## Bell-test experiments rule out additivity of expectations

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

### Re: Bell-test experiments rule out additivity of expectation

minkwe wrote:
gill1109 wrote:He means more than deterministic. He means that after you have conditioned on the values taken by the hidden variables in any particular case, everything else becomes deterministic.

That's what deterministic means.

Yes, but note that he’s not referring to the quantum state. The idea is that the quantum state is just a statistical collective of underlying (hidden) states, for each of which deterministic rules apply. Linearity of certain expectation values (linearities predicted by QM) might hold ‘on average’ without it holding within each hidden deterministic state separately.
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### Re: Bell-test experiments rule out additivity of expectation

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Let me reproduce here the absolute proof of an unmistakeable mistake in Bell's theorem from my paper in the simplest possible terms:

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### Re: Bell-test experiments rule out additivity of expectation

A hidden variable theory must reproduce the statistical predictions of quantum theory. Quantum theory predicts that when one measures an observable, one gets to see one of the eigenvalues of the corresponding operator. Quantum theory predicts that the outcomes of measuring sigma_+ x sigma_y are not equal to sums of eigenvalues of sigma_x and sigma_y. So a hidden variable theory need not predict this either. Quantum theory does predict that the mean value of measurement outcomes when measuring sigma_x + sigma_y on systems in state rho is the sum of the mean values of outcomes of measuring sigma_x and of measuring sigma_y on systems in state rho. So a hidden variables theory must preserve those linearities of expectations predicted by QM, at the level of states existing in QM. Such properties need not hold for the “microstates” of a hidden variables theory, statistical aggregates of which constitute the QM states. They should only hold “on average”.
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### Re: Bell-test experiments rule out additivity of expectation

gill1109 wrote:
A hidden variable theory must reproduce the statistical predictions of quantum theory. Quantum theory predicts that when one measures an observable, one gets to see one of the eigenvalues of the corresponding operator. Quantum theory predicts that the outcomes of measuring sigma_+ x sigma_y are not equal to sums of eigenvalues of sigma_x and sigma_y. So a hidden variable theory need not predict this either. Quantum theory does predict that the mean value of measurement outcomes when measuring sigma_x + sigma_y on systems in state rho is the sum of the mean values of outcomes of measuring sigma_x and of measuring sigma_y on systems in state rho. So a hidden variables theory must preserve those linearities of expectations predicted by QM, at the level of states existing in QM. Such properties need not hold for the “microstates” of a hidden variables theory, statistical aggregates of which constitute the QM states. They should only hold “on average”.

The above is an invalid argument, popularized by N. David Mermin, but not supported by mathematics. It has been disproven in Appendix A of my paper. I reproduce the relevant text here:

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### Re: Bell-test experiments rule out additivity of expectation

An operator generally has a number of eigenvalues. So I don't understand what you write here. You say "the eigenvalue of the sum..." but script R, S, T, U are sets of eigenvalues. Do you mean to say that *an* eigenvalue of the sum can't be equal to sums of some eigenvalues of the summands?

And the left and right hand side of equation (12) must be equal if script R(lambda_k) etc are numbers.
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### Re: Bell-test experiments rule out additivity of expectation

gill1109 wrote:
An operator generally has a number of eigenvalues. So I don't understand what you write here. You say "the eigenvalue of the sum..." but script R, S, T, U are sets of eigenvalues. Do you mean to say that *an* eigenvalue of the sum can't be equal to sums of some eigenvalues of the summands?

Yes, any one of the possible eigenvalues. The eigenvalue of the sum R + S + T + U cannot be r + s + t + u, where r, s, t, and u are one of the eigenvalues of R, S, T, and U, respectively.

gill1109 wrote:
And the left and right hand side of equation (12) must be equal if script R(lambda_k) etc are numbers.

Yes, that is correct. Arithmetically they are equal. But the equality is physically not correct. You have to snap out of arithmetic to appreciate the mistake both von Neumann and Bell made.
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### Re: Bell-test experiments rule out additivity of expectation

Joy Christian wrote:
gill1109 wrote:
An operator generally has a number of eigenvalues. So I don't understand what you write here. You say "the eigenvalue of the sum..." but script R, S, T, U are sets of eigenvalues. Do you mean to say that *an* eigenvalue of the sum can't be equal to sums of some eigenvalues of the summands?

Yes, any one of the possible eigenvalues. The eigenvalue of the sum R + S + T + U cannot be r + s + t + u, where r, s, t, and u are one of the eigenvalues of R, S, T, and U, respectively.

gill1109 wrote:
And the left and right hand side of equation (12) must be equal if script R(lambda_k) etc are numbers.

Yes, that is correct. Arithmetically they are equal. But the equality is physically not correct. You have to snap out of arithmetic to appreciate the mistake both von Neumann and Bell made.

That is a very interesting point of view. Physicists use mathematics. If the mathematics of the present day is inadequate they invent/discover new mathematical structures. Mathematicians are on the whole very happy with the new concepts and new structures discovered by physicists in the real world. It gives us more fun stuff to do.

You can't "snap out of arithmetic". In this case, you will have to reject the whole conventional mathematical structure of observables = self adjoint operators etc, etc. The just mentioned arithmetic cannot be wrong. But the framework of self-adjoint operators, Born's law, etc etc could be wrong. In fact, if for physical reasons you know that reality operates in local realistic and non-conspiratorial ways, then ordinary QM has to be rejected. Bell himself explained that this was one of the logically available options which followed from his incontrovertible maths.
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### Re: Bell-test experiments rule out additivity of expectation

gill1109 wrote:
You can't "snap out of arithmetic". In this case, you will have to reject the whole conventional mathematical structure of observables = self adjoint operators etc, etc. The just mentioned arithmetic cannot be wrong. But the framework of self-adjoint operators, Born's law, etc etc could be wrong. In fact, if for physical reasons you know that reality operates in local realistic and non-conspiratorial ways, then ordinary QM has to be rejected. Bell himself explained that this was one of the logically available options which followed from his incontrovertible maths.

von Neumann was not stupid. He implicitly accepted my Eq. (9) because it is self-evidently true as far as arithmetic is concerned. But accepting that equation for physics was precisely his mistake, as Bell so eloquently explained in section 3 of the first chapter of his book. What is true arithmetically is not true physically. That is the message of Bell's criticism of von Neumann. But unless one is desperate enough to resort to double standards, the same criticism applies to Bell's acceptance of my Eq. (9) in his own theorem, because physically my Eq.(9) is nonsense.
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### Re: Bell-test experiments rule out additivity of expectation

Joy Christian wrote:von Neumann was not stupid. He implicitly accepted my Eq. (9) because it is self-evidently true as far as arithmetic is concerned. But accepting that equation for physics was precisely his mistake, as Bell so eloquently explained in section 3 of the first chapter of his book. What is true arithmetically is not true physically. That is the message of Bell's criticism of von Neumann. But unless one is desperate enough to resort to double standards, the same criticism applies to Bell's acceptance of my Eq. (9) in his own theorem, because physically Eq. (9) is nonsense.
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I am confused, there is no eq. (9) above, shouldn't it be equation (12)?

In this case, I agree with Richard Gill, equation (12) is confusing. Although I understand what Joy Christians means there must be another way to express it, as it is written is an arithmetic contradiction.
Justo

### Re: Bell-test experiments rule out additivity of expectation

Justo wrote:
Joy Christian wrote:von Neumann was not stupid. He implicitly accepted my Eq. (9) because it is self-evidently true as far as arithmetic is concerned. But accepting that equation for physics was precisely his mistake, as Bell so eloquently explained in section 3 of the first chapter of his book. What is true arithmetically is not true physically. That is the message of Bell's criticism of von Neumann. But unless one is desperate enough to resort to double standards, the same criticism applies to Bell's acceptance of my Eq. (9) in his own theorem, because physically Eq. (9) is nonsense.

I am confused, there is no eq. (9) above, shouldn't it be equation (12)?

In this case, I agree with Richard Gill, equation (12) is confusing. Although I understand what Joy Christians means there must be another way to express it, as it is written is an arithmetic contradiction.

I meant Eq. (9) in my paper: https://arxiv.org/pdf/1704.02876.pdf. Eq. (12) is the negation of Eq. (9).

The confusion arises if one looks at Eq. (12) out of context of the full discussion in my paper. The point is that, in the present context, arithmetical triviality is physical nonsense.
.
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### Re: Bell-test experiments rule out additivity of expectation

Joy Christian wrote:
Justo wrote:
Joy Christian wrote:von Neumann was not stupid. He implicitly accepted my Eq. (9) because it is self-evidently true as far as arithmetic is concerned. But accepting that equation for physics was precisely his mistake, as Bell so eloquently explained in section 3 of the first chapter of his book. What is true arithmetically is not true physically. That is the message of Bell's criticism of von Neumann. But unless one is desperate enough to resort to double standards, the same criticism applies to Bell's acceptance of my Eq. (9) in his own theorem, because physically Eq. (9) is nonsense.

I am confused, there is no eq. (9) above, shouldn't it be equation (12)?

In this case, I agree with Richard Gill, equation (12) is confusing. Although I understand what Joy Christians means there must be another way to express it, as it is written is an arithmetic contradiction.

I meant Eq. (9) in my paper: https://arxiv.org/pdf/1704.02876.pdf. Eq. (12) is the negation of Eq. (9).

The confusion arises if one looks at Eq. (12) out of context of the full discussion in my paper. The point is that, in the present context, arithmetical triviality is physical nonsense.

Unfortunately, arithmetic triviality is arithmetic triviality. Tautology. If quantum mechanics leads to contradiction with arithmetic triviality then quantum mechanics is wrong. Maybe that is what you want to say? Parts of it could be OK but the usual complete package could be plain wrong. That's certainly a logically tenable position to take, as Bell himself said.
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### Re: Bell-test experiments rule out additivity of expectation

gill1109 wrote:
Joy Christian wrote:
Justo wrote:
Joy Christian wrote:von Neumann was not stupid. He implicitly accepted my Eq. (9) because it is self-evidently true as far as arithmetic is concerned. But accepting that equation for physics was precisely his mistake, as Bell so eloquently explained in section 3 of the first chapter of his book. What is true arithmetically is not true physically. That is the message of Bell's criticism of von Neumann. But unless one is desperate enough to resort to double standards, the same criticism applies to Bell's acceptance of my Eq. (9) in his own theorem, because physically Eq. (9) is nonsense.

I am confused, there is no eq. (9) above, shouldn't it be equation (12)?

In this case, I agree with Richard Gill, equation (12) is confusing. Although I understand what Joy Christians means there must be another way to express it, as it is written is an arithmetic contradiction.

I meant Eq. (9) in my paper: https://arxiv.org/pdf/1704.02876.pdf. Eq. (12) is the negation of Eq. (9).

The confusion arises if one looks at Eq. (12) out of context of the full discussion in my paper. The point is that, in the present context, arithmetical triviality is physical nonsense.

Unfortunately, arithmetic triviality is arithmetic triviality. Tautology. If quantum mechanics leads to contradiction with arithmetic triviality then quantum mechanics is wrong. Maybe that is what you want to say? Parts of it could be OK but the usual complete package could be plain wrong. That's certainly a logically tenable position to take, as Bell himself said.

There is nothing wrong with quantum mechanics. The mistake is Bell's and Bell's only. Bell ridiculed von Neumann for accepting Eq. (9) of my paper as a physically meaningful equation and called his mistake a silly mistake. Bell then went on to make the same silly mistake in his own silly theorem and made outrageous metaphysical claims on its basis. But now Bell has become such a sacred cow that Bell-believers are willing to throw out quantum mechanics instead of admitting that their hero made the same silly mistake he ridiculed von Neumann for making.
.
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### Re: Bell-test experiments rule out additivity of expectation

Joy Christian wrote:
gill1109 wrote:
Joy Christian wrote:
Justo wrote:I am confused, there is no eq. (9) above, shouldn't it be equation (12)?

In this case, I agree with Richard Gill, equation (12) is confusing. Although I understand what Joy Christians means there must be another way to express it, as it is written is an arithmetic contradiction.

I meant Eq. (9) in my paper: https://arxiv.org/pdf/1704.02876.pdf. Eq. (12) is the negation of Eq. (9).

The confusion arises if one looks at Eq. (12) out of context of the full discussion in my paper. The point is that, in the present context, arithmetical triviality is physical nonsense.

Unfortunately, arithmetic triviality is arithmetic triviality. Tautology. If quantum mechanics leads to contradiction with arithmetic triviality then quantum mechanics is wrong. Maybe that is what you want to say? Parts of it could be OK but the usual complete package could be plain wrong. That's certainly a logically tenable position to take, as Bell himself said.

There is nothing wrong with quantum mechanics. The mistake is Bell's and Bell's only. Bell ridiculed von Neumann for accepting Eq. (9) of my paper as a physically meaningful equation and called his mistake a silly mistake. Bell then went on to make the same silly mistake in his own silly theorem and made outrageous metaphysical claims on its basis. But now Bell has become such a sacred cow that Bell-believers are willing to throw out quantum mechanics instead of admitting that their hero made the same silly mistake he ridiculed von Neumann for making.

I don't throw out quantum mechanics. I'm saying that if you want to be logically consistent, you are obliged to throw it out. But you can also be happy with just contradicting yourself!
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### Re: Bell-test experiments rule out additivity of expectation

gill1109 wrote:
Joy Christian wrote:
There is nothing wrong with quantum mechanics. The mistake is Bell's and Bell's only. Bell ridiculed von Neumann for accepting Eq. (9) of my paper as a physically meaningful equation and called his mistake a silly mistake. Bell then went on to make the same silly mistake in his own silly theorem and made outrageous metaphysical claims on its basis. But now Bell has become such a sacred cow that Bell-believers are willing to throw out quantum mechanics instead of admitting that their hero made the same silly mistake he ridiculed von Neumann for making.

I don't throw out quantum mechanics. I'm saying that if you want to be logically consistent, you are obliged to throw it out. But you can also be happy with just contradicting yourself!

This is pure nonsense.

Logic dictates that Bell's theorem is self-contradictory and therefore it must be thrown out, lock, stock, and two smoking barrels.

.
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### Re: Bell-test experiments rule out additivity of expectation

Joy Christian wrote:
gill1109 wrote:
Joy Christian wrote:
There is nothing wrong with quantum mechanics. The mistake is Bell's and Bell's only. Bell ridiculed von Neumann for accepting Eq. (9) of my paper as a physically meaningful equation and called his mistake a silly mistake. Bell then went on to make the same silly mistake in his own silly theorem and made outrageous metaphysical claims on its basis. But now Bell has become such a sacred cow that Bell-believers are willing to throw out quantum mechanics instead of admitting that their hero made the same silly mistake he ridiculed von Neumann for making.

I don't throw out quantum mechanics. I'm saying that if you want to be logically consistent, you are obliged to throw it out. But you can also be happy with just contradicting yourself!

This is pure nonsense.

Logic dictates that Bell's theorem is self-contradictory and therefore it must be thrown out, lock, stock, and two smoking barrels.

I think you are polluting this thread with illogical nonsense. How about a Zoom meeting to discuss things face to face?
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### Re: Bell-test experiments rule out additivity of expectation

gill1109 wrote:
How about a Zoom meeting to discuss things face to face?

I would rather be dead in a ditch than have any kind of meeting with you.
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### Re: Bell-test experiments rule out additivity of expectation

Joy Christian wrote:
gill1109 wrote:How about a Zoom meeting to discuss things face to face?

I would rather be dead in a ditch than have any kind of meeting with you.

Such hatred is not healthy.
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### Re: Bell-test experiments rule out additivity of expectation

gill1109 wrote:
Joy Christian wrote:
gill1109 wrote:How about a Zoom meeting to discuss things face to face?

I would rather be dead in a ditch than have any kind of meeting with you.

Such hatred is not healthy.

It is not hatred. It is contempt. And for the new readers of this forum, it stems from these reasons: viewtopic.php?f=6&t=183&p=5995#p5995.
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Joy Christian
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### Re: Bell-test experiments rule out additivity of expectation

Joy Christian wrote:It is not hatred. It is contempt. And for the new readers of this forum, it stems from these reasons: viewtopic.php?f=6&t=183&p=5995#p5995.
.

Personal attacks are unjustifiable. If you think someone published an incorrect result, the right thing is to write another paper or a comment explaining the error.
Neither do I agree with the retraction at Annals of Physics.
They could have issued a paper by an invited author explaining the presumed errors.
It is not good for science to reject ideas because they go against established wisdom. The errors should be explained.
Justo

### Re: Bell-test experiments rule out additivity of expectation

Justo wrote:
Joy Christian wrote:
It is not hatred. It is contempt. And for the new readers of this forum, it stems from these reasons: viewtopic.php?f=6&t=183&p=5995#p5995.

Personal attacks are unjustifiable. If you think someone published an incorrect result, the right thing is to write another paper or a comment explaining the error.
Neither do I agree with the retraction at Annals of Physics.
They could have issued a paper by an invited author explaining the presumed errors.
It is not good for science to reject ideas because they go against established wisdom. The errors should be explained.

Thank you, Justo.

By the way, the paper retracted by Annals of Physics is now republished in IEEE Access: https://ieeexplore.ieee.org/document/8836453 (I did have to change the title to avoid confusion).
.
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