Mode regularization, time slicing, Weyl ordering, and phase space path integrals for quantum-mechanical nonlinear sigma models

A simple, often invoked, regularization scheme of quantum mechanical path integrals in curved space is mode regularization: one expands fields into a Fourier series, performs calculations with only the first M modes, and at the end takes the limit M --> infinity. This simple scheme does not manifestly preserve reparametrization invariance of the target manifold: particular noncovariant terms of order (h) over bar(2) must be added to the action in order to maintain general coordinate invariance. Regularization by time slicing requires a different set of terms of order (h) over bar(2) which can be derived from Weyl ordering of the Hamiltonian. With these counterterms both schemes give the same answers to all orders of loops. As a check we perform the three-loop calculation of the trace anomaly in four dimensions in both schemes. We also present a diagrammatic proof of Matthews' theorem that phase space and configuration space path integrals are equal.