On the local structure of the representation of a local gauge group

TL;DR

This paper investigates the local structure of energy representations of gauge groups acting on Boson Fock spaces, revealing that the commutant remains small even when the representation is reducible, with implications for quantum field theory.

Contribution

It clarifies the local structure of energy representations of gauge groups and shows the smallness of the commutant even in reducible cases, extending understanding in quantum field theory contexts.

Findings

Irreducibility depends on manifold dimension

Even reducible representations have a small commutant

Support property of functions is key to analysis

Abstract

This paper clarifies the local structure of the energy representation of a local gauge group. The group to be considered is a smooth map from a manifold into a compact Lie group. It acts on a Boson Fock spaces generated by connection 1-forms on the manifold. The irreduciblity depends on the dimensionality of a manifold. However, we showed that even if the representation is reducible(for example, dim=1 case) the commutant is very small. The discussion is based on the support property of the function, which is analogous to the algebraic Quantum Field Theory.