Trace anomalies and the string-inspired definition of quantum-mechanical path integrals in curved space

We consider quantum-mechanical path integrals for non-linear sigma models o n a circle defined by the string-inspired method of Strassler, where one co nsiders periodic quantum fluctuations about a center-of-mass coordinate. In this approach one finds incorrect answers for the local trace anomalies of the corresponding n-dimensional field theories in curved space. The quantu m field theory approach to the quantum-mechanical path-integral, where fluc tuations are not periodic but vanish at the endpoints, yields the correct a nswers. We explain these results by a detailed analysis of general coordina te invariance in both methods. Both approaches can be derived from the same operator expression and the integrated trace anomalies in both schemes agr ee. In the string-inspired method the integrands are not invariant under ge neral coordinate transformations and one is therefore not permitted to use Riemann normal coordinates. (C) 1999 Elsevier Science B.V. All rights reser ved.