Nonlinear quantum equations: Classical field theory

An exact classical field theory for nonlinear quantum equations is presented herein. It has been applied recently to a nonlinear Schr¨ odinger equation, and it is shown herein to hold also for a nonlinear generalization of the Klein-Gordon equation. These generalizations were carried by introducing nonlinear terms, characterized by exponents depending on an index q, in such a way that the standard, linear equations, are recovered in the limit q → 1. The main characteristic of this field theory consists on the fact that besides the usual (x,t), a new field (⃗x,t) needs to be introduced in the Lagrangian, as well. The field (⃗x,t), which is defined by means of an additional equation, becomes∗(⃗x,t) only when q → 1. The solutions for the fields (x,t) and (⃗x,t) are found herein, being expressed in terms of a q-plane wave; moreover, both field equations lead to the relation E2 = p2c2 + m2c4, for all values of q. The fact that such a classical field theory works well for two very distinct nonlinear quantum equations, namely, the Schr¨ odinger and Klein-Gordon ones, suggests that this procedure should be appropriate for a wider class nonlinear equations. It is shown that the standard global gauge invariance is broken as a consequence of the nonlinearity.

2013