Wilson Action of Lattice Gauge Fields with An Additional Term from Noncommutative Geometry

TL;DR

This paper develops a noncommutative geometric framework for lattice gauge fields, deriving a generalized action that extends the Wilson action with an additional term linked to non-unitarity of link variables.

Contribution

It introduces a noncommutative geometric approach to lattice gauge theories, resulting in a generalized action including a new term beyond the traditional Wilson action.

Findings

Reproduces the Wilson action as a special case

Identifies an additional term related to non-unitarity of link variables

Provides a geometric interpretation of lattice gauge actions

Abstract

Differential structure of lattices can be defined if the lattices are treated as models of noncommutative geometry. The detailed construction consists of specifying a generalized Dirac operator and a wedge product. Gauge potential and field strength tensor can be defined based on this differential structure. When an inner product is specified for differential forms, classical action can be deduced for lattice gauge fields. Besides the familiar Wilson action being recovered, an additional term, related to the non-unitarity of link variables and loops spanning no area, emerges.