In orthodox quantum mechanics, it has virtually become the custom to identify properties of a physical system with operationally testable propositions about the system. The causes and consequences of this practice are explored mathematically in this paper. Among other things, it is found that such an identification imposes severe constraints on the admissible states of the physical system.

The purpose of this paper is to sketch an attack on the general problem of representing a composite physical system in terms of its constituent parts. For quantum-mechanical systems, this is traditionally accomplished by forming either direct sums or tensor products of the Hilbert spaces corresponding to the component systems. Here, a more general mathematical construction is given which includes the standard quantum-mechanical formalism as a special case.

We first present a realistic framework for quantum probability theory based on the path integral formalism of quantum mechanics and illustrate this framework by constructing a model that describes a quantum particle evolving in a discrete space-time lattice. We then present a finite model for describing the internal dynamics of “elementary particles” and show that this model gives the standard particle classification scheme and successfully predicts particle masses.

Using the mathematical notion of an entity to represent states in quantum and classical mechanics, we show that, in a strict sense, proper superpositions are possible in classical mechanics.

We develop the concept of quantum probability based on ideas of R. Feynman. The general guidelines of quantum probability are translated into rigorous mathematical definitions. We then compare the resulting framework with that of operational statistics. We discuss various relationship between measurements and define quantum stochastic processes. It is shown that quantum probability includes both conventional probability theory and traditional quantum mechanics. Discrete quantum systems, transition amplitudes, and discrete Feynman amplitudes are treated. We close with some examples that illustrate previously (...) defined concepts. (shrink)