Moyal Planes are Spectral Triples

TL;DR

This paper extends the axioms of spectral triples to nonunital cases, focusing on Moyal planes, and explores their applications in noncommutative quantum field theory, including Wick monomials and the Connes--Lott action.

Contribution

It introduces new axioms for nonunital spectral triples and applies them to Moyal planes, advancing the mathematical framework for noncommutative geometry in quantum physics.

Findings

Extended spectral triple axioms to nonunital cases.

Analyzed Moyal planes as spectral triples.

Applied to noncommutative Wick monomials and Connes--Lott action.

Abstract

Axioms for nonunital spectral triples, extending those introduced in the unital case by Connes, are proposed. As a guide, and for the sake of their importance in noncommutative quantum field theory, the spaces $\R^{2N}$ endowed with Moyal products are intensively investigated. Some physical applications, such as the construction of noncommutative Wick monomials and the computation of the Connes--Lott functional action, are given for these noncommutative hyperplanes.