Stochastic quantization of finite dimensional systems with electromagnetic interactions

We study the stochastic quantization of finite dimensional systems via path-wise calculus of variations with the mean discretized classical action in the general case of electromagnetic interactions. We show that there exists a unique choice of the mean discretized action corresponding to the minimal classical magnetic coupling and derive the general equations of motion by means of a path-wise stochastic calculus of variations. In the case of purely scalar interactions, the total mean energy of the system (which gives the usual quantum mechanical expectation of the Hamiltonian in the canonical limit) works as a Lyapunov functional and the system relaxes on the canonical solutions, represented by Nelson's diffusions, which act as an attracting set. We show that, in presence of a minimal magnetic coupling, the mean energy is no longer a Lyapunov functional. We construct for a simple example, a new Lyapunov functional, and we show that the system can reach the dynamical equilibrium also by absorbing energy from the external magnetic field.