  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 $M \to \infty$. This simple scheme does not manifestly
preserve reparametrization invariance of the target manifold: particular
noncovariant terms of order $\hbar^2$ 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 $\hbar^2$ 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.
