Foundations for Quantum Mechanics

Abstract

A discussion is given of the structure of a physical theory and an ``ideal form'' for such a theory is proposed. The essential feature is that all concepts should be defined in operational terms. Quantum (and classical) mechanics is then formulated in this way (the formulation being, however, restricted to the kinematical theory). This requires the introduction of the concept of a mixed test, related to a pure test (or ``question'') just as a mixed state is related to a pure state. In the new formulation, the primitive concepts are not states and observables but certain operationally accessible mixed states and tests called physical. The notion of a C*‐system is introduced; each such system is characterized by a certain C*‐algebra. The structure of a general C*‐system is then studied, all concepts being defined in terms of physical states and tests. It is shown first how pure states and tests can be so defined. The quantum analog of the phase space of classical mechanics is then constructed and on it is built a mathematical structure, called a q‐topology, which is a quantum analog of the topology of classical phase space. Mathematically, a q‐topology is related to a noncommutative C*‐algebra as an ordinary topology is related to a commutative C*‐algebra. Some properties of the q‐topology of a C*‐system are given. An appendix contains some physically motivated examples illustrating the theory.