Quantum stochastic processes and the Feynman path integral for a single spinless particle

Abstract

A general scheme of a theory of the quantum stochastic processes is outlined in analogy to the theory of the classical stochastic processes. Special attention is paid to the simplest possible case of physical interest, namely, to the description of a single, spinless non-relativistic particle moving in the presence of an external electromagnetic field. We develop further, in comparison to our previous work, the idea of quantum stochastic processes by introducing explicitly random variables describing the position of a particle at different times. The transition amplitude, which before played the leading role, may be expressed in terms of these variables and some complex-valued average operations. These operations substitute the measure theoretical notions used in the theory of classical stochastic processes. We have shown that the solutions of chauchy's problem for the general Schrödinger equations may be expressed as, what we call, quantum averages of some functionals on the process. As one knows these solutions are usually written in the form of heuristic Feynman path integrals. Therefore, our formulae give a mathematical meaning to the Feynman path integrals. A factor-ordering ambiguity is analysed and removed by a convention concerning the stochastic integrals in volved.