Probability and quantum mechanics the conceptual foundations of stochastic mechanics

Abstract

We analyze the peculiar features of probabilistic concepts in quantum mechanics, with particular enphasis on the role of coherence properties of the quantum-mechanical description. It is recognized that the paradoxical aspects of quantum mechanics can be interpreted as having rise from improper use of classical probabilistic concepts, in particular the definition of conditional expectations.

Then the program of stochastic quantization is presented from the point of view of avoiding these difficulties. Here the kinematics of the system is described through stochastic differential equations, thus employing standard concepts of classical probability theory. On the other hand the dynamics is specified through stochastic variational principles, based on dynamical actions containing averages with respect to all possible initial configurations of the system. In this way, the resulting programming equation is of the Hamilton-Jacobi type with additional terms depending on the density. As a consequence we find a peculiar organization of the probabilistic content of the theory, fully equivalent to quantum coherence. In fact the programming equation leads to Schrödinger equation and one can observe typical "interference" of probabilities, according to standard (even if improper) quantum mechanical terminology.

Finally we show that a more refined concept of conditional expectation is needed in a probabilistic frame employing dynamical conditions based on averaged stochastic actions, as here. In fact the physical processes correspond to probability measures which make extremal the action. It is easy to check that extremalization followed by standard mathematical conditioning is not equivalent to extremalize over conditioned trajectories. The first pattern is spontaneously followed by researchers led by the primarily philosophical interest of giving realistic interpretations to quantum mechanics. Then paradoxical aspects arise. But it is the second pattern which is really involved in the quantum measurement problem.

Our general conclusion is that classical probability theory can be employed for the description of quantum systems, but the dynamical structure must take into account the peculiar features of quantum mechanics, well incorporated in the form of stochastic variational principles.

The resulting general probabilistic scheme may well merge in the future with the approach based on the development of the new theory, called quantum probability, the main topic of this Conference.