Quasi-Local Formulation of Non-Abelian Finite-Element Gauge Theory

TL;DR

This paper develops a quasi-local formulation for non-Abelian gauge theories on a lattice, making the Dirac and Yang-Mills equations local but resulting in a nonunitary scheme, with applications to current anomalies.

Contribution

It introduces a local scheme for non-Abelian gauge theories that builds on Matsuyama's approach, extending finite-element methods to a quasi-local framework.

Findings

Successfully implements a local Dirac equation using link operators.

Calculates current anomalies in 2D and 4D space-time.

The scheme is generally nonunitary, highlighting limitations of the approach.

Abstract

Recently it was shown how to formulate the finite-element equations of motion of a non-Abelian gauge theory, by gauging the free lattice difference equations, and simultaneously determining the form of the gauge transformations. In particular, the gauge-covariant field strength was explicitly constructed, locally, in terms of a path ordered product of exponentials (link operators). On the other hand, the Dirac and Yang-Mills equations were nonlocal, involving sums over the entire prior lattice. Earlier, Matsuyama had proposed a local Dirac equation constructed from just the above-mentioned link operators. Here, we show how his scheme, which is closely related to our earlier one, can be implemented for a non-Abelian gauge theory. Although both Dirac and Yang-Mills equations are now local, the field strength is not. The technique is illustrated with a direct calculation of the current anomalies in two and four space-time dimensions. Unfortunately, unlike the original finite-element proposal, this scheme is in general nonunitary.