Nonlinear dirac equations

Abstract

In this thesis, we describe two approaches to generalise the linear Dirac equation. The first is an axiomatic approach, where we demand the nonlinear generalisations to satisfy several physical properties possessed by the linear equation. The second is an information-theoretic approach, where we use some information-theoretic arguments to generalise the linear Dirac equation. Unlike the formal case, in the latter approach, nonlinearity is not demanded at the onset but is a consequence together with Lorentz violation.

As for applications, we first apply these generalised equations to study neutrino oscillations where we find that the results are sensitive to the individual neutrino masses, unlike the case of the conventional theory. Secondly, we derive the non-relativistic limit of these generalisations and discover that the relativistic corrections regularise some potential singularities. We also find that these corrections are enhanced at the nodes, if any, of the wavefunctions which allow us to test quantum nonlinearity in future experiments that are sensitive to the nodes in the wavefunctions. Thirdly, we discuss the discrete symmetries possessed by these generalisations which may suggest possible new source of CP violation or origin of matter-antimatter asymmetry.

In summary, we are able to generate a class of nonlinear Dirac equations that is more general than previous constructs in the literatures. We provide an argument that quantum linearity may have a close relation to space-time symmetry. Lastly, the various applications we have discussed serve as tests for quantum nonlinearity in future experiments.