Groenewold-Moyal Product, \alpha^\star-Cohomology, and Classification of Translation-Invariant Non-Commutative Structures

TL;DR

This paper thoroughly studies alpha_star-cohomology, establishing a unique harmonic form in each class that classifies translation-invariant non-commutative structures and shows their physical equivalence to Groenewold-Moyal quantum field theories.

Contribution

It provides an algebraic classification of translation-invariant non-commutative structures using alpha_star-cohomology and proves their equivalence to Groenewold-Moyal theories.

Findings

Existence of a unique harmonic 2-cocycle in each cohomology class.

Classification of translation-invariant non-commutative structures.

Equivalence of general translation-invariant non-commutative QFTs to Groenewold-Moyal QFTs.

Abstract

The theory of alpha_star-cohomology is studied thoroughly and it is shown that in each cohomology class there exists a unique 2-cocycle, the harmonic form, which generates a particular Groenewold-Moyal star product. This leads to an algebraic classification of translation-invariant non-commutative structures and shows that any general translationinvariant non-commutative quantum field theory is physically equivalent to a Groenewold- Moyal non-commutative quantum field theory.