As indicated by recent studies (Nelson, 1979; Guerra, 1981) stochastic mechanics (its name is probably due to Guerra) is a very simple and clear theory, with enormous possibilities of physical and mathematical effectiveness in the exploration of properties of quantum mechanical systems, especially when a large number of degrees of freedom are involved, as the experience of the Euclidean methods in constructive quantum field theory has shown. The main advantage of stochastic mechanics, from the point of view of mathematical control, relies on the possibility of exploiting the well-developed mathematical methods of probability theory and stochastic processes (see Chapter 6). However stochastic mechanics is not free from criticism. We would like to stress a peculiar aspect of stochastic mechanics which is the basis of typical misunderstanding and unjustified criticism. It is remarked that a very natural objection arises in connection with any proposal of exploiting stochastic processes for the formulation of a theoretical scheme related to quantum mechanics. As usual, it is suggested that quantum mechanics may be interpreted as a Markov process, but irreducible to a Brownian-type stochastic motion. Or, more clearly it can be phrased as follows: ‘Stochastic processes are typical expressions of diffusion phenomena and therefore share a substantial time irreversible character, while quantum mechanics is time reversible and therefore cannot have anything to do with stochastic processes’ (see Guerra, 1981).
KeywordsRelativistic Case Schrodinger Equation Stochastic Theory Smoluchowski Equation Stochastic Mechanic
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