Mode regularization of the configuration space path integral in curved space - art. no. 044014

The path integral representation of the transition amplitude for a particle moving in curved space has presented unexpected challenges since the intro duction of path integrals by Feynman fifty years ago. In this paper we disc uss and review mode regularization of the configuration space path integral , and present a three loop computation of the transition amplitude to test with success the consistency of such a regularization. The key features of the method are the use of the "Lee-Yang" ghost fields, which guarantee a co nsistent treatment of the nontrivial path integral measure at higher loops, and an effective potential specific to mode regularization which arises at the two loop order. We also perform the computation of the transition ampl itude using the regularization of the path integral by time discretization, which also makes use of Lee-Yang ghost fields and needs its own specific e ffective potential. This computation is shown to reproduce the same final r esult as the one performed in mode regularization. [S0556-2821(99)02216-X].