Stochastic quantization for a system of N identical Bose particles

We apply Stochastic Quantization to a system of N interacting identical Bosons in an external potential Phi, by means of a general stationary-action principle. The collective motion is described in terms of a Markovian diffusion on R^(3N), with joint density Rho and entangled current velocity field V, in principle of non-gradient form, related one to the other by the continuity equation. Dynamical equations relax to those of canonical quantization, in some analogy with Parisi-Wu stochastic quantization. Thanks to the identity of particles, the one-particle marginal densities rho, in the physical space R^3, are all the same and it is possible to give, under mild conditions, a natural definition of the single-particle current velocity, which is related to rho by the continuity equation in R^3. The motion of single particles in the physical space comes to be described in terms of a non-Markovian three-dimensional diffusion with common density rho and, at least at dynamical equilibrium, common current velocity v. The three-dimensional drift is perturbed by zero-mean terms depending on the whole configuration of the N-bosons interacting system. Finally we discuss in detail under which conditions the one-particle dynamical equations, which in their general form allow rotational perturbations, can be particularized, up to a change of variables, to Gross-Pitaevskii equations.