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Algebraic inversion of the Dirac equation for the vector potential in the non-Abelian case


Inglis, SM and Jarvis, PD, Algebraic inversion of the Dirac equation for the vector potential in the non-Abelian case, Journal of Physics A: Mathematical and Theoretical, 45, (46) Article 465202. ISSN 1751-8113 (2012) [Refereed Article]

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Copyright 2012 IOP Publishing Ltd

DOI: doi:10.1088/1751-8113/45/46/465202


We study the Dirac equation for spinor wavefunctions minimally coupled to an external field, from the perspective of an algebraic system of linear equations for the vector potential. By analogy with the method in electromagnetism, which has been well-studied, and leads to classical solutions of the Maxwell–Dirac equations, we set up the formalism for non-Abelian gauge symmetry, with the SU(2) group and the case of four-spinor doublets. An extended isospincharge conjugation operator is defined, enabling the hermiticity constraint on the gauge potential to be imposed in a covariant fashion, and rendering the algebraic system tractable. The outcome is an invertible linear equation for the non-Abelian vector potential in terms of bispinor current densities. We show that, via application of suitable extended Fierz identities, the solution of this system for the non-Abelian vector potential is a rational expression involving only Pauli scalar and Pauli triplet, Lorentz scalar, vector and axial vector current densities, albeit in the non-closed form of a Neumann series.

Item Details

Item Type:Refereed Article
Research Division:Mathematical Sciences
Research Group:Mathematical physics
Research Field:Algebraic structures in mathematical physics
Objective Division:Expanding Knowledge
Objective Group:Expanding knowledge
Objective Field:Expanding knowledge in the physical sciences
UTAS Author:Inglis, SM (Mr Shaun Inglis)
UTAS Author:Jarvis, PD (Dr Peter Jarvis)
ID Code:80438
Year Published:2012
Web of Science® Times Cited:5
Deposited By:Mathematics and Physics
Deposited On:2012-10-31
Last Modified:2013-07-02

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