Note on the Gauge Fixing in Gauge Theory

In the absence of Gribov complications, the modified gauge fixing in gauge theory ∫ DAμ{exp[−SYM (Aμ)− ∫ f(Aμ)dx]/ ∫ Dg exp[− ∫ f(Aμ)dx]} for example, f(Aμ) = (1/2)(Aμ) 2, is identical to the conventional Faddeev-Popov formula ∫ DAμ{δ(D δf(Aν ) δAμ )/ ∫ Dgδ(Dμ δf(A g ν ) δA g μ )} exp[−SYM (Aμ)] 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 YangMills 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.