University of Tasmania
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The manifestly gauge invariant Maxwell-Dirac equations

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posted on 2023-05-27, 10:29 authored by Shaun Inglis
We study the Maxwell-Dirac equations, which model the fermionic relativistic electrodynamics in the case where the fermion field is itself the source of the electromagnetic field. This system is formulated by exploiting that fact that the Dirac equation can be algebraically inverted, and the resulting expression for the vector potential in terms of the spinor fields can be directly substituted into Maxwell's equations. We work in a formalism where the physical states are described by a set of tensor fields, formed from bilinear combinations of (non-Grassmann) spinor fields and Dirac matrices. This results in a set of manifestly gauge invariant equations that lack such unphysical degrees of freedom. Through the use of Fierz expansions on quadratic spinor combinations, and their associated identities, a large set of interrelationships between bilinear fields can be obtained. This permits the description of the Maxwell-Dirac system in terms of tensor current densities, and their quadratic Fierz identities and continuity constraints. The resulting set of self-coupled Maxwell-Dirac equations is mathematically intractable without further constraint. We show how demanding invariance of the bilinear tensor fields under the action of arbitrary subgroups of the Poincarè group of rotations, translations and boosts reduces the equations to the point where they are more manageable. In this thesis, we demonstrate in detail how the Maxwell-Dirac equations reduce under several example subgroups. We also develop the gauge invariant bilinear formalism for the stress-energy tensor, which can be used to calculate physical quantities such as the momentum and mass-energy corresponding to a Maxwell-Dirac solution. The calculation is approached from two independent points of view, namely the Belinfante method and the variational method from general relativity, which we find to be in agreement. Finally, by analogy with the method in electromagnetism, we extend the algebraic inversion of the Dirac equation to the case where the spinors are isospin doublets, and the gauge field corresponds to the non-Abelian group SU(2). Following the definition of non-Abelian bilinears and Fierz identities, the inverted form itself is given formally, with the application of a Neumann series required for an explicit expression.


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Copyright 2015 the author Chapter 5 appears to be, in part, the equivalent of a post-print version of an article published as: Inglis, S. M., Jarvis, P. D. 2016. Maxwell‚Äö-Dirac stress‚Äö-energy tensor in terms of Fierz bilinear currents, Annals of physics, 366, 57-75 Chapter 7 appears to be, in part, the equivalent of a post-print version of an article published as: Inglis, S. M., Jarvis, P. D. 2012. Algebraic inversion of the Dirac equation for the vector potential in the non-Abelian case, Journal of physics A: Mathematical and theoretical, 45(46), 1-19

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