  Quantum mechanical models with extended supersymmetry find interesting
applications in worldline approaches to relativistic field theories. In this
paper we consider one-dimensional nonlinear sigma models with O(N) extended
supersymmetry on the worldline, which are used in the study of higher spin
fields on curved backgrounds. We calculate the transition amplitude for
euclidean times (i.e. the heat kernel) in a perturbative expansion, using both
canonical methods and path integrals. The latter are constructed using three
different regularization schemes, and the corresponding counterterms that
ensure scheme independence are explicitly identified.
