Nonlinear quantum equations: Classical field theory
- 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)
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 Ψ(x(vector sign),t), a new field Φ(x(vector sign),t) needs to be introduced in the Lagrangian, as well. The field Φ(x(vector sign),t), which is defined by means of an additional equation, becomes Ψ{sup *}(x(vector sign),t) only when q→ 1. The solutions for the fields Ψ(x(vector sign),t) and Φ(x(vector sign),t) are found herein, being expressed in terms of a q-plane wave; moreover, both field equations lead to the relation E{sup 2}=p{sup 2}c{sup 2}+m{sup 2}c{sup 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.
- OSTI ID:
- 22217889
- Journal Information:
- Journal of Mathematical Physics, Vol. 54, Issue 10; Other Information: (c) 2013 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
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