Gauge equivalence among quantum nonlinear many body systems

TL;DR

This paper introduces a nonlinear gauge transformation method to classify and simplify U(1)-invariant nonlinear Schrödinger equations, transforming complex nonlinearities into real ones and analyzing coupled systems with gauge fields.

Contribution

The paper develops a nonlinear gauge transformation technique to convert complex nonlinear Schrödinger equations into real form, extending it to coupled systems and gauge field interactions.

Findings

Transformations simplify nonlinear Schrödinger equations to real form.

Method applies to coupled systems with Hermitian and anti-Hermitian matrices.

Applicable to gauge field coupled equations, preserving final form.

Abstract

Transformations performing on the dependent and/or the independent variables are an useful method used to classify PDE in class of equivalence. In this paper we consider a large class of U(1)-invariant nonlinear Schr\"odinger equations containing complex nonlinearities. The U(1) symmetry implies the existence of a continuity equation for the particle density $\rho\equiv|\psi|^2$ where the current ${\bfm j}_{_\psi}$ has, in general, a nonlinear structure. We introduce a nonlinear gauge transformation on the dependent variables $\rho$ and ${\bfm j}_{\psi}$ which changes the evolution equation in another one containing only a real nonlinearity and transforms the particle current ${\bfm j}_{_\psi}$ in the standard bilinear form. We extend the method to U(1)-invariant coupled nonlinear Schr\"odinger equations where the most general nonlinearity is taken into account through the sum of an Hermitian matrix and an anti-Hermitian matrix. By means of the nonlinear gauge transformation we change the nonlinear system in another one containing only a purely Hermitian nonlinearity. Finally, we consider nonlinear Schr\"odinger equations minimally coupled with an Abelian gauge field whose dynamics is governed, in the most general fashion, through the Maxwell-Chern-Simons equation. It is shown that the nonlinear transformation we are introducing can be applied, in this case, separately to the gauge field or to the matter field with the same final result. In conclusion, some relevant examples are presented to show the applicability of the method.