Note on the gauge fixing in gauge theory

In the absence of Gribov complications, the modified gauge fixing in gauge theory integral DA(mu) ({exp[-S-YM(A(mu)) - integral f(A(mu))dx]/integral D g exp[-integral f(A(mu)(g))dx]} for example, f(A mu) = (1/2)(A(mu))(2), is identical to the conventional Faddeev-Popov formula integral DA(mu){delta(D -mu delta f(A(nu))/delta A(mu))/integral Dg delta(D-mu delta f(A(nu)(g))/de lta A(mu)(g))}exp[-S-YM(A mu)] if one takes into account the variation of t he 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 the ory, 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. (C) 2000 Else vier Science B.V. All rights reserved.