Noncommutative physics on Lie algebras, Z_2^n lattices and Clifford algebras

TL;DR

This paper surveys noncommutative geometries derived from Lie algebras, explores differential structures on these spaces, and provides exact quantization of noncommutative gauge theories on finite lattices.

Contribution

It introduces methods to perform differential geometry on noncommutative spaces from Lie algebra deformations and computes their cohomology and gauge theory solutions.

Findings

Computed noncommutative de Rham cohomology for Clifford algebras.

Quantized noncommutative U(1)-Yang-Mills theory on Z2×Z2.

Analyzed moduli of solutions to Maxwell's equations in noncommutative settings.

Abstract

We survey noncommutative spacetimes with coordinates being enveloping algebras of Lie algebras. We also explain how to do differential geometry on noncommutative spaces that are obtained from commutative ones via a Moyal-product type cocycle twist, such as the noncommutative torus, $\theta$-spaces and Clifford algebras. The latter are noncommutative deformations of the finite lattice $(\Z_2)^n$ and we compute their noncommutative de Rham cohomology and moduli of solutions of Maxwell's equations. We exactly quantize noncommutative U(1)-Yang-Mills theory on $\Z_2\times\Z_2$ in a path integral approach.