Nonlinear Dirac Equations

TL;DR

This paper develops nonlinear extensions of the Dirac equation that maintain key physical properties, classifies possible nonlinearities, and explores their symmetries and potential physical applications.

Contribution

It introduces a systematic construction and classification of nonlinear Dirac equations preserving essential symmetries and properties, including explicit examples and symmetry analyses.

Findings

Nonlinear Dirac equations can preserve locality, separability, probability conservation, and Poincaré invariance.

Explicit examples of nonlinear equations with various symmetry properties are provided.

The constructed equations are not gauge equivalent to the linear Dirac equation.

Abstract

We construct nonlinear extensions of Dirac's relativistic electron equation that preserve its other desirable properties such as locality, separability, conservation of probability and Poincar\'e invariance. We determine the constraints that the nonlinear term must obey and classify the resultant non-polynomial nonlinearities in a double expansion in the degree of nonlinearity and number of derivatives. We give explicit examples of such nonlinear equations, studying their discrete symmetries and other properties. Motivated by some previously suggested applications we then consider nonlinear terms that simultaneously violate Lorentz covariance and again study various explicit examples. We contrast our equations and construction procedure with others in the literature and also show that our equations are not gauge equivalent to the linear Dirac equation. Finally we outline various physical applications for these equations.