  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
$\rho=|\psi|^2$.
