Introduction to Nelson Stochastic Mechanics as a Model for Quantum Mechanics

Abstract

We give a short presentation of Nelson stochastic mechanics, as a generalization of classical mechanics, based on the theory of stochastic processes and stochastic variational principles. Stochastic mechanics can be connected to quantum mechanics through a very simple physical interpretation scheme. From this point of view, stochastic mechanics can be seen as a quantization procedure for mechanical systems, different, but physically equivalent, to the usual operator quantization. Then we deal with the problems related to the possibility of considering stochastic mechanics as a complete physical theory. Through a discrete generalization, we show how the main features of the postulated underlying Brownian motion, at the origin of quantum fluctuations, can be derived also as a consequence of stochastic variational principles. We also discuss the problem of formulating stochastic mechanics in representations different from the configuration representation, and show how the different representations, related by unitary transformations in ordinary quantum mechanics, are connected in stochastic mechanics through stochastic measure preserving transformations. Finally, we show how the basic aspects of the measurement problem, in particular the wave packet reduction, can be interpreted in the frame of generalized stochastic mechanics. Here, the wave function collapse is not instantaneous, but is ruled by a well defined dynamical scheme, with a time asymptotic relaxation behavior. Moreover, the relaxation is slower, if the result of the measurement is more uncertain. Finally, we discuss the possibility of embedding stochastic mechanics into a generalized scheme of Schrodinger stochastic processes. 1