Hidden Variables and the Propensity Theory of Probability

L. E. Ballentine

M. D. Srinivas

Peter J. Clark

Aldo F. G. Solis-Labastida, Melina Gastélum Vargas, Jorge G. Hirsch

The violation of Bell inequalities is often interpreted as showing that, if hidden variables exist, they must be contextual and non local. But they can also be explained questioning the probability space employed, or the validity of the Kolmogorov axioms. In this article we explore the additional constrains which can be deduced from two widely used objetive probability theories: frequentism and propensities. One of the strongest objections in the deduction of one version of Bell inequalities goes about the probability space, which assumes the existence of values for the output of the experiment in each run, while only two of the four values can be measured each time, making them counterfactual. It is shown that frequentism rejects the possibility of using counterfactual situations, while long-run propensities allow their use. In this case the introduction of locality and contextuality does not help to explain the violation, and an alternative explanation could point to a failure of the probability. Single case propensities were designed to associate probabilities to single events, but they need to be conditional to the whole universe, and do not have a clear link with the observed relative frequencies. It heavily limits their use.

Niall Shanks

Geoffrey Paul Hellman

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.