Note on the Gauge Fixing in Gauge Theory

Abstract

In the absence of Gribov complications, the modified gauge fixing in gauge theory $\int \mathcal{D}{A}_{\mu }\{\exp [-S_{YM}(A_{\mu })-\int f(A_{\mu })dx]/\int \mathcal{D}g\exp [-\int f(A_{\mu }^g)dx]\}$ for example, $f(A_{\mu })=(1/2)(A_{\mu }{)}^2$, is identical to the conventional Faddeev-Popov formula $\int \mathcal{D}{A}_{\mu }\{\delta (D^{\mu }\frac{\delta f(A_{\nu })}{\delta A_{\mu }})/\int \mathcal{D}g\delta (D^{\mu }\frac{\delta f(A_{\nu }^g)}{\delta A_{\mu }^g})\}\exp [-S_{YM}(A_{\mu })]$ if one takes into account the variation of the gauge field along the entire gauge orbit. Despite of its quite different appearance,the modified formula defines a local and BRST invariant theory and thus ensures unitarity at least in perturbation theory. In the presence of Gribov complications, as is expected in non-perturbative Yang-Mills theory, the modified formula is equivalent to the conventional formula but not identical to it:Both of the definitions give rise to non-local theory in general and thus the unitarity is not obvious. Implications of the present analysis on the lattice regularization are briefly discussed.

Comments: 12 pages