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

We introduce a nonlinear and noncanonical gauge transformation which allows the reduction of a complex nonlinearity, contained in a Schrödinger equation, into a real one. This Schrödinger equation describes a canonical system, whose kinetics is governed by a generalized Exclusion-Inclusion Principle. The transformation can be easily generalized and used in order to reduce complex nonlinearities into real ones for a wide class of nonlinear Schrödinger 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± i (t) ∆x2 ρi(t)[1+κρi±1(t)]. (1) The factor 1 + κρi±1(t) means that the transition probability depends on the particle population ρi±1(t) of the arrival site. If κ > 0 the π± i (t) introduces an inclusion effect. In fact, the population at the arrival site i± 1 stimulates the transition and π± i (t) increases linearly with ρi±1(t). In the case κ < 0 the π± i (t) takes into account an exclusion effect. If the arrival site is empty ρi(t) = 0, the π± i (t) depends only on the population of the starting point. If the arrival site is populated 0 < ρi(t) ≤ ρmax the transition is inhibited. The Pauli master equation can be written as follows: dρi(t) dt = π i−1(t)+π − i+1(t)−π i (t)−π− i (t). (2) Copyright c © 2001 by G Kaniadakis, A Lavagno, P Quarati and A M Scarfone 162 G Kaniadakis, A Lavagno, P Quarati and A M Scarfone If we define: ji(t) = [ π i (t)− π− i+1(t) ] ∆x, (3) which represents a discrete current, the master equation can be written as: dρi(t) dt + ∆ji(t) ∆x = 0, (4) where ∆ji(t) = ji+1(t)− ji(t). (5) Equation (4) represents a forward continuity equation in a discrete one-dimensional space. Analogously we can obtain a backward continuity equation. The half of the sum of these equations, in the limit ∆x → 0 gives the following continuity equation: ∂ρ(t, x) ∂t + ∂j(t) ∂x = 0, (6) where j(t, x) assumes the form: j(t, x) = u(t, x)[1+κρ(t, x)]ρ(t, x). (7) We note that the current velocity u(t, x) depends on the nature of the particle interaction while the factor 1+κρ(t, x) takes into account the generalized Exclusion-Inclusion Principle (EIP) [1]. In Ref. [2, 3, 4] was recently considered by us the nonlinear canonical model defined by the Lagrangian density: L = i 2 ( ψ∗ ∂ψ ∂t − ψ ∗ ∂t ) − 2 2m ∣∣∇ψ ∣∣2−UEIP [ψ,ψ∗]−V ψ∗ψ, (8)