Extended SUSY quantum mechanics: transition amplitudes and path integrals
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.
In this paper the $\omega$-limit behaviour of trajectories of solutions of ordinary differential equations is studied by methods of an axiomatic theory of solution spaces. We prove, under very general assumptions, semi-invariance of $\omega$-limit sets and a Poincar'{e}-Bendixon type theorem.