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Fierz bilinear formulation of the Maxwell-Dirac equations and symmetry reductions
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 PhysicsVolume
348Pagination
176-222ISSN
0003-4916Department/School
School of Natural SciencesPublisher
Academic Press Inc Elsevier SciencePlace of publication
525 B St, Ste 1900, San Diego, USA, Ca, 92101-4495Rights statement
Copyright 2014 ElsevierRepository Status
- Restricted