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Philosophy of Science

Counterfactual Reasoning in the Bell-EPR Paradox

Malcolm R. Forster

Published Mar 1, 1986• 8 citations
Publisher

Abstract

Skyrms's formulation of the argument against stochastic hidden variables in quantum mechanics using conditionals with chance consequences suffers from an ambiguity in its “conservation“ assumption. The strong version, which Skyrms needs, packs in a “no-rapport” assumption in addition to the weaker statement of the “experimental facts.“ On the positive side, I argue that Skyrms's proof has two unnoted virtues (not shared by previous proofs): (1) it shows that certain difficulties that arise for deterministic hidden variable theories that exploit a non-classical probability theory extend to the stochastic case; (2) the use of counterfactual conditionals relates the Bell puzzle to Dummett's (1976) discussion of realism in quantum mechanics.

DOI
10.1086/289299

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Skyrms's formulation of the argument against stochastic hidden variables in quantum mechanics using conditionals with chance consequences suffers from an ambiguity in its “conservation“ assumption. The strong version, which Skyrms needs, packs in a “no-rapport” assumption in addition to the weaker statement of the “experimental facts.“ On the positive side, I argue that Skyrms's proof has two unnoted virtues (not shared by previous proofs): (1) it shows that certain difficulties that arise for deterministic hidden variable theories that exploit a non-classical probability theory extend to the stochastic case; (2) the use of counterfactual conditionals relates the Bell puzzle to Dummett's (1976) discussion of realism in quantum mechanics.
Two EPR arguments are reviewed, for their own sake, and for the purpose of clarifying the status of “stochastic“ hidden variables. The first is a streamlined version of the EPR argument for the incompleteness of quantum mechanics. The role of an anti-instrumentalist (“realist”) interpretation of certain probability statements is emphasized. The second traces out one horn of a central foundational dilemma, the collapse dilemma; complex modal reasoning, similar to the original EPR, is used to derive determinateness (of all spin components of two spin-½ particles in the singlet state) from just (a form of) weak locality, result definiteness, and an assumption on propensities based on conservation. Theories meeting these conditions are therefore constrained by the Bell inequalities. Neither controversial assumptions of “strong locality” (“factorability”) nor of determinism are employed in the derivation. The categories of “stochastic hidden variables” are then analyzed; one can focus on “quasi-definite” theories, without loss of generality. A means of excluding these is proposed, based on a demand that certain ideal cases be accurately treated. Theorems from quantum measurement theory, sometimes cited as showing that such cases are not physically possible, are found inapplicable.
Most physicists agree that the Einstein–Podolsky–Rosen–Bell paradox exemplifies much of the strange behavior of quantum mechanics, but argument persists about what assumptions underlie the paradox. To clarify what the debate is about, we employ a simple and well-known thought experiment involving two correlated photons to help us focus on the logical assumptions needed to construct the EPR and Bell arguments. The view presented in this paper is that the minimal assumptions behind Bell’s inequality are locality and counterfactual definiteness but not scientific realism, determinism, or hidden variables as are often suggested. We further examine the resulting constraints on physical theory with an illustration from the many-worlds interpretation of quantum mechanics—an interpretation that we argue is deterministic, local, and realist but that nonetheless violates the Bell inequality.
We argue that for the proof of Bell's theorem no assumptions about realism or free will are necessary. The key formula E(AB∣a,b)=∫A(a,b,λ)B(a,b,λ)ρ(λ)dλE(AB|a,b) = \int A(a,b,λ)B(a,b,λ)ρ(λ) dλE(AB∣a,b)=∫A(a,b,λ)B(a,b,λ)ρ(λ)dλ follows from the logic of plausible reasoning (the objective Bayesian interpretation of probability theory) taken alone, without any further assumptions about realism. The space ΛΛΛ, usually interpreted as some space of `hidden variables', can be constructed for an arbitrary `field of discourse' using Stone's theorem. The rejection of superdeterminism follows from logical independence -- the non-existence of information which suggest a dependence -- of the free decisions of the experimenters from everything else. To prove the Bell inequality it is, then, sufficient to reduce this to E(AB∣a,b)=∫A(a,λ)B(b,λ)ρ(λ)dλ.E(AB|a,b) = \int A(a,λ)B(b,λ)ρ(λ) dλ.E(AB∣a,b)=∫A(a,λ)B(b,λ)ρ(λ)dλ. This follows for space-like separated measurements from Einstein causality alone. So, the consequence of the violation of Bell's inequality is that Einstein causality has been empirically falsified, without any loopholes left. We consider and reject the idea of weakening the logic of plausible reasoning could be used to create a loophole. Finally, we discuss what can be used to replace Einstein causality. While weak (signal) Einstein causality holds, it follows either from a notion of strong causality or from human inability to prepare states which violate quantum uncertainty, which makes it uninteresting for fundamental considerations. A notion of strong causality has either unacceptable causal loops or requires a hidden preferred frame. Given that a hidden preferred frame is completely compatible with modern physics, it is argued that this choice is preferable.
Bell's theorem proves the incompatibility between quantum mechanics and local realistic hidden-variable theories. In this paper we show that, contrary to a common belief, the theoretical proof of Bell's theorem is not affected by counterfactual reasoning. Then, we demonstrate that the experimental verification of this theorem may be affected in an unknowable way by our ignorance about the probability distribution of the hidden variables. Our study is based on the standard theory of random variables, and lays the groundwork for a critical rethinking of Bell's theorem and its consequences.
Abstract One of the Bell’s assumptions in the original derivation of his inequalities was the hypothesis of locality, i.e., the absence of the influence of two remote measuring instruments on one another. That is why violations of these inequalities observed in experiments are often interpreted as a manifestation of the nonlocal nature of quantum mechanics, or a refutation of a local realism. It is well known that the Bell’s inequality was derived in its traditional form, without resorting to the hypothesis of locality and without the introduction of hidden variables, the only assumption being that the probability distributions are nonnegative. This can therefore be regarded as a rigorous proof that the hypothesis of locality and the hypothesis of existence of the hidden variables not relevant to violations of Bell’s inequalities. The physical meaning of the obtained results is examined. Physical nature of the violation of the Bell inequalities is explained under new EPR-B nonlocality postulate. We show that the correlations of the observables involved in the Bohm–Bell type experiments can be expressed as correlations of classical random variables. The revisited Bell type inequality in canonical notatons reads 〈AB〉 + 〈A’B〉 + 〈AB’〉 − 〈A’B’〉 ⩽ 6.
The EPR paradox is shown to be a result of the wrong definitions in quantum mechanics of such physical quantities as momentum and position. The corrected definitions show that momentum and position of particles can be precisely defined simultaneously and uncertainty relations do not prevent it. Therefore, with correct definition of physical quantities in quantum mechanics, the paradox disappears. The Bohm Aharonov version of the EPR paradox is illustrated by the example of measuring the polarization of photon pairs. The common view of the two photon state radiated by a source is presented. The common Bell’s inequality is discussed and the simplest version of it for a specific hidden variable is derived. A possibility of an experimental study of whether a photon and its polarization are preexistent before measurements is considered.