Interpretations of Probability in Quantum Mechanics

Abstract

Taking for granted that the mathematical apparatus for describing probabilities in quantum mechanics is well-understood via work of von Neumann, Lüders, Mackey, and Gleason, we present an overview of different interpretations of “probability” in quantum mechanics bearing on physics and experiment, with the aim of clarifying the meaning and place of so-called “objective” interpretations of quantum probability. The dichotomy “objective/subjective” is unfortunate, we argue, as we should distinguish two different dimensions integral to the concept of probability. The first concerns the values of probability functions, viz. what the real numbers measure, e.g. relative frequencies of experimental outcomes, or strengths of physical dispositions (objective-1), vs. degrees of belief of idealized agents (subjective-1), etc. But a second dimension is also important, concerning the domain of definition, the “events” or “bearers” of probability, what the probabilities are probabilities of. Relative frequencies of what, described how, or strengths of dispositions to do what, described how, degrees of belief in what, etc. Reminding ourselves of the quantum mechanical phenomenon of incompatible observables, we recall that contradictions are standardly avoided by describing probabilities as pertaining to “measurement outcomes” rather than “possessed properties”: thereby, subjective elements are introduced into the very description of the “events”. (Interpretations qualify as objective-2 if they avoid “bad words” like “measurement” as primitive, in favor of possessed properties or physical interactions; as subjective-2 if such terms are employed in an essential way.) This leads to a two-by-two matrix of interpretative possibilities. The remainder of our talk consists in filling in the blanks (which the reader is invited to try for him/herself) and providing commentary on the relative advantages and disadvantages, which go to the heart of the problem of interpreting quantum theory. Given our scheme, it turns out that “objective version of Copenhagen” makes good sense; this is one locus of “propensities”, which can be made sense of, we claim, along the lines of pre-hidden-variables Bohm (his 1951 text), not to be confused with Popper. We close by noting a serious deficiency in recent Bayesian approaches to quantum probability (lying in the subj-1, subj.-2 quadrant), viz. its explanatory impoverishment. But I’ve already given too many hints.