Summary
A local hidden-variable theory is possible in view of quantum mechanicsonly if it satisfies bilinearity, symmetry and rotational invariance. If, however, a local hidden-variable theory satisfies all these algebraic properties, then Bell’s inequalities are not theorems of this theory. Therefore, violation of Bell’s inequalities by quantum mechanics is inconsequential for locality.
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References
Y. Koç:Linearity and Local Hidden Variable Theories, ISS/Ph 89-02, Boğaziçi University, Istanbul, Turkey.
Y. Koç:Bilinearity, Quantum Mechanics and Local Hidden Variable Theories, ISS/Ph 89-04, Boğaziçi University, Istanbul, Turkey.
Y. Koç:Local Hidden Variable Theories and «Bell’s theorem»: A Reconsideration, ISS/Ph 90-01, Boğaziçi University, Istanbul, Turkey.
In the present paper, the term «spin expectation value function» indicates that of two correlated particles.
Bell (ref. [6,7]) does not consider these algebraic properties.
J. S. Bell:Physics,1, 195 (1964).
J. S. Bell:Introduction to the hidden variable question, inFoundations of Quantum Mechanics, edited byB. D’Espagnat (Academic Press, 1971), p. 171.
In the present paper, we consider only [7]; however, our arguments can be restated with regard to [6] as well.
These averages are taken over the variable λ.
The function\(\bar A \bar B\) is the product of the factorized, functions\(\bar A\) and\(\bar B\) in eq. (1).
In the present paper,R 3 denotes the three-dimensional Euclidean space andR denotes the class of real numbers.
Bell’s inequality is a theorem of a local hidden variable theory if, and only if Bell’s inequality yields a true statement (e.g., 1≤2) forevery value of the hidden-variable expectation value functions in the inequality. Obviously, if Bell’s inequality is not a theorem ofsome local hidden-variable theory, then violation of Bell’s inequality by quantum mechanics is inconsequential for locality.
In definition 1, bilinearity appears in a restricted sense; however, that is sufficient for our purposes in the present paper.
E H denotes the expectation value function of the product of the spins of two correlated particles which are prepared in the singlet state; we therefore considerE H as a mapping fromR 3×R 3 intoR.
Let us also consider the Clauser-Horne inequality ([16], p. 528, inequality (4)). Clauser and Horne argue that Bell’s generalized inequality (BGI H ) is a corollary of their inequality (4) (please see: «First, (B2) is a corollary of (4)», p. 533, 1974). Theorem 1 and the lemma, however, entail that no inequality (e.g., Clauser-Horne inequality) which implies Bell’s generalized inequality (BGI H ) as a corollary can be a theorem of a local stochastic hidden-variable theory which statisfies bilinearity, symmetry and rotational invariance.
J. F. Clauser andM. A. Horne:Phys. Rev. D,10, 526 (1974).
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Koç, Y. The local expectation value function and Bell’s inequalities. Nuov Cim B 107, 961–971 (1992). https://doi.org/10.1007/BF02899297
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DOI: https://doi.org/10.1007/BF02899297