Emergence of the Haar measure in the standard functional integral representation of the Yang-Mills partition function

TL;DR

This paper demonstrates that the standard Yang-Mills path integral inherently produces a gauge-invariant partition function through the Haar measure, clarifying its role in non-perturbative gauge invariance.

Contribution

It reveals how the Haar measure naturally emerges in the Yang-Mills path integral via the Faddeev-Popov determinant, ensuring gauge invariance.

Findings

The path integral yields a gauge-invariant partition function with explicit gauge projection.

The Haar measure appears from the Faddeev-Popov determinant in a maximal Abelian gauge.

Gauge invariance is maintained non-perturbatively through this measure.

Abstract

The conventional path integral expression for the Yang-Mills transition amplitude with flat measure and gauge-fixing built in via the Faddeev-Popov method has been claimed to fall short of guaranteeing gauge invariance in the non-perturbative regime. We show, however, that it yields the gauge invariant partition function where the projection onto gauge invariant wave functions is explicitly performed by integrating over the compact gauge group. In a variant of maximal Abelian gauge the Haar measure arises in the conventional Yang-Mills path integral from the Faddeev-Popov determinant.