Gauge Theory and a Dirac Operator on a Noncommutative Space

TL;DR

This paper introduces a Dirac operator for noncommutative spaces to facilitate gauge theory quantization and explores its potential to address the cosmological constant problem.

Contribution

It presents a novel Dirac operator acting as a line element in noncommutative geometry and constructs the Dixmier trace for regularized infinite-dimensional matrices.

Findings

Constructed a Dirac operator for noncommutative spaces

Developed the Dixmier trace for regularization

Proposed a new approach to the cosmological constant problem

Abstract

As a tool to carry out the quantization of gauge theory on a noncommutative space, we present a Dirac operator that behaves as a line element of the canonical noncommutative space. Utilizing this operator, we construct the Dixmier trace, which is the regularized trace for infinite-dimensional matrices. We propose the possibility of solving the cosmological constant problem by applying our gauge theory on the noncommutative space.