MODE REGULARIZATION, TIME SLICING, WEYL ORDERING, AND PHASE-SPACE PATH-INTEGRALS FOR QUANTUM-MECHANICAL NONLINEAR SIGMA-MODELS - ART. NO. 044002

A simple, often invoked, regularization scheme of quantum mechanical p ath integrals in curved space is mode regularization: one expands fiel ds into a Fourier series, performs calculations with only the first M modes, and at the end takes the limit M --> infinity. This simple sche me does not manifestly preserve reparametrization invariance of the ta rget manifold: particular noncovariant terms of order (h) over bar(2) must be added to the action in order to maintain general coordinate in variance. Regularization by time slicing requires a different set of t erms 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. W e also present a diagrammatic proof of Matthews' theorem that phase sp ace and configuration space path integrals are equal.