Note on the Gauge Fixing in Gauge Theory

TL;DR

This paper compares a modified gauge fixing approach to the conventional Faddeev-Popov method in gauge theory, showing their equivalence without Gribov issues and discussing implications for non-perturbative regimes and lattice regularization.

Contribution

It demonstrates that the modified gauge fixing formula is equivalent to the Faddeev-Popov method in absence of Gribov problems and discusses its properties and implications in non-perturbative contexts.

Findings

Modified gauge fixing is equivalent to Faddeev-Popov without Gribov complications.

The modified formula defines a local, BRST invariant, and unitary theory in perturbation theory.

In presence of Gribov issues, the formulas are equivalent but not identical, leading to non-local theories.

Abstract

In the absence of Gribov complications, the modified gauge fixing in gauge theory $ \int{\cal D}A_{\mu}\{\exp[-S_{YM}(A_{\mu})-\int f(A_{\mu})dx] /\int{\cal 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 $\int{\cal D}A_{\mu}\{\delta(D^{\mu}\frac{\delta f(A_{\nu})}{\delta A_{\mu}})/\int {\cal D}g\delta(D^{\mu}\frac{\delta f(A_{\nu}^{g})} {\delta A_{\mu}^{g}})\}\exp[-S_{YM}(A_{\mu})]$ 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.