as von Neumann, Bell, Wigner, or Shimony, to truly understand what is meant by "hidden variables." But that is true about any science. All editors of physics journals seek approval of only certified physicists. Why? Because there are very good reasons for that.

But I digress. The rules that must be respected by any hidden variable theory are precisely known since the work of von Neumann and others in the early 1930s. The main rule is so simple that even uncertified non-physicists can understand it. The rule is that, unlike in quantum theory, every observable in a hidden variable theory must have a definite value, which must be an eigenvalue of the corresponding quantum mechanical operator. That is it. The rule is hardly mysterious or imprecise. Therefore the mistake in Richard D. Gill's second unnumbered equation I have quoted above is also crystal clear. The RHS of his equation is not summing over the correct eigenvalue. The correct eigenvalue, for the current case, is given in Eq. (35) of my paper.

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Re: The Mistakes by Bell and von Neumann are Identical

Postby gill1109 » Mon Jul 06, 2020 10:42 am

Dear Joy, you are repeating von Neumann's error! You are confusing physical categories and mathematical categories. Von Neumann was a mathematician, not a physicist, that is why he fell into this error. He confused the mathematical model for the physical reality, because he only really knew about the mathematical model. You are a physicist and make the same error, but for the complementary reason. Bell was a real scientist and a real meta-physicist, who understood the distinction.

For a real physicist, the ultimate authority is nature. For a real mathematician, the ultimate authority is mathematical truth (abstract logic). Not many people reach the summits of both kinds of science. Bell was such a seldom person. He led a double life. Most mere mortals are trapped in one or the other. Statisticians are people who live on the interface. Nobody loves us. In fact, everyone hates us. But they all need us.

Physical observables are represented by Hermitean operators. But not every Hermitean operator represents a physical observable.
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Re: The Mistakes by Bell and von Neumann are Identical

Postby Joy Christian » Mon Jul 06, 2020 11:32 am

gill1109 wrote:
Physical observables are represented by Hermitean operators. But not every Hermitean operator represents a physical observable.

Wrong!

According to Hilbert space quantum mechanics --- which is universally accepted by all physicists --- the correspondence between Hermitian operators and observables is one-to-one.

But that is not the main problem. The main problem is that Bell's theorem is based on the assumption that the sum of expectation values is equal to the expectation value of the sum.

This assumption is not valid for hidden variable theories. Therefore Bell's theorem against local theories is as invalid as von Neumann's theorem against general hidden variable theories.

By the way, the correct spelling is Hermitian, not Hermitean.

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Re: The Mistakes by Bell and von Neumann are Identical

Postby gill1109 » Fri Jul 10, 2020 10:51 am

Joy Christian wrote:
gill1109 wrote:Physical observables are represented by Hermitian [typo corrected] operators. But not every Hermitian operator represents a physical observable.

Wrong!

According to Hilbert space quantum mechanics --- which is universally accepted by all physicists --- the correspondence between Hermitian operators and observables is one-to-one.

But that is not the main problem. The main problem is that Bell's theorem is based on the assumption that the sum of expectation values is equal to the expectation value of the sum.

This assumption is not valid for hidden variable theories. Therefore Bell's theorem against local theories is as invalid as von Neumann's theorem against general hidden variable theories.

Well, it is not universally agreed by physicists that the correspondence between Hermitian operators and observables is one-to-one. Only so-called "dynamical variables" are considered to be *observables*, and which Hermitian operators correspond to dynamical variables/observables depends on whether or not they satisfy transformation laws corresponding to changes of frames of reference of different observers. At least, that is what is written in the Wikipedia page on "Observable". If you disagree, find some reliable source which supports your point of view, and edit the article. It includes numerous references. You will have to rewrite most standard textbooks on quantum mechanics. Of course, disproving Bell's theorem should lead to the same result - all those books need to be rewritten.

But this is irrelevant to the main point in the OP. Let me mention an example. Let's take a look at sigma_x and sigma_z. They don't commute so can't be measured at the same time. Their sum is another 2x2 self-adjoint matrix. It is easily checked that it equals sqrt 2 times the matrix with orthonormal rows (1/sqrt 2, 1/sqrt2) and (1/sqrt 1, -1/sqrt 2). So it's another spin matrix! Let me call it sigma_u. Exercise to the reader: figure out what is the unit 3-vector u.

So we have sigma_x + sigma_z = sqrt 2 sigma_u and none of those three spin matrices can be measured at the same time. If the system is in the state rho (2x2 density matrix) then we find that trace(rho sigma_x) + trace(rho sigma_z) = sqrt 2 trace(rho sigma_u). The three expressions "trace rho sigma..." are the expectation values of the results of measuring the corresponding observable on a system in state rho. One could write <sigma_x> + <sigma_z> = sqrt 2 <sigma_u> and possibly add a subscript "rho" to the three "expectation operators" <...> to indicate the state of the system, whose dynamic variables (observables) one could consider observing, though not simultaneously. One can observe any one of the three as many times as one likes, but not simultaneously, on many new *preparations* of the same state.

The sum of expectation values is equal to the expectation value of the sum. J.C. writes "That is not valid for hidden variable theories". Oh yes, it is! Hidden variable theories reproduce exactly the predictions of quantum mechanics by the extension of the "universe" of quantum mechanics to a hidden universe of classical variables which underlie or explain in a classical statistical fashion, the probabilistic predictions of QM. QM tells us that when we observe sigma_x we get to observe +1 or -1 and we can compute the expected value of the measurement (ie the average of what we would see if we repeated the preparation of the state many, many times). So QM tells us the probabilities of the two outcomes of that measurement but does not explain, particle by particle, how it arises, from a deterministic evolution from the initial conditions of particle and measurement apparatus, which quite naturally vary from repetition to repetition. "Mere statistical variation" of "hidden", ie not directly observable, initial conditions were expected by Einstein to *explain* the random variation of outcomes of a measurement of the same observable on a system in the same "state". Einstein said "God does not toss dice" but actually, dice outcomes are deterministic, explained by classical physics, and Einstein did think that quantum dice were actually no different from classical dice. It's just that we can't (yet) directly see the true underlying dynamical variables of the system, nor (evidently) can we (yet) control them.

There is another problem. sigma_x and sigma_y when observed both produce outcomes +/- 1. One would expect measurement of their sum to produce outcomes which could be -2, 0 or +2. But when we measure their sum we get to observe +/- sqrt 2!

Sum of expectations equals expectation of sum is certainly true, but outcomes of sum are sums of outcomes of summands is not true!QM is definitely a bit weird.

This is a kind of toy version of the Kochen-Specker theorem.

By the way, take a look at the movie https://www.youtube.com/watch?v=XDpurdHKpb8, Infinite Potential: The Life and Ideas of David Bohm, to catch a glimpse of how the physical establishment silenced David Bohm, who showed that there was a hidden variable theory which reproduced quantum theory. Einstein called Bohm his spiritual son. Oppenheimer, who had been David's mentor and supporter, got some top physicists together to find the mistakes in Bohm's theory. They failed. So they decided to shun Bohm. He fled the US and moved to Israel and later London. Nobody cited his papers. He was silenced.
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Re: The Mistakes by Bell and von Neumann are Identical

Postby Joy Christian » Sat Jul 11, 2020 11:37 am

gill1109 wrote:
The sum of expectation values is equal to the expectation value of the sum. J.C. writes "That is not valid for hidden variable theories". Oh yes, it is!

All of your comments above are less than relevant except the one I have quoted. By claiming that the rule "sum of expectation values is equal to the expectation value of the sum" is valid for hidden variable theories, you are disagreeing not only with me but also with Einstein, Hermann, Siegel, Jauch and Piron, Kochen and Specker, Mermin and Schack, Shimony, as well as your hero John S. Bell himself. The references to the works of all of these authors can be found in my paper. It is our opinion versus yours. Let the physics community decide who is right.

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Re: The Mistakes by Bell and von Neumann are Identical

Postby Joy Christian » Sun Jul 12, 2020 2:18 am

gill1109 wrote:
The sum of expectation values is equal to the expectation value of the sum. J.C. writes "That is not valid for hidden variable theories". Oh yes, it is!

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

Postby gill1109 » Mon Jul 13, 2020 11:35 am

Joy Christian wrote:
gill1109 wrote:The sum of expectation values is equal to the expectation value of the sum. J.C. writes "That is not valid for hidden variable theories". Oh yes, it is!

All of your comments above are less than relevant except the one I have quoted. By claiming that the rule "sum of expectation values is equal to the expectation value of the sum" is valid for hidden variable theories, you are disagreeing not only with me but also with Einstein, Hermann, Siegel, Jauch and Piron, Kochen and Specker, Mermin and Schack, Shimony, as well as your hero John S. Bell himself. The references to the works of all of these authors can be found in my paper. It is our opinion versus yours. Let the physics community decide who is right.

You cite papers. Please cite papers with page number and line number or formula numbers. The space of random variables on a probability space is a linear space. If we restrict attention to bounded random variables, the expectation of a sum is a sum of expectation values. The space of self-adjoint operators on a Hilbert space is a linear space. Fix a density matrix rho. The mapping from bounded self-adjoint operators to real numbers defined by A |--> trace rho A is a linear map from operators to real numbers.

In fact, *all* normalised and positive linear maps from bounded measurable functions to real numbers are expectation values w.r.t. some probability measure (Hahn-Banach theorem, I think), and all normalised and positive linear maps from bounded self-adjoint operators to real numbers are quantum expectation values w.r.t. some density matrix. This is all part of the basic theory of Banach and Hilbert spaces. I very much doubt that Einstein, Hermann, Siegel, Jauch and Piron, Kochen and Specker, Mermin and Schack, Shimony, or John S. Bell, disagreed with these fundamental mathematical facts.

Perhaps your notion of a "hidden variable" is something outside of existing standard mathematics. Already, Pitowsky suggested that a hidden variable theory could be resurrected by using non-measurable functions whose expectation values quite simply did not exist. The law of large numbers would fail. Observed averages would forever fluctuate and never converge. But you do assume in all your papers that long run averages do converge, so this escape route is not available for you.
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Re: The Mistakes by Bell and von Neumann are Identical

Postby FrediFizzx » Mon Jul 13, 2020 12:12 pm

gill1109 wrote:... so this escape route is not available for you.

Why would Joy need that particular escape route? His model is classical. Which you seem to have forgotten.
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Re: The Mistakes by Bell and von Neumann are Identical

Postby Joy Christian » Mon Jul 13, 2020 12:32 pm

gill1109 wrote:
Joy Christian wrote:
gill1109 wrote:The sum of expectation values is equal to the expectation value of the sum. J.C. writes "That is not valid for hidden variable theories". Oh yes, it is!

All of your comments above are less than relevant except the one I have quoted. By claiming that the rule "sum of expectation values is equal to the expectation value of the sum" is valid for hidden variable theories, you are disagreeing not only with me but also with Einstein, Hermann, Siegel, Jauch and Piron, Kochen and Specker, Mermin and Schack, Shimony, as well as your hero John S. Bell himself. The references to the works of all of these authors can be found in my paper. It is our opinion versus yours. Let the physics community decide who is right.

You cite papers. Please cite papers with page number and line number or formula numbers. The space of random variables on a probability space is a linear space. If we restrict attention to bounded random variables, the expectation of a sum is a sum of expectation values. The space of self-adjoint operators on a Hilbert space is a linear space. Fix a density matrix rho. The mapping from bounded self-adjoint operators to real numbers defined by A |--> trace rho A is a linear map from operators to real numbers.

In fact, *all* normalised and positive linear maps from bounded measurable functions to real numbers are expectation values w.r.t. some probability measure (Hahn-Banach theorem, I think), and all normalised and positive linear maps from bounded self-adjoint operators to real numbers are quantum expectation values w.r.t. some density matrix. This is all part of the basic theory of Banach and Hilbert spaces. I very much doubt that Einstein, Hermann, Siegel, Jauch and Piron, Kochen and Specker, Mermin and Schack, Shimony, or John S. Bell, disagreed with these fundamental mathematical facts.

Perhaps your notion of a "hidden variable" is something outside of existing standard mathematics. Already, Pitowsky suggested that a hidden variable theory could be resurrected by using non-measurable functions whose expectation values quite simply did not exist. The law of large numbers would fail. Observed averages would forever fluctuate and never converge. But you do assume in all your papers that long run averages do converge, so this escape route is not available for you.

You still haven't understood the extremely elementary point --- universally accepted by those well versed in foundations of quantum mechanics and explicitly stated by Einstein, Hermann, Siegel, Jauch and Piron, Kochen and Specker, Mermin and Schack, Shimony, and Bell --- that the sum of expectation values is not equal to the expectation value of the sum for hidden variable theories when the corresponding quantum mechanical operators do not commute. Note that none of these authors disagree with von Neumann's claim that the sum of expectation values is equal to the expectation value of the sum for the quantum mechanical expectation values. Their disagreement with von Neumann on this point only concerns his extension of this rule to hidden variable theories, for which, as they rightly point out, it does not hold. Either read my paper to understand their elementary point or read sections 3, 4, and 5 of the very first chapter of Bell's book. Bell explains the point more lucidly than anyone else who has written on the subject before my paper was published on the arXiv.

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Re: The Mistakes by Bell and von Neumann are Identical

Postby gill1109 » Tue Jul 14, 2020 12:38 am

FrediFizzx wrote:
gill1109 wrote:... so this escape route is not available for you.

Why would Joy need that particular escape route? His model is classical. Which you seem to have forgotten.

No Fred, I haven't forgotten that. Joy's classical model reproduces the predictions of quantum mechanics. The prediction of quantum mechanics is that the sum of expectations is the expectation of the sum. So if Joy has classical variables which "underlie" the quantum variables, the same will be true for them.

They may be "hidden", in other words, not directly observable, but from a mathematical point of view, hidden variables are just random variables defined on a probability space, which also supports the random variables representing the "observables".

Yes, it is a subtle point!
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Re: The Mistakes by Bell and von Neumann are Identical

Postby gill1109 » Tue Jul 14, 2020 12:48 am

Joy Christian wrote:
gill1109 wrote:
Joy Christian wrote:
gill1109 wrote:The sum of expectation values is equal to the expectation value of the sum. J.C. writes "That is not valid for hidden variable theories". Oh yes, it is!

All of your comments above are less than relevant except the one I have quoted. By claiming that the rule "sum of expectation values is equal to the expectation value of the sum" is valid for hidden variable theories, you are disagreeing not only with me but also with Einstein, Hermann, Siegel, Jauch and Piron, Kochen and Specker, Mermin and Schack, Shimony, as well as your hero John S. Bell himself. The references to the works of all of these authors can be found in my paper. It is our opinion versus yours. Let the physics community decide who is right.

You cite papers. Please cite papers with page number and line number or formula numbers. The space of random variables on a probability space is a linear space. If we restrict attention to bounded random variables, the expectation of a sum is a sum of expectation values. The space of self-adjoint operators on a Hilbert space is a linear space. Fix a density matrix rho. The mapping from bounded self-adjoint operators to real numbers defined by A |--> trace rho A is a linear map from operators to real numbers.

In fact, *all* normalised and positive linear maps from bounded measurable functions to real numbers are expectation values w.r.t. some probability measure (Hahn-Banach theorem, I think), and all normalised and positive linear maps from bounded self-adjoint operators to real numbers are quantum expectation values w.r.t. some density matrix. This is all part of the basic theory of Banach and Hilbert spaces. I very much doubt that Einstein, Hermann, Siegel, Jauch and Piron, Kochen and Specker, Mermin and Schack, Shimony, or John S. Bell, disagreed with these fundamental mathematical facts.

Perhaps your notion of a "hidden variable" is something outside of existing standard mathematics. Already, Pitowsky suggested that a hidden variable theory could be resurrected by using non-measurable functions whose expectation values quite simply did not exist. The law of large numbers would fail. Observed averages would forever fluctuate and never converge. But you do assume in all your papers that long run averages do converge, so this escape route is not available for you.

You still haven't understood the extremely elementary point --- universally accepted by those well versed in foundations of quantum mechanics and explicitly stated by Einstein, Hermann, Siegel, Jauch and Piron, Kochen and Specker, Mermin and Schack, Shimony, and Bell --- that the sum of expectation values is not equal to the expectation value of the sum for hidden variable theories when the corresponding quantum mechanical operators do not commute. Note that none of these authors disagree with von Neumann's claim that the sum of expectation values is equal to the expectation value of the sum for the quantum mechanical expectation values. Their disagreement with von Neumann on this point only concerns his extension of this rule to hidden variable theories, for which, as they rightly point out, it does not hold. Either read my paper to understand their elementary point or read sections 3, 4, and 5 of the very first chapter of Bell's book. Bell explains the point more lucidly than anyone else who has written on the subject before my paper was published on the arXiv.

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I repeat: you cite a whole list of authors. Please cite specific papers with page number and line number or formula numbers. I have read those authors and read your papers too. Nowhere does anyone else say anything like what you say, at all.

Alternatively, do the calculations yourself. Come up with a hidden variables model for a single spin half system under state (density matrix) rho. Homework: prove that sigma_x + sigma_z = sqrt 2 sigma_u where u is some direction in 3D space. Check that in QM, the sum of the expectations equals the expectation of the sum, *whatever* the state rho. If your hidden variables theory reproduces the probability distributions of measurements of sigma_x, sigma_y and sigma_u, it will also reproduce their expectation values. Your hidden variables theory will support three random variables X, Z and U, and you will find that E(X) + E(Z) = sqrt 2 E(U).

You can also arrange that X, Z and U only take on the values +/- 1. Bell himself showed that in Hilbert space dimension 2 there are hidden variables models. The Kochen Specker theorem requires at least dimension 3. Bell's theorem requires at least dimension 2 x 2.
gill1109
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Re: The Mistakes by Bell and von Neumann are Identical

Postby Joy Christian » Tue Jul 14, 2020 1:39 am

gill1109 wrote:
I repeat: you cite a whole list of authors. Please cite specific papers with page number and line number or formula numbers. I have read those authors and read your papers too. Nowhere does anyone else say anything like what you say, at all.

Denial is not going to get you anywhere. As I said before, read section 3 of the first chapter of Bell's book. That is specific enough. Also, the following are explicit comments from Einstein:

Image

You can find the above comments from Einstein on page 89 of the collection of papers by Shimony, cited as Ref. [6] in my paper. Here, by "a state not acknowledged by quantum mechanics" Einstein means a dispersion-free state, or a state with no statistical character, like a state supplemented with hidden variables. And by observables "not simultaneously measurable" Einstein means non-commuting observables. What Einstein is saying in these comments is that for non-commuting observables within hidden variable theories the sum of expectation values need not be equal to the expectation value of the sum as von Neumann had assumed. But Bell assumes exactly the same thing in his own famous theorem, and therefore his theorem is also invalid.

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