Dynamical States and the Conventionality of (Non-) Classicality

Abstract

Itamar Pitowsky long championed the view that quantum mechanics (QM) is best understood as a non-classical probability theory. Here, I want to offer some modest caveats. One is that QM is best seen, not as a new probability theory, but as something narrower, namely, a particular probabilistic theory-roughly, a class of probabilistic models, selected from within a much more general framework. It is this general framework that, if anything, deserves to be regarded as a "non-classical" probability theory. However, as I will also argue, this framework represents a very conservative extension of classical probability theory, essentially just eliding a tacit, and contingent, assumption in the latter that measurements or experiments can always be performed together. Moreover, for individual probabilistic models, and even for probabilistic theories, the distinction between "classical" and a "non-classical" is largely a conventional one, bound up with the question of what one means by the state of a system. In fact, for systems with a high degree of symmetry, including quantum mechanics, it is possible to interpret general probabilistic models as having a perfectly classical probabilistic structure, but an additional dynamical structure: states, rather than corresponding simply to probability measures, are represented as certain probability measure-valued functions on the system's symmetry group, and thus, as fundamentally dynamical objects. Conversely, a classical probability space equipped with reasonably well-behaved family of such "dynamical states" can be interpreted as a generalized probabilistic model in a canonical way. It is noteworthy that this "dynamical" representation is not a conventional hiddenvariables representation, and the question of what one means by "non-locality" in this setting is not entirely straightforward.

Quantum logic in the context of the mathematical foundation of quantum mechanics

“…В этой связи примечательны работы Макки [Макки, 1965]. Как сказано в Стенфордской энциклопедии по философии, Макки изложил математический аппарат квантовой механики как «обобщенную теорию вероятности, в основе которой лежит логика экспериментальных высказываний, или, в его терминологии, «вопросов», имеющая структуру сигма ортомодулярного частично упорядоченного множества» [Wilce, 2021].…”