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


Inglis, S and Jarvis, P, Fierz bilinear formulation of the Maxwell-Dirac equations and symmetry reductions, Annals of Physics, 348 pp. 176-222. ISSN 0003-4916 (2014) [Refereed Article]

Copyright Statement

Copyright 2014 Elsevier

DOI: doi:10.1016/j.aop.2014.05.017


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.

Item Details

Item Type:Refereed Article
Keywords:quantum mechanics, special relativity, mathematical physics, Maxwell-Dirac equations, Fierz identities, relativistic quantum mechanics, symmetry reduction, magnetic monopole
Research Division:Mathematical Sciences
Research Group:Mathematical physics
Research Field:Mathematical aspects of quantum and conformal field theory, quantum gravity and string theory
Objective Division:Expanding Knowledge
Objective Group:Expanding knowledge
Objective Field:Expanding knowledge in the physical sciences
UTAS Author:Inglis, S (Mr Shaun Inglis)
UTAS Author:Jarvis, P (Dr Peter Jarvis)
ID Code:92675
Year Published:2014
Web of Science® Times Cited:2
Deposited By:Mathematics and Physics
Deposited On:2014-06-25
Last Modified:2017-11-01

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