A generalized nonlinear Schrödinger equation: Classical field-theoretic approach

¹ Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro - RJ, Brazil
² Santa Fe Institute - 1399 Hyde Park Road, Santa Fe, NM 87501, USA

^a fdnobre@cbpf.br

Received: 3 October 2011
Accepted: 9 January 2012

Abstract

An exact classical field theory, for a recently proposed nonlinear generalization of the Schrödinger equation, is presented. In this generalization, a nonlinearity depending on an index q appears in the kinetic term, such that the free-particle linear Schrödinger equation is recovered in the limit q→1. It is shown that besides the usual , a new field must be introduced, which becomes only when q→1. In analogy to the linear case, is interpreted as the probability density for finding the particle at time t, in a given position inside an arbitrary finite volume V, for any q. The possible physical consequences are discussed, and, in particular, the important property that the fields and do not interact with light.