Conformal Motions and the Duistermaat-Heckman Integration Formula

We derive a geometric integration formula for the partition function of a classical dynamical system and use it to show that corrections to the WKB approximation vanish for any Hamiltonian which generates conformal motions of some Riemannian geometry on the phase space. This generalizes previous cases where the Hamiltonian was taken as an isometry generator. We show that this conformal symmetry is similar to the usual formulations of the Duistermaat-Heckman integration formula in terms of a supersymmetric Ward identity for the dynamical system. We present an explicit example of a localizable Hamiltonian system in this context and use it to demonstrate how the dynamics of such systems differ from previous examples of the Duistermaat-Heckman theorem. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. Understanding the circumstances under which a partition function is given exactly by its semi-classical approximation reveals connections between classical mechanics and the geometry and topology of the associated phase space. The Duistermaat-Heckman theorem [1]–[3] has been the basis of so-called localization theory in both mathematics and physics [4]–[12] (see [13] for a recent review) and it provides geometric criteria for the exactness of the semiclassical approximation for the finite-dimensional partition function Z(T ) = ∫