Gauge Theory and Dirac Operator on Noncommutative Space II -Minkowskian and Euclidean Cases-

TL;DR

This paper extends the construction of Dirac operators and integrals from noncommutative tori to Minkowskian and Euclidean spaces, proposing a geometric framework for gauge theory in noncommutative geometry.

Contribution

It introduces a method to generalize Dirac operators and integrals to Minkowskian and Euclidean noncommutative spaces, advancing the geometric understanding of gauge theories.

Findings

Extended Dirac operator construction to Minkowskian and Euclidean cases

Proposed a geometric notion of gauge theory in noncommutative spaces

Established a framework for noncommutative geometry applications in physics

Abstract

In the preceding paper [arXiv:hep-th/0604217], we construct the Dirac operator and the integral on the canonical noncommutative space. As a matter of fact, they are ones on the noncommutative torus. In the present article, we introduce the method to extend to the Minkowskian and Euclidean cases. As a concluding remark, we present a geometrical notion of our gauge theory.