A Mathematical Definition of Path Integrals on Symplectic Manifolds

TL;DR

This paper provides a rigorous mathematical framework for defining path integrals on symplectic and Poisson manifolds, with a focus on coherent state path integrals and examples involving Kähler manifolds with constant Bergman kernels.

Contribution

It introduces a formal mathematical definition of path integrals on symplectic and Poisson manifolds, highlighting computable cases like certain Kähler manifolds.

Findings

Defined path integrals relevant to quantization of symplectic manifolds

Identified Kähler manifolds with constant Bergman kernel as key examples

Provided computable instances of the proposed path integral framework

Abstract

We give a mathematical definition of some path integrals, emphasizing those relevant to the quantization of symplectic manifolds (and more generally, Poisson manifolds) $\unicode{x2013}$ in particular, the coherent state path integral. We show that K\"{a}hler manifolds provide many computable examples and we emphasize those whose Bergman kernel is constant along the diagonal.