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American Journal of Physics

Probability and quantum mechanics

Daniel J. Velleman

Published Nov 1, 1998• 4 citations
Publisher

Abstract

We discuss the use of probability theory in quantum mechanics, and suggest that there are some interpretations of quantum mechanics in which probabilities may be used inappropriately.
DOI
10.1119/1.19007

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This is an attempt to clarify certain concepts related to a debate on the interpretation of quantum mechanics, a debate between Khrennikov on the one hand and Stacey and Schack on the other hand. Central to this debate is the notion of quantum probabilities. I first take up the probability concept in the QBist school, as seen from my point of view, and then give my own arguments for the Born formula for calculating quantum probabilities. In that connection, I also sketch some consequences of my approach toward the foundation and interpretation of quantum theory. I discuss my general views on QBism as a possible alternative interpretation of quantum mechanics before I give some final remarks.
The topic of probabilty in quantum mechanics is rather vast, and in this article, we shall choose to discuss it from the perspective of whether and in what sense quantum mechanics requires a generalisation of the usual (Kolmogorovian) concept of probability. We shall focus on the case of finite-dimensional quantum mechanics (which is analogous to that of discrete probability spaces), partly for simplicity and partly for ease of generalisation. While we shall largely focus on formal aspects of quantum probability (in particular the non-existence of joint distributions for incompatible observables), our discussion will relate also to notorious issues in the interpretation of quantum mechanics. Indeed, whether quantum probability can or cannot be ultimately reduced to classical probability connects rather nicely to the question of 'hidden variables' in quantum mechanics. An abridged version of this essay will appear in A. Hajek and C. Hitchcock (eds.), 'The Oxford Handbook of Probability and Philosophy', Oxford University Press. The present version includes in particular many more footnotes and references and the Appendix with the proofs of the Lemma and Proposition of Section 5.
The abstract theory of probability and its interpretation are briefly reviewed, and it is explicitly demonstrated that the formalism of quantum mechanics satisfies the axioms of probability theory. This refutes the suggestions which have occasionally been made that ‘‘classical’’ probability theory does not apply to quantum mechanics. Several erroneous applications of probability theory to quantum mechanics are examined, and the nature of the errors are exposed. It is urged that more attention be given to probability theory in the physics curriculum.
Born’s probabilistic interpretation of quantum mechanics is commonly assumed to rely on classical probability theory in the sense of Kolmogorov. In this paper we argue that this assumption is methodologically unjustified. We show that quantum probabilities naturally arise from an analytic method fundamentally different from the synthetic approach underlying classical probability theory. In the analytic framework, the whole (the totality of alternatives) is primary, while probabilities of individual alternatives are derived quantities. Interference appears at the level of the realization probability of the whole, whereas conditional probabilities of alternatives remain additive within a given context. We demonstrate that many well-known quantum paradoxes originate from the inappropriate application of synthetic probabilistic reasoning. The proposed formulation is consistent with the Born rule, its extensions developed in our earlier works, and with Birkhoff–von Neumann quantum logic. Classical probability theory appears only as a degenerate formal limit and is not methodologically universal.