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

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.