Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 023, 20 pages      arXiv:0707.1553      https://doi.org/10.3842/SIGMA.2009.023

Nonlinear Dirac Equations

Wei Khim Ng and Rajesh R. Parwani
Department of Physics, National University of Singapore, Kent Ridge, Singapore

Received November 12, 2008, in final form February 23, 2009; Published online February 27, 2009

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 Poincaré 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.

Key words: nonlinear Dirac equation; Lorentz violation.

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