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Fierz bilinear formulation of the Maxwell-Dirac equations and symmetry reductions

journal contribution
posted on 2023-05-18, 01:41 authored by Shaun Inglis, Peter JarvisPeter Jarvis
We study the Maxwell–Dirac equations in a manifestly gauge invariant presentation using only the spinor bilinear scalar and pseudoscalar densities, and the vector and pseudovector currents, together with their quadratic Fierz relations. The internally produced vector potential is expressed via algebraic manipulation of the Dirac equation, as a rational function of the Fierz bilinears and first derivatives (valid on the support of the scalar density), which allows a gauge invariant vector potential to be defined. This leads to a Fierz bilinear formulation of the Maxwell tensor and of the Maxwell–Dirac equations, without any reference to gauge dependent quantities. We show how demanding invariance of tensor fields under the action of a fixed (but arbitrary) Lie subgroup of the Poincaré group leads to symmetry reduced equations. The procedure is illustrated, and the reduced equations worked out explicitly for standard spherical and cylindrical cases, which are coupled third order nonlinear PDEs. Spherical symmetry necessitates the existence of magnetic monopoles, which do not affect the coupled Maxwell–Dirac system due to magnetic terms cancelling. In this paper we do not take up numerical computations.

History

Publication title

Annals of Physics

Volume

348

Pagination

176-222

ISSN

0003-4916

Department/School

School of Natural Sciences

Publisher

Academic Press Inc Elsevier Science

Place of publication

525 B St, Ste 1900, San Diego, USA, Ca, 92101-4495

Rights statement

Copyright 2014 Elsevier

Repository Status

  • Restricted

Socio-economic Objectives

Expanding knowledge in the physical sciences