A Note on Invariant Measure on the Local Gauge Group

TL;DR

This paper proves that a finite translationally invariant measure cannot exist on the local gauge group of smooth functions from Euclidean space to a matrix Lie group, highlighting a fundamental limitation in gauge theory measure theory.

Contribution

It establishes the non-existence of finite invariant measures on the local gauge group, providing a key theoretical insight into gauge group measure properties.

Findings

No finite invariant measure exists on the local gauge group

The result applies to any matrix Lie group G

Implications for gauge theory and measure theory

Abstract

In this paper we investigated the problem of the existence of invariant meaures on the local gauge group. We prove that it is impossible to define a {\it finite} translationally invariant measure on the local gauge group $C^{\infty}({\bf R}^n,G)$(where $G$ is an arbitrary matrix Lie group).