Semi-classical quantum mechanics and stochastic calculus of variations

Abstract

Within the framework of the general theory of stochastic calculus of variations, we examine mainly the notion of second variation in the stochastic mechanics of E. Nelson, a representative of quantum mechanics in which the concept of path for particles keep a sense. We show that the two approaches used in classical calculus of variation to know if a path is not only an extremum but also the minimum of the action, namely, the local one (weak minimum) and the global one (strong minimum), can be generalized to include the quantum-mechanical paths. Thus, we can prove that locally, a solution of the classical equation of motion is really the minimum, even in a large class of quantum paths containing the semi-classical trajectories. By introducing a stochastic version of the excess function of Weierstrass, we show the analogous global property. There, of course, one can speak of the principle of least action in a strict sense. Several explicit examples are discussed.