Discrete stochastic variational principles and quantum mechanics

Abstract

We consider stochastic variational principles for random processes taking values on a discrete configuration space. For a suitable time-reversal-invariant choice of the stochastic action, the resulting programming equations can be related to the Schrödinger equation for a discrete system. If the discrete system is considered as an approximation of a continuous system, then the limit reproduces Nelson's stochastic mechanics and allows one to derive the assumptions on the random noise acting on the system. In fact, the variational principle also gives information about the osmotic behavior of the process. Finally, we show that there are also critical processes whose behavior can be interpreted as a model for quantum measurement, because they relax in time to mixtures of processes. Therefore, stochastic variational principles can provide a very simple conceptual model simulating quantum behavior, both from the point of view of unperturbed time evolution and of the measurement phenomena leading to wave-function collapse.