
SIGMA 5 (2009), 023, 20 pages arXiv:0707.1553
http://dx.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 nonpolynomial 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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