Towards a Nonperturbative Path Integral in Gauge Theories

TL;DR

This paper introduces a modified path integral approach for gauge theories with curved orbit spaces, addressing Gribov ambiguities and operator ordering issues, and applies it to lattice gauge theory to explore the mass gap problem.

Contribution

It develops a nonperturbative path integral formalism for gauge theories with curved orbit spaces, ensuring gauge invariance and resolving Gribov and operator ordering issues.

Findings

Provides a modified path integral consistent with gauge invariance.

Addresses the Gribov problem in non-Euclidean orbit spaces.

Applies the formalism to lattice gauge theory and discusses the mass gap.

Abstract

We propose a modification of the Faddeev-Popov procedure to construct a path integral representation for the transition amplitude and the partition function for gauge theories whose orbit space has a non-Euclidean geometry. Our approach is based on the Kato-Trotter product formula modified appropriately to incorporate the gauge invariance condition, and thereby equivalence to the Dirac operator formalism is guaranteed by construction. The modified path integral provides a solution to the Gribov obstruction as well as to the operator ordering problem when the orbit space has curvature. A few explicit examples are given to illustrate new features of the formalism developed. The method is applied to the Kogut-Susskind lattice gauge theory to develop a nonperturbative functional integral for a quantum Yang-Mills theory. Feynman's conjecture about a relation between the mass gap and the orbit space geometry in gluodynamics is discussed in the framework of the modified path integral.