Nonlinear Dirac Equations

Abstract

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.

Information and Particle Physics

“…The positivity constraint on the information measure, to be discussed below, turns out to be very restrictive. We emphasize that unlike [17], here we do not start by demanding nonlinearity, but rather find it as one of the unavoidable consequences of an information-theoretic generalisation.…”

“…In Ref. [17] nonlinear extensions of Dirac's equation were constructed axiomatically by requiring the extension to preserve various desirable properties of the original linear equation. Such equations, which generalise previous versions [18,19,20,21,22], have since been used to study potential corrections to the standard neutrino oscillation probabilities [23] and may be relevant also in the condensed matter context [24].…”

“…However, one drawback of the approach in Ref. [17] is that the constraints were not sufficiently restrictive, resulting in large classes of nonlinearities. This is because Ref.…”

“…This is because Ref. [17] did not use information-theoretic arguments as in Ref. [9]: While in the non-relativistic case one may start from a well-motivated classical system and obtain the linear or nonlinear information-theoretic extensions, in the relativistic case the corresponding classical starting point is unknown or ambiguous.…”

“…In this paper we adopt a strategy different from Ref. [17] to obtain generalisations of linear quantum evolution equations for spinors. As mentioned at the end of Ref.…”