Nonlinear Schroedinger Equations within the Nelson Quantization Picture

TL;DR

This paper introduces a new class of nonlinear Schrödinger equations derived from a generalized classical kinetics approach, which can be linearized in stationary states, expanding the Nelson quantization framework for interacting particle systems.

Contribution

It generalizes previous Nelson quantization methods to a broader class of nonlinear Schrödinger equations based on nonlinear classical kinetics.

Findings

Derived a new class of NLSEs from classical kinetics.

Proposed a transformation to linearize NLSEs in stationary states.

Extended Nelson quantization to interacting particle systems.

Abstract

We present a class of nonlinear Schroedinger equations (NLSEs) describing, in the mean field approximation, systems of interacting particles. This class of NLSEs is obtained generalizing expediently the approach proposed in Ref. [G.K. Phys. Rev. A 55, 941 (1997)], where a classical system obeying to an exclusion-inclusion principle is quantized using the Nelson stochastic quantization. The new class of NLSEs is obtained starting from the most general nonlinear classical kinetics compatible with a constant diffusion coefficient D=\hbar/2m. Finally, in the case of s-stationary states, we propose a transformation which linearizes the NLSEs here proposed.