Towards a nonperturbative path integral in gauge theories

We propose a modification of the Faddeev-Popov procedure to construct a pat h integral representation for the transition amplitude and the partition fu nction for gauge theories whose orbit space has a non-Euclidean geometry. O ur approach is based on the Kato-Trotter product formula modified appropria tely to incorporate the gauge invariance condition, and thereby equivalence to the Dirac operator formalism is guaranteed by construction. The modifie d 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 ex plicit, examples are given to illustrate new features of the formalism deve loped. The method is applied to the Kogut-Susskind lattice gauge theory to develop a nonperturbative functional integral for a quantum Yang-Mills theo ry. Feynman's conjecture about a relation between the mass gap and the orbi t space geometry in gluodynamics is discussed in the framework of the modif ied path integral. (C) 1999 Published by Elsevier Science B.V. All rights r eserved.