Nonstandard (non-σ-additive) probabilities in algebraic quantum field theory

Abstract

Traditionally, physicists deduce the observational (physical) meaning of probabilistic predictions from the implicit assumption that thewell-defined events whose probabilities are 0 never occur. For example, the conclusion that in a potentially infinite sequence of identical experiments with probability 0.5 (like coin tossing) the frequency of heads tends to 0.5 follows from the theorem that sequences for which the frequencies do not tend to 0.5 occur with probability 0. Similarly, the conclusion that in quantum mechanics, measuring a quantity always results in a number from its spectrum is justified by the fact that the probability of getting a number outside the spectrum is 0. In the mid-60s, a consistent formalization of this assumption was proposed by Kolmogorov and Martin-Löf, who defined arandom element of a probability space as an element that does not belong to any definable set of probability 0 (definable in some reasonable sense). This formalization is based on the fact that traditional probability measures are σ-additive, i.e., that the union of countably many sets of probability 0 has measure 0. In quantum mechanics with infinitely many degrees of freedom (e.g., in quantum field theory) and in statistical physics one must often consider non-σ-additive measures, for which the Martin-Löf's definition does not apply. Many such measures can be defined as “limits” of standard probability distributions. In this paper, we formalize the notion of a random element for such finitely-additive probability measures, and thus explain the observational (physical) meaning of such probabilities.