Ultraproducts of quantum mechanical systems

Abstract

The study of ultraproducts for various spaces is motivated by an interest in methods of non-standard mathematical analysis, which operates on infinitesimal (or infinitely large) sequences as if they were numbers. Onone hand, a space obtained as a set-theoretic ultraproduct of a sequence of spaces becomes very «rich». On the other hand, it loses some attractive properties of factors. In particular, it has no a natural Hausdorff topology generated by its factors, and the natural $\sigma$-algebra of its measurable subsets is not countably generated.If a space «is embedded» into its ultrapower with the preservation of required properties, then the usage of the ultraproduct technique gives some advantages in proving many «standard» assertions.In order to preserve various properties of factors, we need to change the construction of an ultraproduct. For example, by changing the construction of an ultraproduct, it becomes possible to preserve the Hausdorff topology, the structure of a normed space, the structure of operator algebras, von Neumann algebras, and so on.In this paper we discuss the stochastic properties of the so-called quantum mechanical systems in a rather abstract form. Such systems (structures) arise in probability theory, in the theory of operator algebras and in the theory of topological vector spaces. The ultraproducts for sequences of such structures are also defined, and certain properties of these ultraproducts are investigated.The notion of an observable on an event structure is an analogue of a random variable defined on a probability space. An observable is naturally given in the ultraproduct of quantum mechanical systems which is defined in the present paper. We study its probabilistic characteristics. Moreover, ultraproducts of quantum logics are also considered within the framework of ultraproducts for quantum mechanical systems.