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

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.Comment: 14 pages, LaTeX, no figure

On the Yang-Mills wave functional in Coulomb gauge

We investigate the dependence of the Yang-Mills wave functional in Coulomb gauge on the Faddeev-Popov determinant. We use a Gaussian wave functional multiplied by an arbitrary power of the Faddeev-Popov determinant. We show, that within the resummation of one-loop diagrams the stationary vacuum energy is independent of the power of the Faddeev-Popov determinant and, furthermore, the wave functional becomes field-independent in the infrared, describing a stochastic vacuum. Our investigations show, that the infrared limit is rather robust against details of the variational ans\"atze for the Yang-Mills wave functional. The infrared limit is exclusively determined by the divergence of the Faddeev-Popov determinant at the Gribov horizon.Comment: 9 pages, no figure