A generalized nonlinear Schrödinger equation: Classical field-theoretic approach

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 $\Psi(\vec{x},t) $, a new field $\Phi(\vec{x},t) $ must be introduced, which becomes $\Psi^{\ast}(\vec{x},t) $ only when q→1. In analogy to the linear case, $\rho(\vec{x},t)={1\over 2V}[\Psi(\vec{x},t) \Phi(\vec{x},t) +\Psi^{\ast}(\vec{x},t) \Phi^{\ast}(\vec{x},t)] $ is interpreted as the probability density for finding the particle at time t, in a given position $\vec{x} $ inside an arbitrary finite volume V, for any q. The possible physical consequences are discussed, and, in particular, the important property that the fields $\Psi(\vec{x},t) $ and $\Phi(\vec{x},t) $ do not interact with light