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Cambridge University Press eBooks

Free variables and local causality

James S. Bell

Published Jun 3, 2004• 63 citations
Publisher

Abstract

It has been argued that quantum mechanics is not locally causal and cannot be embedded in a locally causal theory. That conclusion depends on treating certain experimental parameters, typically the orientations of polarization filters, as free variables. Roughly speaking it is supposed that an experimenter is quite free to choose among the various possibilities offered by his equipment. But it might be that this apparent freedom is illusory. Perhaps experimental parameters and experimental results are both consequences, or partially so, of some common hidden mechanism. Then the apparent non-locality could be simulated. This possibility is the starting point of a paper by Clauser, Home and Shimony (CHS hereafter), which is valuable in particular for a careful mathematical formulation of the assumption which excludes such a conspiracy. In this connection they severely criticize my own ‘theory of local beables’ (B hereafter). Much of their criticism is perfectly just. In B there were jumps in the argument, and the assumption in question was not stated at the appropriate place, but only later and inadequately. However, I do not agree with CHS that this assumption, when carefully formulated, is an unreasonable one. I will organize these remarks around the three phrases in which I belatedly formulated the hypothesis in B, Section 8.

DOI
10.1017/cbo9780511815676.014

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It has been argued that quantum mechanics is not locally causal and cannot be embedded in a locally causal theory. That conclusion depends on treating certain experimental parameters, typically the orientations of polarization filters, as free variables. Roughly speaking it is supposed that an experimenter is quite free to choose among the various possibilities offered by his equipment. But it might be that this apparent freedom is illusory. Perhaps experimental parameters and experimental results are both consequences, or partially so, of some common hidden mechanism. Then the apparent non-locality could be simulated. This possibility is the starting point of a paper by Clauser, Home and Shimony (CHS hereafter), which is valuable in particular for a careful mathematical formulation of the assumption which excludes such a conspiracy. In this connection they severely criticize my own ‘theory of local beables’ (B hereafter). Much of their criticism is perfectly just. In B there were jumps in the argument, and the assumption in question was not stated at the appropriate place, but only later and inadequately. However, I do not agree with CHS that this assumption, when carefully formulated, is an unreasonable one. I will organize these remarks around the three phrases in which I belatedly formulated the hypothesis in B, Section 8.
a) Bell tries to formulate more explicitly a notion of “local causality”: correlations between physical events in different space-time regions should be explicable in terms of physical events in the overlap of the backward light cones. It is shown that ordinary relativistic quantum field theory is not locally causal in this sense, and cannot be embedded in a locally causal theory. b) Clauser, Home and Shimony criticize several steps in Bell's argument that any theory of local “beables” is incompatible with quantum mechanics. It is contended that the Clauser-Horne derivation of a Bell-type inequality circumvents his weak steps. The Clauser-Horne derivation must assume that there are no undetected correlations between choices of controllable variables in two space-like separated regions. Methodological considerations support this assumption. c) In response to criticism by Shimony, Home, and Clauser, Bell tries to clarify the argument of “The theory of local beables”, and to defend as permissible the hypothesis of free variables. d) Bell's reply to an earlier criticism by Shimony, Clauser, and Home is answered. The convergence of Bell's position towards theirs is noted.
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.
In a recent note, Hance and Hossenfelder (arXiv:2211.01331) recall that "locally causal completions of quantum mechanics are possible, if they violate the assumption [called statistical independence or measurement independence] that the hidden variables do not in any way depend on measurement settings" and that, consequently, the experimental violations of Bell inequalities "show that maintaining local causality requires violating statistical independence". However, Hance and Hossenfelder also argue that "we should (...) look for independent experimental evidence that can distinguish the two different options: non-locality and statistical independence, or locality and violations of statistical independence" and that "the unwillingness to consider theories without statistical independence may be the reason we do not yet have a locally causal theory for the foundations of physics that is consistent with general relativity". Here, we recall that there is a third option, namely, rejecting that measurement outcomes are governed in any way by hidden variables. Moreover, we argue that some recent results in the search for principles singling out the sets of quantum correlations for Bell and Kochen-Specker contextuality scenarios point out that this third option is scientifically more plausible and answers the question of why and how nature produces quantum nonlocality.
The study of hidden-variable interpretations of quantum mechanics was transformed by two papers of Bell. In his 1966 paper in Reviews of Modern Physics [1] he analysed the argument of von Neumann [2] and the more powerful arguments of Jauch and Piron [3] and of Gleason [4], all leading to similar conclusions about the nonassignability of simultaneous values to all the observables of a nontrivial quantum system; and he showed that these arguments do not exclude what may be called “contextualistic” hidden-variable theories, in which the value of an observable O is allowed to depend not only upon the hidden state λ, but also upon the set C of compatible observables measured along with O . In Bell's 1964 paper in Physics [5] (written later than the other, in spite of the earlier publication date) he gave a new kind of argument against a hidden-variable interpretation of quantum mechanics, by showing that no hidden-variables theory which satisfies a certain condition of locality can agree with all the statistical predictions of quantum mechanics. The main purpose of this lecture is to discuss an experiment, designed by Clauser, Horne, Holt and myself [6], which is based on Bell's proof of the incompatibility of quantum mechanics and local hidden-variable theories. Before doing this, however, I wish to make several comments.
Bell’s theorem is purported to demonstrate the impossibility of a local “hidden variable” theory underpinning quantum mechanics. It relies on the well-known assumption of ‘locality’, and also on a little-examined assumption called ‘statistical independence ’ (SI). Violations of this assumption have variously been thought to suggest “backward causation”, a “conspiracy ” on the part of nature, or the denial of “free will”. It will be shown here that these are spurious worries, and that denial of SI simply implies nonlocal correlation between spacelike degrees of freedom. Lorentz-invariant theories in which SI does not hold are easily constructed: two are exhibited here. It is conjectured, on this basis, that quantum-mechanical phenomena may be modeled by a local theory after all. 1
As was remarked two years ago at a symposium of the PSA on locality and hidden variables (Shimony 1981), the Bell theorems together with experiments supporting the quantum correlations (violating the inequalities of the theorems) provide a rather remarkable opportunity for philosophy. Shimony summed this up in the nice phrase, “experimental metaphysics”. For, as he pointed out, arguments based on Bell-type theorems, concluding that certain experiments should exhibit correlations obeying a certain inequality, generally employ premises of a metaphysical character, e.g., that the systems in question are deterministic (in the experimental observables). If, in fact, the experimental predictions are violated, and if one has reason to believe any other premises involved, then one must give up the metaphysical hypothesis.