The locally covariant Dirac field

TL;DR

This paper formulates the free Dirac field as a locally covariant quantum field theory, exploring geometric and cohomological aspects, and analyzing properties like Hadamard states and stress-energy relations.

Contribution

It provides a representation independent construction of the Dirac field in curved spacetime and clarifies the role of cohomology in the theory's uniqueness.

Findings

The theory is unique when restricted to observable algebras.

Established properties of the Dirac field in curved spacetime.

Connected relative Cauchy evolution to stress-energy tensor commutators.

Abstract

We describe the free Dirac field in a four dimensional spacetime as a locally covariant quantum field theory in the sense of Brunetti, Fredenhagen and Verch, using a representation independent construction. The freedom in the geometric constructions involved can be encoded in terms of the cohomology of the category of spin spacetimes. If we restrict ourselves to the observable algebra the cohomological obstructions vanish and the theory is unique. We establish some basic properties of the theory and discuss the class of Hadamard states, filling some technical gaps in the literature. Finally we show that the relative Cauchy evolution yields commutators with the stress-energy-momentum tensor, as in the scalar field case.