A general scheme for microscopic theories

Abstract

We discuss from an operational point of view some fundamental concepts of the micro-scopic physical theories, with the aim of providing a background for a successive investigation of the microscopic space-time structure. We consistently develop the remark that a frame of reference is determined by physical objects which may interact with the objects under investigation. As it is not clear that a state can be prepared by means of physical operations, we do not use the concept of physical state for the foundation of the theory. In our approach, the primitive concepts are the measurement procedures, following which one gets a numerical result, and the transformation procedures, which have the aim of building a frame of reference. We discuss several rules which allow us to define new procedures in terms of known procedures. The statistical laws of physics are formulated in terms of an order relation between measurement procedures, which defines also an equivalence relation. The equivalence classes of measurement procedures are called measurements. We define also equivalence classes of transformation procedures, called transformations. The mathematical structure of the set of measurements and of the set of transformations is discussed in detail. We consider measurements with an arbitrary finite number of possible results, as this enables us to give a rigorous definition of compatibility. Finally, we point out that all the physical theories necessarily contain ideal measurements and transformations which do not correspond to any known physical procedure. The introduction of these ideal objects permits a considerable simplification of the mathematical structure of the theory, but reduces its physical content.