Conformal Motions and the Duistermaat-Heckman Integration Formula

TL;DR

This paper introduces a geometric integration formula for classical systems, showing that conformal symmetries lead to vanishing corrections in the WKB approximation, extending the Duistermaat-Heckman formula to conformal motions.

Contribution

It generalizes the Duistermaat-Heckman formula to systems with conformal symmetries, providing new insights into the geometric structure of classical dynamical systems.

Findings

Corrections to WKB vanish for Hamiltonians generating conformal motions.

The conformal symmetry relates to a supersymmetric Ward identity.

An explicit example demonstrates differences from traditional Duistermaat-Heckman applications.

Abstract

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.