Institutionalized Denial of the Disproof of Bell's Theorem

[Joy Christian]

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More than fourteen years ago my first paper disproving Bell's theorem appeared on the arXiv, on the 20th of March 2007: https://arxiv.org/abs/quant-ph/0703179.

The reception of the paper was icy cold from my colleagues at the Perimeter Institute in Canada. I was a visiting professor at the Institute at the time. My host at the Institute was Lucien Hardy, who has his own Bell-type theorem, and then there is the GHZ theorem. So some at the Institute shifted the goalpost and claimed: Locality is dead because we have Hardy's theorem and GHZ theorem. Not surprisingly, my visiting position at the Institute was not renewed and I returned to Oxford in September 2007. My paper was rejected by PRL on dubious grounds.

Two years later, in 2009, I posted the disproofs of Hardy's and GHZ's theorems on the arXiv: https://arxiv.org/abs/0904.4259. Hardly anyone noticed and my paper was rejected by PRD.

One of the reasons for the rejection of my paper by PRD was what I had written in its Introduction. For amusement, let me reproduce the introductory paragraph here:

No-go theorems in physics are often founded on unjustified, if tacit assumptions, and Bell’s theorem is no exception.
It is no different, in this respect, from von Neumann’s theorem rejecting all hidden variables [1], or Coleman-Mandula
theorem neglecting supersymmetry [2]. Despite being in plain sight, the unjustified assumptions underlying the latter
two theorems seemed so innocuous to many that they escaped detection for decades. In the case of Coleman-Mandula
theorem—which concerned combining spacetime and internal symmetries—it took a truly imaginative development
of supersymmetry to finally bring about recognition of its limited veracity. In the curious case of von Neumann’s
theorem, however, even an explicit counterexample—namely, the pilot wave theory [3][4]—did not discourage a series
of similarly misguided “impossibility proofs” for decades [5]. Thus ensued over half a century of false belief that no
such completion of quantum mechanics is possible, even in principle. Unfortunately, as is evident from the widespread
belief in Bell’s theorem, such examples of institutionalized denial are not confined to the history of physics. Just as in
the premises of von Neumann and Coleman-Mandula theorems, the unjustified assumption underlying Bell’s theorem
is also in plain sight—in the very first equation of Bell’s paper [6]—and yet it has received little attention.

It took another nine years before my disproof on GHZ theorem was finally published in a Royal Society journal Open Science: https://royalsocietypublishing.org/doi/ ... sos.180526.

Meanwhile, my formal argument against Bell's original theorem has become quite sophisticated: https://arxiv.org/pdf/1704.02876.pdf.
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[Joy Christian]

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I forgot to mention that the sixth and seventh papers of mine in the following series are in preparation at the moment, in case someone thought I have had enough of this stuff:

(1) https://link.springer.com/article/10.10 ... 014-2412-2

(2) https://royalsocietypublishing.org/doi/ ... sos.180526

(3) https://ieeexplore.ieee.org/document/8836453

(4) https://ieeexplore.ieee.org/document/9226414

(5) https://ieeexplore.ieee.org/document/9418997
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[Justo]

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Meanwhile, my formal argument against Bell's original theorem has become quite sophisticated: https://arxiv.org/pdf/1704.02876.pdf.

Dear Christian
I agree in part and also disagree with other parts of your paper.
I do not believe in institutionalized conspirations against the truth, but I do believe that scientists, and in particular physicists, rely too much on dogmatic wisdom and a more critical attitude is badly needed.
It is true that physicists, in general, do not need to employ rigorous mathematics. Otherwise, they would become mathematicians.
However, consciously not using mathematical rigor does not mean we are allowed to violate the laws of logical inference to achieve convenient results at whatever cost.

I share your amazement concerning the "oversight" you explain in section III. Your equation (21) is usually justified by using counterfactual reasoning and that just can't be right if we are going to claim that the Bell inequality makes sense.
It is fair, however, to point out that neither John Bell nor CHSH ever mentioned or explicitly used counterfactual reasoning. It is also fair to acknowledge that such criticisms deserve to be explained instead of being ignored.
Two papers that also give good explanations of this problem are,
- Guillaume Adenier. Foundations of Probability and Physics, chapter Refutation of Bell’s Theorem, pages 29–38. World Scientific, 2001.
- W. De Baere. On the significance of Bell’s inequality for hidden-variables theories.
Lettere Al Nuovo Cimento, 39(11):234–238, 1984.

In particular, De Baere points out that to achieve your equation (21) with experimental data we need an extra hypothesis he termed the "Reproducibility Hypothesis".

I wrote two papers containing strong criticisms against "counterfactual definiteness" and surprisingly they were accepted for publication(after many problems with reviewers). One of them was recently published https://doi.org/10.1142/S0219749921500180. This contains two theorems proving the irrelevance of counterfactual reasoning when assuming "elements of physical reality".
The second paper is forthcoming in another journal. It contains a different approach and shows that De Baere's Reproducibility Hypothesis is not an independent assumption.

I wish you luck with the publication of your piece. In my opinion, you should try to look for funding to publish it in a "serious" open access journal.

[Joy Christian]

I share your amazement concerning the "oversight" you explain in section III. Your equation (21) is usually justified by using counterfactual reasoning and that just can't be right if we are going to claim that the Bell inequality makes sense.

Thank you for your comments.

My sociological assessment of the "Institutionalized Denial" is based on my experience of how the community has treated me and my Disproof for the past fourteen years. Only a handful of physicists have actually bothered to understand what I have been arguing with regard to my quaternionic 3-sphere model for quantum correlations (and Richard D. Gill, who has spent over ten years of his time attacking me and my work on a nearly daily basis, is not one of them).

As regards to my formal argument against Bell's theorem and your comment I have quoted above, it has nothing to do with the use, misuse, or un-use of counterfactual definiteness.

My argument is exceedingly simple to understand. The assumption of the additivity of expectation values in the derivation of any Bell-type theorem is simply not a valid assumption for any hidden variable theory, regardless of its specific characteristics such as local realism. This problem was identified by Einstein, Hermann, Bell, and many others. For observables that are not simultaneously measurable (such as those involved in the Bell-test experiments), the eigenvalue of a linear sum of operators such as R + S is not a linear sum of the individual eigenvalues of R and S, such as r + s. Consequently, the additivity of expectation values is not a valid assumption within any hidden variable theory. This, in essence, is the blunder made by Bell in all of his papers concerning his "theorem." It is quite an ironic blunder, for he himself was one of the people who exposed the same blunder in the context of von Neumann's earlier theorem.
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[gill1109]

I share your amazement concerning the "oversight" you explain in section III. Your equation (21) is usually justified by using counterfactual reasoning and that just can't be right if we are going to claim that the Bell inequality makes sense.

Thank you for your comments.
...

My argument is exceedingly simple to understand. The assumption of the additivity of expectation values in the derivation of any Bell-type theorem is simply not a valid assumption for any hidden variable theory, regardless of its specific characteristics such as local realism. This problem was identified by Einstein, Hermann, Bell, and many others. For observables that are not simultaneously measurable (such as those involved in the Bell-test experiments), the eigenvalue of a linear sum of operators such as R + S is not a linear sum of the individual eigenvalues of R and S, such as r + s. Consequently, the additivity of expectation values is not a valid assumption within any hidden variable theory. This, in essence, is the blunder made by Bell in all of his papers concerning his "theorem." It is quite an ironic blunder, for he himself was one of the people who exposed the same blunder in the context of von Neumann's earlier theorem.

Indeed, eigenvalues of sums of operators are not necessarily sums of eignvalues of the summands. But that is irrelevant.

Suppose R + S = T where R, S and T do not commute with one another.
By quantum mechanics <T> = <R + S> = <R> + <S>
Suppose we had a hidden variables theory which reproduced the probability distributions of measurements of quantum mechanical observables. This theory would give us a Kolmogorov probability space on which are defined random variables X, Y and Z, whose probability distributions are the same as the probability distributions of measurements of observables R, S and T. Hence their expectation values will also coincide: E(X) = <R>, E(Y) = <S>, E(Z) = <T>.
Hence E(X) + E(Y) = E(Z).
I hope that Justo will respond to this argument and tell us if he thinks it is wrong or right.

I spent 10 years trying to point out to Joy that he was making some elementary mistakes in his mathematics. He should be grateful that he has readers who carefully attend to the mathematical details of his works. Obviously, nobody likes it when someone else finds errors in their work, but it is an essential part of science. It shouldn't be taken personally. I found Joy's pioneering use of geometric algebra fascinating and exciting and his fundamental idea that quantum correlations arise from the geometry of space-time could well be absolutely right. In order to promote it, little mathematical quibbles about details like the above, need to be sorted out in a polite and indeed courteous way. I think both Joy and I are making progress in acquiring these social skills.

His three papers in IEEE Access and RSOS receive polite "Comments" by me and he has the opportunity to publish a "Reply" to each one. That's a win-win situation. I was delighted with the breakthrough he achieved by getting these papers published at last. They prove that there is *no* institutionalized denial of his disproof.

[Joy Christian]

Indeed, eigenvalues of sums of operators are not necessarily sums of eignvalues of the summands. But that is irrelevant.

Suppose R + S = T where R, S and T do not commute with one another.
By quantum mechanics <T> = <R + S> = <R> + <S>
Suppose we had a hidden variables theory which reproduced the probability distributions of measurements of quantum mechanical observables. This theory would give us a Kolmogorov probability space on which are defined random variables X, Y and Z, whose probability distributions are the same as the probability distributions of measurements of observables R, S and T. Hence their expectation values will also coincide: E(X) = <R>, E(Y) = <S>, E(Z) = <T>.
Hence E(X) + E(Y) = E(Z).

The above argument is manifestly wrong. Z has to be an eigenvalue of the operator R + S for the theory to be a hidden variable theory, by the very meaning of "a hidden variable theory."
But the eigenvalue Z cannot be a sum of one of the eigenvalues of R and one of the eigenvalues of S when R and S cannot be simultaneously measured, as in the Bell-test experiments.

Besides, Gill's argument is based on maintaining double standards in physics --- one standard for the mere mortals like von Neumann and another standard for Gill's demigod John S. Bell.

I spent 10 years trying to point out to Joy that he was making some elementary mistakes in his mathematics.

There are no mistakes of any kind in my work. Not a single one. On the other hand, there are extremely silly mathematical mistakes in almost all of Gill's papers, which I have repeatedly exposed many times in the past. See, for example, the following two papers and references therein:

(1) https://www.academia.edu/38423874/Refut ... ls_Theorem

(2) https://ieeexplore.ieee.org/document/9418997

The claim of mistakes in my work is the cheap trick Gill has been using to undermine my work because my work destroys his fanatical belief in Bell's nonsensical and disproved "theorem."
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[gill1109]

Indeed, eigenvalues of sums of operators are not necessarily sums of eignvalues of the summands. But that is irrelevant.

Suppose R + S = T where R, S and T do not commute with one another.
By quantum mechanics <T> = <R + S> = <R> + <S>
Suppose we had a hidden variables theory which reproduced the probability distributions of measurements of quantum mechanical observables. This theory would give us a Kolmogorov probability space on which are defined random variables X, Y and Z, whose probability distributions are the same as the probability distributions of measurements of observables R, S and T. Hence their expectation values will also coincide: E(X) = <R>, E(Y) = <S>, E(Z) = <T>.
Hence E(X) + E(Y) = E(Z).

The above argument is manifestly wrong. Z has to be an eigenvalue of the operator R + S for the theory to be a hidden variable theory, by the very meaning of "a hidden variable theory."
But the eigenvalue Z cannot be a sum of one of the eigenvalues of R and one of the eigenvalues of S when R and S cannot be simultaneously measured, as in the Bell-test experiments.

Besides, Gill's argument is based on maintaining double standards in physics --- one standard for the mere mortals like von Neumann and another standard for Gill's demigod John S. Bell.

I spent 10 years trying to point out to Joy that he was making some elementary mistakes in his mathematics.

There are no mistakes of any kind in my work. Not a single one. On the other hand, there are extremely silly mathematical mistakes in almost all of Gill's papers, which I have repeatedly exposed many times in the past. See, for example, the following two papers and references therein:

(1) https://www.academia.edu/38423874/Refut ... ls_Theorem

(2) https://ieeexplore.ieee.org/document/9418997

The claim of mistakes in my work is the cheap trick Gill has been using to undermine my work because my work destroys his fanatical belief in Bell's nonsensical and disproved "theorem."
.

Please give us *your* definition of a hidden variables theory. Clearly, it differs from mine. I think mine is clear enough. If you disagree, let me know.

“Now we have a contradiction. Now we can make some progress”

[Joy Christian]

Please give us *your* definition of a hidden variables theory.

The universally accepted definition of a hidden variable theory is provided by Bell in the first chapter of his book. I quote from his Section 3:

But for a dispersion free state [of hidden variable thoories] (which has no statistical character) the expectation value of an observable must equal one of its eigenvalues.

Thus, in your notation, Z must be an eigenvalue of R + S. In other words, in your notation, X must be an eigenvalue of R, Y must be an eigenvalue of S, and Z must be an eigenvalue of R + S.

However, when R and S are non-commuting, Z is not equal to X + Y. Therefore, in a hidden variable theory E(X) + E(Y) = E(X + Y) is not a valid assumption in general.

To put this differently, your equation E(X) + E(Y) = E(Z) is a valid equation as long as Z is not assumed to be equal to X + Y, because R and S do not commute in the Bell-test experiments.
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[Justo]

Indeed, eigenvalues of sums of operators are not necessarily sums of eigenvalues of the summands. But that is irrelevant.
.
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I hope that Justo will respond to this argument and tell us if he thinks it is wrong or right.

You are right, it is irrelevant concerning the Bell inequality. Bell did not assume that, and as Joy observed, it would have been very ironic had he assumed that since it is precisely the "silly error" Von Neumann committed and that Bell pointed out.

I have previously agreed with Joy Christian when he attacked the counterfactual definiteness hypothesis as meaningless. However, now he shifted to another alleged error committed by Bell and I no longer agree with him.

I don't have an opinion on Joy's models using Geometric Algebra because I am very limited and do not understand that language. However, when he frames his attacks within the elementary mathematics needed to formulate the Bell inequality, it becomes easy to spot the mistakes.

Somehow I sympathize with Joy because I also maintain a c****pot claim: the irrelevance of the counterfactual definiteness assumption unfairly attributed to John Bell and CHSH and that 95% or so of scientists seem to believe in.
Just like Joy, I am also absolutely convinced my claim is correct. Just like Joy, I am also having a hard time publishing my two papers explaining the problem from two different angles. I have finally made it but it was not easy. However, I do not see any "conspiracy". It is just the way one would expect it to be: in my case there is a 95% percent chance the reviewer will reject it and find it utterly mistaken.

[Joy Christian]

Indeed, eigenvalues of sums of operators are not necessarily sums of eigenvalues of the summands. But that is irrelevant.

You are right, it is irrelevant concerning the Bell inequality. Bell did not assume that, and as Joy observed, it would have been very ironic had he assumed that since it is precisely the "silly error" Von Neumann committed and that Bell pointed out.

Bell did assume E(X) + E(Y) = E(X + Y) to derive his inequality. Without this assumption, Bell inequalities cannot be derived.

But the assumption E(X) + E(Y) = E(X + Y) is not a valid assumption within any hidden variable theory, which, by definition, involves only dispersion-free states without statistical character.

The correct equality is E(X) + E(Y) = E(Z), where X must be an eigenvale of the operator R, Y must be an eigenvalue of the operator S, and Z must be an eigenvlaue of the operetor R + S.

Thus corrected, and thereby local realism implemented correctly in Bell's argument, the absolute bound on the CHSH correlator works out to be 2\/2 instead of 2, as I have demonstrated in https://arxiv.org/abs/1704.02876. Consequently, what is ruled out by the Bell-test experiments is not local realism but the unjustified assumption of the additivity of expectation values.
.

[Justo]

Indeed, eigenvalues of sums of operators are not necessarily sums of eigenvalues of the summands. But that is irrelevant.

You are right, it is irrelevant concerning the Bell inequality. Bell did not assume that, and as Joy observed, it would have been very ironic had he assumed that since it is precisely the "silly error" Von Neumann committed and that Bell pointed out.

Bell did assume E(X) + E(Y) = E(X + Y) to derive his inequality. Without this assumption, Bell inequalities cannot be derived.

But the assumption E(X) + E(Y) = E(X + Y) is not a valid assumption within any hidden variable theory, which, by definition, involves only dispersion-free states without statistical character.

The correct equality is E(X) + E(Y) = E(Z), where X must be an eigenvale of the operator R, Y must be an eigenvalue of the operator S, and Z must be an eigenvlaue of the operetor R + S.

Thus corrected, and thereby local realism implemented correctly in Bell's argument, the absolute bound on the CHSH correlator works out to be 2\/2 instead of 2, as I have demonstrated in https://arxiv.org/abs/1704.02876. Consequently, what is ruled out by the Bell-test experiments is not local realism but the unjustified assumption of the additivity of expectation values.
.

To be more precise let us consider the four operators in the CHSH inequality, . If we define a fifth operator T by , then it is correct to say that the sum of eigenvalues of the O_i is not equal to the igenvalues to T.
However, in a Bell test experiment, the operator T is not an issue. I do not know, and probably nobody knows, what the "physical" operator T means apart from knowing that it is mathematically defined as the sum of the O_i.
We know what the O_i physically represent and we can measure them. That is all we need to know to test the CHSH inequality. Hidden variables make predictions for each O_i and so does quantum mechanics. Operator T is not an issue. By the way, it is an interesting problem to find out what we have to measure to obtain the eigenvalues of T and calculate its values. But whatever they are, quantum mechanics predicts that <T>=<O_1>+<O_2>+<O_3>+<O_4>. However, the last prediction is not an issue in a Bell test experiment.
In a Bell test experiment, we are not comparing the values of the eigenvalues of T with values of the eigenvalues of the O_i. We do not know the eigenvalues of T and we do not need to know them.
The Bell inequality makes predictions for four different operators or experiments and the sum or their values, nothing more, nothing less.

Dear Joy, if you manage to publish your article, I will write a comment explaining just that (unless it is open access in which case I won't because I can't pay for it).

[Joy Christian]

Indeed, eigenvalues of sums of operators are not necessarily sums of eigenvalues of the summands. But that is irrelevant.

You are right, it is irrelevant concerning the Bell inequality. Bell did not assume that, and as Joy observed, it would have been very ironic had he assumed that since it is precisely the "silly error" Von Neumann committed and that Bell pointed out.

Bell did assume E(X) + E(Y) = E(X + Y) to derive his inequality. Without this assumption, Bell inequalities cannot be derived.

But the assumption E(X) + E(Y) = E(X + Y) is not a valid assumption within any hidden variable theory, which, by definition, involves only dispersion-free states without statistical character.

The correct equality is E(X) + E(Y) = E(Z), where X must be an eigenvale of the operator R, Y must be an eigenvalue of the operator S, and Z must be an eigenvlaue of the operetor R + S.

Thus corrected, and thereby local realism implemented correctly in Bell's argument, the absolute bound on the CHSH correlator works out to be 2\/2 instead of 2, as I have demonstrated in https://arxiv.org/abs/1704.02876. Consequently, what is ruled out by the Bell-test experiments is not local realism but the unjustified assumption of the additivity of expectation values.
.

To be more precise let us consider the four operators in the CHSH inequality, . If we define a fifth operator T by , then it is correct to say that the sum of eigenvalues of the O_i is not equal to the igenvalues to T.
However, in a Bell test experiment, the operator T is not an issue. I do not know, and probably nobody knows, what the "physical" operator T means apart from knowing that it is mathematically defined as the sum of the O_i.
We know what the O_i physically represent and we can measure them. That is all we need to know to test the CHSH inequality. Hidden variables make predictions for each O_i and so does quantum mechanics. Operator T is not an issue. By the way, it is an interesting problem to find out what we have to measure to obtain the eigenvalues of T and calculate its values. But whatever they are, quantum mechanics predicts that <T>=<O_1>+<O_2>+<O_3>+<O_4>. However, the last prediction is not an issue in a Bell test experiment.
In a Bell test experiment, we are not comparing the values of the eigenvalues of T with values of the eigenvalues of the O_i. We do not know the eigenvalues of T and we do not need to know them.
The Bell inequality makes predictions for four different operators or experiments and the sum or their values, nothing more, nothing less.

Dear Joy, if you manage to publish your article, I will write a comment explaining just that (unless it is open access in which case I won't because I can't pay for it).

Justo, whether or not the operator T is actually measured is irrelevant. What matters is that there is one-to-one correspondence between operators and observables in quantum theory, and that T can be measured at least in principle. A hidden variable theory is a theory where one must be able to assign pre-existing values to all observables of a physical system (albeit only contextually), whether or not they are actually measured. That is what a hidden variable theory means. If, instead, Bell inequality does what you say it does, then it has nothing to do with Nature, quantum theory, local realism, or hidden variable theory. Then it is simply an empty mathematical result of no value to physics. I have discussed this more carefully in my paper.
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[gill1109]

Indeed, eigenvalues of sums of operators are not necessarily sums of eigenvalues of the summands. But that is irrelevant.

You are right, it is irrelevant concerning the Bell inequality. Bell did not assume that, and as Joy observed, it would have been very ironic had he assumed that since it is precisely the "silly error" Von Neumann committed and that Bell pointed out.

Bell did assume E(X) + E(Y) = E(X + Y) to derive his inequality. Without this assumption, Bell inequalities cannot be derived.

But the assumption E(X) + E(Y) = E(X + Y) is not a valid assumption within any hidden variable theory, which, by definition, involves only dispersion-free states without statistical character.

The correct equality is E(X) + E(Y) = E(Z), where X must be an eigenvale of the operator R, Y must be an eigenvalue of the operator S, and Z must be an eigenvlaue of the operetor R + S.

Thus corrected, and thereby local realism implemented correctly in Bell's argument, the absolute bound on the CHSH correlator works out to be 2\/2 instead of 2, as I have demonstrated in https://arxiv.org/abs/1704.02876. Consequently, what is ruled out by the Bell-test experiments is not local realism but the unjustified assumption of the additivity of expectation values.
.

To be more precise let us consider the four operators in the CHSH inequality, . If we define a fifth operator T by , then it is correct to say that the sum of eigenvalues of the O_i is not equal to the igenvalues to T.
However, in a Bell test experiment, the operator T is not an issue. I do not know, and probably nobody knows, what the "physical" operator T means apart from knowing that it is mathematically defined as the sum of the O_i.
We know what the O_i physically represent and we can measure them. That is all we need to know to test the CHSH inequality. Hidden variables make predictions for each O_i and so does quantum mechanics. Operator T is not an issue. By the way, it is an interesting problem to find out what we have to measure to obtain the eigenvalues of T and calculate its values. But whatever they are, quantum mechanics predicts that <T>=<O_1>+<O_2>+<O_3>+<O_4>. However, the last prediction is not an issue in a Bell test experiment.
In a Bell test experiment, we are not comparing the values of the eigenvalues of T with values of the eigenvalues of the O_i. We do not know the eigenvalues of T and we do not need to know them.
The Bell inequality makes predictions for four different operators or experiments and the sum or their values, nothing more, nothing less.

Dear Joy, if you manage to publish your article, I will write a comment explaining just that (unless it is *not* open access in which case I won't because I can't pay for it).

Thank you, Justo!

I admire you for exploring your own "c****pot" claim and I'm glad you got it published at last. It is a good thing to carefully discuss all possible objections to Bell's work and what came of it, and for that purpose, it is vitally important that people pick up and explore "weird" ideas, outside of the mainstream. The mainstream is busy polishing existing expertise and making tiny incremental steps; mainly busy with publishing more and more papers, which nobody will ever read, in order to maintain academic careers, get money to build bigger research groups, getting publicity in order to get more money and influence...

[Justo]

I admire you for exploring your own "c****pot" claim and I'm glad you got it published at last. It is a good thing to carefully discuss all possible objections to Bell's

Thank you, Richard. However, I did it only for personal reasons. I know it will have zero impact for two reasons. People who get to read it and believe in CFD will dismiss it as c****pot material, wondering how something that wrong could have ever been published.

On the other hand, the guys who really know about quantum foundations are up to something completely different and don't pay attention to trivial misunderstandings. I realize that recently watching a Youtube conversation between Mathew Leifer and Robert Speakers.

[gill1109]

*Matthew Leifer and Robert Spekkens*

[Justo]

*Matthew Leifer and Robert Spekkens*

Pardon my English. During the conversation, Rob Spekkens mentions the ironic mistake of believing that, by discarding realism in local realism, one can get rid of quantum nonlocality.

[Joy Christian]

*Matthew Leifer and Robert Spekkens*

Pardon my English. During the conversation, Rob Spekkens mentions the ironic mistake of believing that, by discarding realism in local realism, one can get rid of quantum nonlocality.

I do not find Leifer's or Spekkens's opinions interesting. Regardless of whether quantum mechanics is nonlocal, there is no such thing as nonlocality in Nature. I have proved that decisively in my work in the past fourteen years (which began at the Perimeter Institute in 2007). And just as Bell did, I take realism for granted. Without realism, there is no point in doing science.
.

[gill1109]

*Matthew Leifer and Robert Spekkens*

Pardon my English. During the conversation, Rob Spekkens mentions the ironic mistake of believing that, by discarding realism in local realism, one can get rid of quantum nonlocality.

It’s not a mistake, it’s a strategy. Many worlds theory denies reality and solves all problems of quantum foundations, by entirely severing its connection to the world we find ourselves in.

BTW it was a matter of spelling not English which I commented on!

[minkwe]

Indeed, eigenvalues of sums of operators are not necessarily sums of eigenvalues of the summands. But that is irrelevant.

You are right, it is irrelevant concerning the Bell inequality. Bell did not assume that, and as Joy observed, it would have been very ironic had he assumed that since it is precisely the "silly error" Von Neumann committed and that Bell pointed out.

Bell did assume E(X) + E(Y) = E(X + Y) to derive his inequality. Without this assumption, Bell inequalities cannot be derived.

But the assumption E(X) + E(Y) = E(X + Y) is not a valid assumption within any hidden variable theory, which, by definition, involves only dispersion-free states without statistical character.

The correct equality is E(X) + E(Y) = E(Z), where X must be an eigenvale of the operator R, Y must be an eigenvalue of the operator S, and Z must be an eigenvlaue of the operetor R + S.

Thus corrected, and thereby local realism implemented correctly in Bell's argument, the absolute bound on the CHSH correlator works out to be 2\/2 instead of 2, as I have demonstrated in https://arxiv.org/abs/1704.02876. Consequently, what is ruled out by the Bell-test experiments is not local realism but the unjustified assumption of the additivity of expectation values.
.

To be more precise let us consider the four operators in the CHSH inequality, . If we define a fifth operator T by , then it is correct to say that the sum of eigenvalues of the O_i is not equal to the igenvalues to T.
However, in a Bell test experiment, the operator T is not an issue. I do not know, and probably nobody knows, what the "physical" operator T means apart from knowing that it is mathematically defined as the sum of the O_i.
We know what the O_i physically represent and we can measure them. That is all we need to know to test the CHSH inequality. Hidden variables make predictions for each O_i and so does quantum mechanics. Operator T is not an issue. By the way, it is an interesting problem to find out what we have to measure to obtain the eigenvalues of T and calculate its values. But whatever they are, quantum mechanics predicts that <T>=<O_1>+<O_2>+<O_3>+<O_4>. However, the last prediction is not an issue in a Bell test experiment.
In a Bell test experiment, we are not comparing the values of the eigenvalues of T with values of the eigenvalues of the O_i. We do not know the eigenvalues of T and we do not need to know them.
The Bell inequality makes predictions for four different operators or experiments and the sum or their values, nothing more, nothing less.

Dear Joy, if you manage to publish your article, I will write a comment explaining just that (unless it is open access in which case I won't because I can't pay for it).

Justo,
Can you please demonstrate the violation of the CHSH inequality by QM? Please write it out in full mathematically so we can see. And then explain what that expression means in QM.

[Justo]

Justo,
Can you please demonstrate the violation of the CHSH inequality by QM? Please write it out in full mathematically so we can see. And then explain what that expression means in QM.

I can't write out for you a detailed calculation. It is a standard textbook calculation. It gives , assuming a and b are unit vectors. Normally, the calculation in papers discussing the Bell inequalities is not included since it is an elementary unpolemical result. Using the appropriate orientations for a_1, a_2, b_1, and b_2 after elementary trigonometry we find


I suppose Joy would agree with me on this.