A Derivation of Geometric Quantization via Feynman's Path Integral on Phase Space

TL;DR

This paper derives the geometric quantization of symplectic manifolds using Feynman's path integral on phase space, connecting physical path integral methods with mathematical quantization frameworks.

Contribution

It introduces a novel derivation of geometric quantization from path integrals, incorporating states with negative norm and analyzing ambiguities in path integral definitions.

Findings

Path integral formulation reproduces geometric quantization.

States with negative norm are incorporated into the framework.

Ambiguities in path integral prescriptions affect symplectomorphism quantization.

Abstract

We derive the geometric quantization program of symplectic manifolds, in the sense of both Kostant-Souriau and Weinstein, from Feynman's path integral formulation on phase space. The state space we use contains states with negative norm and polarized sections determine a Hilbert space. We discuss ambiguities in the definition of path integrals arising from the distinct Riemann sum prescriptions and its consequence on the quantization of symplectomorphisms.