Non-Abelian Finite-Element Gauge Theory

TL;DR

This paper develops a complete finite-element lattice formulation for non-Abelian gauge fields coupled to fermions, ensuring exact gauge invariance through an iterative approach that constructs equations of motion and field strength.

Contribution

It introduces an iterative method to formulate gauge-invariant equations of motion for non-Abelian gauge theories on a finite-element lattice in four dimensions.

Findings

Exact lattice gauge invariance achieved

Interaction terms are nonlocal but systematically constructed

Field strength expressed via path-ordered exponentials

Abstract

We complete the formulation of the equations of motion of a non-Abelian gauge field coupled to fermions on a finite-element lattice in four space-time dimensions. This is accomplished by a straightforward iterative approach, in which successive interaction terms are added to the Dirac and Yang-Mills equations of motion, and to the field strength, in order to preserve lattice gauge invariance exactly, yielding a series in powers of $ghA$. Here $g$ is the coupling constant, $h$ is the lattice spacing, and $A$ is the gauge potential. Gauge transformations of the potentials are determined simultaneously. The interaction terms in the equations of motion are nonlocal, and can be expressed either by an iterative formula or by a difference equation. On the other hand, the field strength is locally constructed from the potentials in terms of a path-ordered product of exponentials.