On an Axiomatization of Path Integral Quantization and its Equivalence to Berezin's Quantization

TL;DR

This paper axiomatizes path integral quantization for symplectic manifolds and proves its equivalence to Berezin's quantization, applying it to Riemann surfaces and Poisson structures.

Contribution

It establishes an axiomatic framework for path integral quantization and demonstrates its equivalence to Berezin's quantization, extending to Poisson manifolds.

Findings

Path integral quantization is equivalent to Berezin's quantization.

Quantization of Riemann surfaces of constant curvature using path integrals.

Application to Poisson structures on the sphere.

Abstract

We axiomatize path integral quantization of symplectic manifolds. We prove that this path integral formulation of quantization is equivalent to an abstract operator formulation, ie. abstract coherent state (or Berezin) quantization. We use the corresponding path integral of Poisson manifolds to quantize all complete Riemann surfaces of constant non$\unicode{x2013}$positive curvature and some Poisson structures on the sphere.