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.