Note on the Gauge Fixing in Gauge Theory

Abstract

In the absence of Gribov complications, the modified gauge fixing in gauge theory $ \int{\cal D}A_μ\{\exp[-S_{YM}(A_μ)-\int f(A_μ)dx] /\int{\cal D}g\exp[-\int f(A_μ^{g})dx]\}$ for example, $f(A_μ)=(1/2)(A_μ)^{2}$, is identical to the conventional Faddeev-Popov formula $\int{\cal D}A_μ\{δ(D^μ\frac{δf(A_ν)}{δA_μ})/\int {\cal D}gδ(D^μ\frac{δf(A_ν^{g})} {δA_μ^{g}})\}\exp[-S_{YM}(A_μ)]$ 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.