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

TL;DR

This paper develops an algebraic method to invert the non-abelian Dirac equation, expressing the gauge potential in terms of spinor currents, extending techniques from electromagnetism to non-abelian gauge theories.

Contribution

It introduces a formalism for algebraic inversion of the non-abelian Dirac equation, including an extended isospin-charge conjugation operator and a solution expressed via Fierz identities.

Findings

Derived an invertible linear equation for the non-abelian gauge potential.

Expressed the gauge potential as a rational function of current densities.

Extended the algebraic inversion method from electromagnetism to non-abelian gauge symmetry.

Abstract

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 isospin-charge 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.