Dirac Operators on Noncommutative Curved Spacetimes

TL;DR

This paper explores the construction of Dirac operators in twist-deformed noncommutative curved spacetimes, proposing axioms, analyzing multiple candidates, and developing a quantum field theory framework for noncommutative Dirac fields.

Contribution

It introduces a set of axioms for noncommutative Dirac operators, compares multiple constructions, and applies the framework to specific noncommutative spacetimes like Minkowski and AdS.

Findings

Multiple Dirac operator candidates satisfy the axioms.

Symmetric spacetimes yield coinciding operators.

Formal self-adjointness selects a preferred operator.

Abstract

We study the notion of a Dirac operator in the framework of twist-deformed noncommutative geometry. We provide a number of well-motivated candidate constructions and propose a minimal set of axioms that a noncommutative Dirac operator should satisfy. These criteria turn out to be restrictive, but they do not fix a unique construction: two of our operators generally satisfy the axioms, and we provide an explicit example where they are inequivalent. For highly symmetric spacetimes with Drinfeld twists constructed from sufficiently many Killing vector fields, all of our operators coincide. For general noncommutative curved spacetimes we find that demanding formal self-adjointness as an additional condition singles out a preferred choice among our candidates. Based on this noncommutative Dirac operator we construct a quantum field theory of Dirac fields. In the last part we study noncommutative Dirac operators on deformed Minkowski and AdS spacetimes as explicit examples.