  The path integral representation of the transition amplitude for a particle
moving in curved space has presented unexpected challenges since the
introduction of path integrals by Feynman fifty years ago. In this paper we
discuss 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. Key features of the
method are the use of the Lee-Yang ghost fields, which guarantee a consistent
treatment of the non-trivial path integral measure at higher loops, and an
effective potential specific to mode regularization which arises at two loops.
We also perform the computation of the transition amplitude using the
regularization of the path integral by time discretization, which also makes
use of Lee-Yang ghost fields and needs its own specific effective potential.
This computation is shown to reproduce the same final result as the one
performed in mode regularization.
