The foundations of probability and quantum mechanics (Q2366247)

Starting with two interpretations of probability, viz. relative frequencies of attributes in infinite sequences, and betting quotients under weak coherence, the author establishes upper and lower probabilities [à la \textit{I. J. Good's} paper, Logic, Methodology and Philosophy of Science, Proc. 1960 internat. Congr. 319-329 (1962; Zbl 0192.021)]. This leads in turn to the adoption of an orthomodular partially ordered set as the domain of definition of a point-valued probability distribution. The main aim of the paper is to investigate the relevance of this approach to quantum mechanics, and it is shown that ``there can be no phase-space representation of the vast majority of quantum systems employing upper and lower probabilities''. The author concludes by proposing a generalized conception of probability that permits both objective and subjective interpretations of quantum mechanics.