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 Schrödinger 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 \documentclass[12pt]{minimal}\begin{document}$\Psi (\vec{x},t)$\end{document} Ψ ( x ⃗ , t ) , a new field \documentclass[12pt]{minimal}\begin{document}$\Phi (\vec{x},t)$\end{document} Φ ( x ⃗ , t ) needs to be introduced in the Lagrangian, as well. The field \documentclass[12pt]{minimal}\begin{document}$\Phi (\vec{x},t)$\end{document} Φ ( x ⃗ , t ) , which is defined by means of an additional equation, becomes \documentclass[12pt]{minimal}\begin{document}$\Psi ^{*}(\vec{x},t)$\end{document} Ψ * ( x ⃗ , t ) only when q → 1. The solutions for the fields \documentclass[12pt]{minimal}\begin{document}$\Psi (\vec{x},t)$\end{document} Ψ ( x ⃗ , t ) and \documentclass[12pt]{minimal}\begin{document}$\Phi (\vec{x},t)$\end{document} Φ ( x ⃗ , t ) are found herein, being expressed in terms of a q-plane wave; moreover, both field equations lead to the relation E 2 = p 2 c 2 + m 2 c 4, for all values of q. The fact that such a classical field theory works well for two very distinct nonlinear quantum equations, namely, the Schrödinger 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.