Nonlinear Gauge Transformation for a Quantum System Obeying an Exclusion-Inclusion Principle

Abstract

We introduce a nonlinear and noncanonical gauge transformation which allows the re- duction of a complex nonlinearity, contained in a Schrodinger equation, into a real one. This Schrodinger equation describes a canonical system, whose kinetics is governed by a generalized Exclusion-Inclusion Principle. The transformation can be easily general- ized and used in order to reduce complex nonlinearities into real ones for a wide class of nonlinear Schrodinger equations. We show also that, for one dimensional system and in the case of solitary waves, the above transformation coincides with the one already adopted to study the Doebner-Goldin equation. Let us consider the kinetics of N particles in a one-dimensional discrete space, which is an one-dimensional Markovian chain. The generic site is labeled by the index i (i =0 , ±1, ±2 ,... ); the position at the ith site is xi = i∆x, where ∆x is a constant. We call ρi(t) the occupational probability of the ith site. Let us assume that only transitions to the nearest neighbors are allowed and define the transition probability π ± i (t) from the site i to the site i ± 1 in the following way: π ± i (t )= α ± (t)