Canonical quantization of nonlinear many body systems

We study the quantization of a classical system of interacting particles obeying a recently proposed kinetic interaction principle (KIP) [G. Kaniadakis, Physica A {\bf 296}, 405 (2001)]. The KIP fixes the expression of the Fokker-Planck equation describing the kinetic evolution of the system and imposes the form of its entropy. In the framework of canonical quantization, we introduce a class of nonlinear Schr\"odinger equations (NSEs) with complex nonlinearities, describing, in the mean field approximation, a system of collectively interacting particles whose underlying kinetics is governed by the KIP. We derive the Ehrenfest relations and discuss the main constants of motion arising in this model. By means of a nonlinear gauge transformation of third kind it is shown that in the case of constant diffusion and linear drift the class of NSEs obeying the KIP is gauge-equivalent to another class of NSEs containing purely real nonlinearities depending only on the field $\rho=|\psi|^2$.