Covariant Quantum Fields on Noncommutative Spacetimes

TL;DR

This paper explores covariant quantum fields on noncommutative spacetimes, extending the concept of Poincaré covariance to quantum fields in noncommutative geometry, and discusses implications for different algebraic structures and groups.

Contribution

It formulates covariant quantum fields on noncommutative spacetimes, extending the Drinfel'd twist framework to nonabelian groups and analyzing the resulting nonassociative spacetimes.

Findings

Covariance is preserved for Moyal spacetimes but conflicts with the *-operation in Voros algebra.

Extending Drinfel'd twists to nonabelian groups leads to nonassociative spacetime structures.

Covariance properties are self-reproducing and applicable to products of covariant fields.

Abstract

A spinless covariant field $\phi$ on Minkowski spacetime $\M^{d+1}$ obeys the relation $U(a,\Lambda)\phi(x)U(a,\Lambda)^{-1}=\phi(\Lambda x+a)$ where $(a,\Lambda)$ is an element of the Poincar\'e group $\Pg$ and $U:(a,\Lambda)\to U(a,\Lambda)$ is its unitary representation on quantum vector states. It expresses the fact that Poincar\'e transformations are being unitary implemented. It has a classical analogy where field covariance shows that Poincar\'e transformations are canonically implemented. Covariance is self-reproducing: products of covariant fields are covariant. We recall these properties and use them to formulate the notion of covariant quantum fields on noncommutative spacetimes. In this way all our earlier results on dressing, statistics, etc. for Moyal spacetimes are derived transparently. For the Voros algebra, covariance and the *-operation are in conflict so that there are no covariant Voros fields compatible with *, a result we found earlier. The notion of Drinfel'd twist underlying much of the preceding discussion is extended to discrete abelian and nonabelian groups such as the mapping class groups of topological geons. For twists involving nonabelian groups the emergent spacetimes are nonassociative.