Towards a nonperturbative path integral in gauge theories

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. 1. Motivations. In what follows we consider only gauge theories of a special (YangMills) type where gauge transformations are linear transformations in the total configuration space. Let X be a configuration space which is assumed to be a Euclidean space R unless specified otherwise. It is a representation space of a gauge group G so the action of G on X is given by a linear transformation x → Ω(ω)x. A formal sum over paths for the action invariant under gauge transformations would diverge. To regularize it, Faddeev and Popov have proposed to insert the identity [1]