On Exact Evaluation of Path Integrals

Abstract

We develop a general method to evaluate exactly certain phase space path integrals. Our method is applicable to hamiltonians which are functions of a classical phase space observable that determines the action of a circle on the phase space. Our approach is based on the localization technique, originally introduced to derive the Duistermaat-Heckman integration formula and its path integral generalizations. For this, we reformulate the phase space path integral in an auxiliary field representation that corresponds to a superloop space with both commuting and anticommuting coordinates. In this superloop space, the path integral can be interpreted in terms of a model-independent equivariant cohomology and evaluated exactly in the sense that it localizes into an integral over the original phase space. The final result can be related to equivariant characteristic classes. Curiously, our auxiliary field representation and the corresponding superloop space equivariant cohomology interpretation of the path integral essentially coincides with a superloop space formulation of ordinary Poincaré supersymmetric quantum field theories.