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Algebraic inversion of the Dirac equation for the vector potential in the non-Abelian case
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.
History
Publication title
Journal of Physics A: Mathematical and TheoreticalVolume
45Issue
46Article number
465202Number
465202Pagination
1-19ISSN
1751-8113Department/School
School of Natural SciencesPublisher
Institute of Physics Publishing, Inc.Place of publication
BristolRights statement
Copyright 2012 IOP Publishing LtdRepository Status
- Restricted