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arXiv:quant-ph/0602080 · Optics and Spectroscopy

Nonlocality, Bell’s Ansatz, and probability

Aloysius F. Kracklauer

Published Sep 1, 2007• 8 citations
arXivHTMLPDF

Abstract

Quantum mechanics lacks an intuitive interpretation, which is the cause of a generally formalistic approach to its use. This in turn has led to a certain insensitivity to the actual meaning of many words used in its description and interpretation. Herein, we analyze carefully the possible mathematical meanings of those terms used in analysis of EPR’s contention, that Quantum Mechanics is incomplete, as well as Bell’s work descendant therefrom. As a result, many inconsistencies and errors in contemporary discussions of nonlocality, as well as in Bell’s Ansatz with respect to the laws of probability, are identified. Evading these errors precludes serious conflicts between Quantum Mechanics and both Special Relativity and Philosophy.

DOI
10.1134/s0030400x07090147

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Quantum mechanics lacks an intuitive interpretation, which is the cause of a generally formalistic approach to its use. This in turn has led to a certain insensitivity to the actual meaning of many words used in its description and interpretation. Herein, we analyze carefully the possible mathematical meanings of those terms used in analysis of EPR’s contention, that Quantum Mechanics is incomplete, as well as Bell’s work descendant therefrom. As a result, many inconsistencies and errors in contemporary discussions of nonlocality, as well as in Bell’s Ansatz with respect to the laws of probability, are identified. Evading these errors precludes serious conflicts between Quantum Mechanics and both Special Relativity and Philosophy.
A factual inconsistency of the nonlocality theory is demonstrated via a careful analysis of the conceptual and formal contents of Bell's theorem and the inequalities associated with it. The conceptual inconsistency of the theory follows from the consideration that it is, in principle, impossible to formulate quantum mechanics in terms of local hidden variables. The formal inconsistency of the nonlocality theory follows from the facts that: (i) nonlocality is not a physical property that can be represented by a function or an operator; (ii) it is impossible to derive Bell's inequality in a mathematically consistent way; and (iii) Bell's definition of locality is incorrect. Therefore, Bell's theorem cannot serve as the foundation for the introduction of the nonlocality notion in quantum mechanics.
It is shown that when properly analyzed using principles consistent with the use of a Hilbert space to describe microscopic properties, quantum mechanics is a local theory: one system cannot influence another system with which it does not interact. Claims to the contrary based on quantum violations of Bell inequalities are argued to be incorrect. A specific example traces a violation of the CHSH Bell inequality in the case of a spin-3/2 particle to the noncommutation of certain quantum operators in a situation where (non)locality is not an issue. A consistent histories analysis of what quantum measurements measure, in terms of quantum properties, is used to identify the basic problem with derivations of Bell inequalities: the use of classical concepts (hidden variables) rather than a probabilistic structure appropriate to the quantum domain. A difficulty with the original Einstein-Podolsky-Rosen (EPR) argument for the incompleteness of quantum mechanics is the use of a counterfactual argument which is not valid if one assumes that Hilbert-space quantum mechanics is complete; locality is not an issue. The quantum correlations that violate Bell inequalities can be understood using local quantum common causes. Wave-function collapse and Schr\"odinger steering are calculational procedures, not physical processes. A general Principle of Einstein Locality rules out nonlocal influences between noninteracting quantum systems. Some suggestions are made for changes in terminology that could clarify discussions of quantum foundations and be less confusing to students.
In the 80 years since the seminal Einstein, Podolsky, and Rosen (EPR) paper, physicists and philosophers have mused about the `spooky action at a distance' aspect of quantum mechanics that so bothered Einstein. In his formal analysis of EPR-type entangled quantum states, Bell (1964) concluded that any hidden variable theory designed to reproduce the predictions of quantum mechanics must necessarily be nonlocal and allow superluminal interactions. This doesn't immediately imply that nonlocality is a characteristic feature of quantum mechanics let alone a fundamental property of nature; however, many physicists and philosophers of science do harbor this belief. Experts in the field often use the term `nonlocality' to designate particular non-classical aspects of quantum entanglement and do not confuse the term with superluminal interactions. However, many physicists seem to take the term more literally. I endeavor to disabuse the latter of this notion by emphasizing that the correlations of Bell-type entanglement are a result of ordinary quantum superposition with no need to introduce nonlocality. The conclusion of the EPR paper wasn't that quantum mechanics is nonlocal but rather that it is an incomplete description of reality. For different reasons, many physicists, including me, agree with Einstein that quantum mechanics is necessarily an incomplete description of reality.
It is shown that when properly analyzed using principles consistent with the use of a Hilbert space to describe microscopic properties, quantum mechanics is a local theory: one system cannot influence another system with which it does not interact. Claims to the contrary based on quantum violations of Bell inequalities are shown to be incorrect. A specific example traces a violation of the CHSH Bell inequality in the case of a spin-3/2 particle to the noncommutation of certain quantum operators in a situation where (non)locality is not an issue. A consistent histories analysis of what quantum measurements measure, in terms of quantum properties, is used to identify the basic problem with derivations of Bell inequalities: the use of classical concepts (hidden variables) rather than a probabilistic structure appropriate to the quantum domain. A difficulty with the original Einstein-Podolsky-Rosen (EPR) argument for the incompleteness of quantum mechanics is the use of a counterfactual argument which is not valid if one assumes that Hilbert-space quantum mechanics is complete; locality is not an issue. The quantum correlations that violate Bell inequalities can be understood using local quantum common causes. Wavefunction collapse and Schr\odinger steering are calculational procedures, not physical processes. A general Principle of Einstein Locality rules out nonlocal influences between noninteracting quantum systems. Some suggestions are made for changes in terminology that could clarify discussions of quantum foundations and be less confusing to students.
Between 1964 and 1990, the notion of nonlocality in Bell's papers underwent a profound change as his nonlocality theorem gradually became detached from quantum mechanics, and referred to wider probabilistic theories involving correlations between separated beables. The proposition that standard quantum mechanics is itself nonlocal (more precisely, that it violates ‘local causality’) became divorced from the Bell theorem per se from 1976 on, although this important point is widely overlooked in the literature. In 1990, the year of his death, Bell would express serious misgivings about themathematical form of the local causality condition and leave ill-defined the issue of the consistency between special relativity and violation of the Bell-type inequality. In our view, the significance of the Bell theorem, in both its deterministic and stochastic forms, can only be fully understood by taking into account the fact that a fully Lorentz covariant version of quantum theory, free of action at a distance, can be articulated in the Everett interpretation. Introduction John S. Bell's last word on his celebrated nonlocality theorem and its interpretation appeared in his 1990 paper ‘La nouvelle cuisine’, first published in the year of his untimely death. Bell was careful here to distinguish between the issue of ‘no superluminal signalling’ in quantum theory (both quantum field theory and quantum mechanics) and a principle he first introduced explicitly in 1976 and called ‘local causality’ [1]. In relation to the former, Bell expressed concerns that amplify doubts he had already expressed in 1976. These concerns touch on what is now widely known as the no-signalling theorem in quantum mechanics, and ultimately have to do with Bell's distaste for what he saw as an anthropocentric element in orthodox quantum thinking. In relation to local causality, Bell emphasised that his famous factorizability (no-correlations) condition is not to be seen ‘as the formulation of local causality, but as a consequence thereof’ and stressed how difficult he found it to articulate this consequence. He left the question of any strict inconsistency between violation of factorizability and special relativity theory unresolved, a not insignificant shift from his thinking up to the early 1980s.
We remind the viewpoint that violation of Bell's inequality might be interpreted not only as an evidence of the alternative -- either nonlocality or ``death of reality'' (under the assumption the quantum mechanics is incomplete). Violation of Bell's type inequalities is a well known sufficient condition of incompatibility of random variables -- impossibility to realize them on a single probability space. Thus, in fact, we should take into account an additional interpretation of violation of Bell's inequality -- a few pairs of random variables (two dimensional vector variables) involved in the EPR-Bohm experiment are incompatible. They could not be realized on a single Kolmogorov probability space. Thus one can choose between: a) completeness of quantum mechanics; b) nonlocality; c) `` death of reality''; d) non-Kolmogorovness. In any event, violation of Bell's inequality has a variety of possible interpretations. Hence, it could not be used to derive the unique conclusion on the relation between quantum and classical models.