Endomorphisms and automorphisms of locally covariant quantum field theories

TL;DR

This paper studies the structure of symmetries in locally covariant quantum field theories, showing that under certain conditions, all endomorphisms are automorphisms and classifying these symmetries for free scalar fields.

Contribution

It establishes that locally covariant theories with standard assumptions have only automorphisms as endomorphisms and provides a classification method for these symmetries.

Findings

Endomorphisms of such theories are all automorphisms.

Automorphism groups are compact.

Complete classification of symmetries for free scalar fields.

Abstract

In the framework of locally covariant quantum field theory, a theory is described as a functor from a category of spacetimes to a category of *-algebras. It is proposed that the global gauge group of such a theory can be identified as the group of automorphisms of the defining functor. Consequently, multiplets of fields may be identified at the functorial level. It is shown that locally covariant theories that obey standard assumptions in Minkowski space, including energy compactness, have no proper endomorphisms (i.e., all endomorphisms are automorphisms) and have a compact automorphism group. Further, it is shown how the endomorphisms and automorphisms of a locally covariant theory may, in principle, be classified in any single spacetime. As an example, the endomorphisms and automorphisms of a system of finitely many free scalar fields are completely classified.