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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Violation of Einstein causality in a model quantum system

Morton H. Rubin

Published Jun 15, 1987• 47 citations
Publisher

Abstract

A series of theorems have been proved showing that localization in quantum mechanics is inconsistent with Einstein causality. In this paper a simple model of a localized source and a localized detector is studied. This model does not satisfy the assumptions of the rigorous theorems; nonetheless, it is found that the model violates Einstein causality. The effect is very small and resides in the nonresonant part of the transition probability.

DOI
10.1103/physrevd.35.3836

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A series of theorems have been proved showing that localization in quantum mechanics is inconsistent with Einstein causality. In this paper a simple model of a localized source and a localized detector is studied. This model does not satisfy the assumptions of the rigorous theorems; nonetheless, it is found that the model violates Einstein causality. The effect is very small and resides in the nonresonant part of the transition probability.
Bell's theorem shows that the reasonable relativistic causal principle known as "local causality" is not compatible with the predictions of quantum mechanics. It is not possible maintain a satisfying causal principle of this type while dropping any of the better-known assumptions of Bell's theorem. However, another assumption of Bell's theorem is the use of classical logic. One part of this assumption is the principle of "ontic definiteness", that is, that it must in principle be possible to assign definite truth values to all propositions treated in the theory. Once the logical setting is clarified somewhat, it can be seen that rejecting this principle does not in any way undermine the type of causal principle used by Bell. Without ontic definiteness, the deterministic causal condition known as Einstein Locality succeeds in banning superluminal influence (including signalling) whilst allowing correlations that violate Bell's inequalities. Objections to altering logic, and the consequences for operational and realistic viewpoints, are also addressed.
The conflict between relativistic causality and localizability is analysed in the light of the existence of unsharp localization observables. A theorem due to Schlieder is generalized, showing that the assumption of local commutativity implies the localization observable in question to be unsharp in a strong sense. Furthermore, a recent generalization of a theorem of Lüders is applied to demonstrate that local commutativity is a necessary consequence of Einstein causality even in the case of unsharp observables if they admit local measurements. These findings raise the question of whether localization observables can be measured by means of local operations.
In papers published in the 25 years following his famous 1964 proof, John Bell refined and reformulated his views on locality and causality. Although his formulations of local causality were in terms of probability, he had little to say about that notion. But assumptions about probability are implicit in his arguments and conclusions. Probability does not conform to these assumptions when quantum mechanics is applied to account for the particular correlations Bell argues are locally inexplicable. This account involves no superluminal action and there is even a sense in which it is local, but it is in tension with the requirement that the direct causes and effects of events be nearby. Introduction I never met John Bell, but his writings have supplied me with a continual source of new insights as I read and reread them over 40 years. As I worked toward a rather different understanding of quantum mechanics he was foremost in my mind as a severe but honest critic of such attempts. We all would love to know what Einstein would have made of Bell's theorem. I confess that the deep regret I feel that Bell cannot respond to this paper is sometimes assuaged by a sense of relief. Locality and Local Causality In his seminal 1964 paper [1], John Bell expressed locality as the requirement that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past. [2, p. 14] This seems to require that the result of a measurement would have been the same, no matter what operations had been performed on such a distant system. But suppose the result of a measurement were the outcome of an indeterministic process. Then the result of the measurement might have been different even if exactly the same operations (if any) had been performed on that distant system. So can no indeterministic theory satisfy the locality requirement? Bell felt no need to address that awkward question in his 1964 paper [1], since he took the EPR argument to establish that any additional variables needed to restore locality and causalitywould have to determine a unique result of ameasurement.