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

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

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

Linearity of expectations is now a corollary.

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

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

gill1109 wrote:

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

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

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[quote="Joy Christian"][quote="gill1109"][quote="Joy Christian"]

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

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

Linearity of expectations is now a corollary.[/quote]

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

https://arxiv.org/abs/1704.02876.[/quote]

[quote="gill1109"]

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

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

***