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Journal of Russian Laser Research

Analysis of explicit and implicit assumptions in the theorems of J. Von Neumann and J. Bell

Andrei Yu. Khrennikov

Published May 1, 2007• 14 citations
Publisher
DOI
10.1007/s10946-007-0013-8

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This article contains a review of Nelson's analysis of Bell's theorem. It shows that Bell's inequalities can be violated with a theory of local random variables if one accepts that the outcomes of these variables are not predetermined prior to measurement. The article describes the relation between Bell's theorem and the Strong Free Will theorem of Conway and Kochen. Then, the original articles of Bell are analyzed in detail. Following an article of Faris, it is explained that Bell's work on the hidden variable question in fact describes two separate theorems. Bell's first theorem says that there can be no model for the singlet state where an outcome does not depend locally on the settings of the detector where the outcome was measured. Bell's second theorem shows that Bell's inequalities can be violated by a theory that is either not deterministic, or violates causality in the sense of relativity or the free will assumption of the experimenters. It is shown in detail where Bell implicitly makes the various locality assumptions that Nelson has shown to be necessary for deriving Bell's inequality. The article closes by relating the various assumptions needed to derive Bell's theorem with the reality criterion of EPR
Abstract A widespread view among physicists is that Bell’s theorem rests on an implicit assumption of “classicality,” in addition to locality. According to this understanding, the violation of Bell’s inequalities poses no challenge to locality, but simply reinforces the fact that quantum mechanics is not classical. The paper provides a critical analysis of this view. First we characterize the notion of classicality in probabilistic terms. We argue that classicality thus construed has nothing to do with the validity of classical physics, nor of classical probability theory, contrary to what many believe. At the same time, we show that the probabilistic notion of classicality is not an additional premise of Bell’s theorem, but a mathematical corollary of locality in conjunction with the standard auxiliary assumptions of Bell. Accordingly, any theory that claims to get around the derivation of Bell’s inequalities by giving up classicality, in fact has to give up one of those standard assumptions. As an illustration of this, we look at two recent interpretations of quantum mechanics, Reinhard Werner’s operational quantum mechanics and Robert Griffiths’ consistent histories approach, that are claimed to be local and non-classical, and identify which of the standard assumptions of Bell’s theorem each of them is forced to give up. We claim that while in operational quantum mechanics the Common Cause Principle is violated, the consistent histories approach is conspiratorial. Finally, we relate these two options to the idea of realism, a notion that is also often identified as an implicit assumption of Bell’s theorem.
Schrödinger’s reaction to the EPR paper is less widely known than, say, Bohr’s, and yet our analysis shows that it fits rather nicely with contemporary concerns in foundations of quantum mechanics. Taking the lead both from the EPR paper and from Pauli’s remarks in their correspondence, Schrödinger shows that EPR’s locality considerations lead to the assignment of values to all quantum mechanical observables, but that under apparently mild assumptions this then leads to contradictions of the von Neumann type. This dilemma (as he explicitly calls it) is thus similar to more recent debates between nonlocality on the one hand and no-go results on the other (whether through violation of the Bell inequalities, the Kochen–Specker theorem, or what you will). We shall first look at Schrödinger’s fundamental worries in the years leading up to 1935. The chapter then discusses in detail the direct reaction by Schrödinger to EPR. It will, however, not exhaust our discussion of Schrödinger, who is a recurring character in the book, having poked and prodded his peers on EPR during the whole summer and autumn of 1935.