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arXiv:quant-ph/0612015

A derivation of Bell's inequalities in Wigner form from the fundamental assumption of statistical mechanics

Ian T. Durham

Published Dec 2, 2006• 0 citations
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Abstract

Bell's inequalities in the form given by Wigner are derived from the so-called fundamental assumption of statistical mechanics. I then demonstrate the possible relationship between these inequalities and the second law, particularly if assuming the Pauli exclusion principle dictates the expected outcomes.
DOI
10.48550/arxiv.quant-ph/0612015

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We propose time-dependent Bell inequalities in a Wigner form, which expand the possibilities of experimental verification of Bohr’s principle of complementarity in the relativistic domain and for nonstationary quantum-mechanical systems. Derivation of the proposed inequalities is based entirely on the Kolmogorov axiomatics of probability theory and the locality hypothesis. Violation of the obtained inequalities is considered in the framework of quantum theory based on the example of oscillations of neutral B mesons.
A number of additional assumptions is discussed playing a role in derivations of the Bell inequalities along with the locality condition. An analogy between quantum mechanics and statistical thermodynamics is exploited to demonstrate that the ex istence of conditional probabilities p(ai|,Lambda)generally assumed in stochastic hidden variables theories, may be questionable within the domain of application of quan tum mechanics. It is demonstrated how this could block derivations of the Bell inequalities and of the Kochen-Specker theorem. The analogy also provides a pos sible understanding of the nonlocality of quantum mechanics as evidenced by the Borchers-Schlieder theorem.
The Bell theorem and inequality were derived as consequences of seemingly reasonable physical and statistical hypotheses. Bell’s assumptions were used to deduce cross-correlations of three spin measurements on two entangled particles neglecting non-commutation. The assumed correlation functions, later confirmed for certain quantum measurements, violate the Bell inequality. The present paper reviews a more general derivation of the Bell inequality showing that it is identically satisfied by finite data sets whether deterministic or random, after assuming merely that they exist. It is thereafter concerned with the consequences of this result for interpretations of the inequality that result in its violation. A primary finding is that correlation functions have differing forms due to quantum commutation, non-commutation, and conditions of measurement, and result in satisfaction of the Bell inequality used consistently with its derivation. A stochastic process having the same correlation function for all variable pairs is shown to be inconsistent with experimentally reported data. The logic of the three and four variable inequalities is shown to be similar. Finally the inequalities in probabilities are shown to follow from those in correlations with quantum mechanical results satisfying either when properly implemented.