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

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.

21pp, uses iopart

References in corpus (2)

• Non-Abelian Fluid Dynamics in Lagrangian Formulation
• Algebraic solution for the vector potential in the Dirac equation

Cited by in corpus (5)

• Fierz bilinear formulation of the Maxwell-Dirac equations and symmetry reductions
• Unveiling a spinor field classification with non-Abelian gauge symmetries
• Maxwell-Dirac stress-energy tensor in terms of Fierz bilinear currents
• The self-coupled Einstein-Cartan-Dirac equations in terms of Dirac bilinears
• Gauge invariant formulation of the self-interacting Duffin-Kemmer-Petiau equations