Path integrals for quantum mechanics in curved space (or anomalies from nonlinear sigma models)

We give a complete derivation of path integrals for quantum mechanical nonlinear sigma models. We begin with bosonic systems with an arbitrary Hamiltonian H-circumflex quadratic in momenta. Starting from the matrix element with arbitrary but a priori fixed operator ordering in H-circumflex, we insert complete sets of position and momentum eigenstates, and obtain the discretized propagators and vertices in closed form. These results tell how to evaluate equal-time contractions in the corresponding continuum path integrals, as well as products of distributions. (author)