An approach to the foundations of quantum mechanics, using the concept of a filter as a primitive / Hans Kummer

It is shown that the Jordan structure of non-relativistic quantum theory can be deduced, at least in the finite-dimensional case, from four physically clear postulates. The axiomatic approach to the foundations of quantum mechanics developed here takes the concept of a filter as a starting point. More precisely, the primitive notions of the theory are: probability, filter, composition of filters, and stochastic selection of filters. All filters are assumed to be built from a countable set of primitive filters by composition and stochastic selection. Next, two physically transparent axioms are introduced, which allow the author to define the concept of a system (Definition 3.2). With every system there is associated a quadruple, the so-called σ-quadruple (Definition 3.10), whose first member is a separable complete order unit space. Then, two additional axioms are formulated, which lead to the effect that this space becomes isomorphic to the order unit space underlying a Jordan- Banach algebra, at least in the finite-dimensional case (Corollary 4.23). (Zentralblatt)

Bibliogr. p. [128-129]