Note on the gauge fixing in gauge theory Original Research Article

In the absence of Gribov complications, the modified gauge fixing in gauge theory ∫DAμ{exp[−SYM(Aμ)−∫f(Aμ)dx]/∫Dgexp[−∫f(Aμg)dx]} for example, f(Aμ)=(1/2)(Aμ)2 , is identical to the conventional Faddeev–Popov formula ∫DAμ{δ(Dμδf(Aν)δAμ)/∫Dg δ(Dμδf(Aνg)δAμg)}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 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.