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

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. 1 typeset using PTPTEX.cls 〈Ver.0.9〉 As for the canonical noncommutative space, [x̂, x̂ ] = iθ , μ = 0, 1, · · · , d− 1, η = diag(±1, 1, · · · , 1), (1) since the generators x̂ are represented in terms of the infinite-dimensional matrices, the ordinary trace, the most familiar map from matrices to c-number, as the volume integration is evidently divergent. Therefore we need to introduce the regularized trace which is known as the Dixmier trace. 2) In Ref. 5), we construct a volume integration written as the trace of matrix and a Dirac operator D = 1 2θ ∑d−1 μ=0(x̂ ) which regulates it and plays a role as volume element. Furthermore we show that the gauge fields Aμ(x̂), the functions on the noncommutative space, are expanded in plane waves like Aμ(x̂) = ∑ k∈Z Aμ,ke ikμx̂μ, (2) and they obey the ortho-normality condition: