NONSTANDARD (NON-SIGMA-ADDITIVE) PROBABILITIES IN ALGEBRAIC QUANTUM-FIELD THEORY

Traditionally, physicists deduce the observational (physical) meaning of probabilistic predictions from the implicit assumption that the wel l-defined events whose probabilities are 0 never occur. For example, t he conclusion that in a potentially infinite sequence of identical exp eriments with probability 0.5 (like coin tossing) the frequency of hea ds 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, th e conclusion that in quantum mechanics, measuring a quantity always re sults 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 Kol mogorov and Martin-Lof, who defined a random element of a probability space as an element that does not belong to any definable set of proba bility 0 (definable in some reasonable sense). This formalization is b ased on the fact that traditional probability measures are sigma-addit ive, i.e., that the union of countably many sets of probability 0 has measure 0. In quantum mechanics with infinitely many degrees of freedo m (e.g., in quantum field theory) and in statistical physics one must often consider non sigma-additive measures, for which the Martin-Lof's definition does not apply. Many such measures can be defined as ''lim its'' of standard probability distributions. In this paper, we formali ze the notion of a random element for such finitely-additive probabili ty measures, and thus explain the observational (physical) meaning of such probabilities.