The Berezin-Simon quantization for K\"ahler manifolds and their path integral representations

TL;DR

This paper develops a rigorous real-time path-integral formalism for Berezin-Simon quantization on Kähler manifolds, extending the operator formalism to a path integral approach using advanced mathematical tools.

Contribution

It introduces a rigorous real-time path-integral formulation for Berezin-Simon quantization on Kähler manifolds, bridging operator formalism and path integrals.

Findings

Path-integral formula derived using G"uneysu's extended Feynman--Kac theorem.

Applicable to classical systems with smooth, bounded Hamiltonians on Kähler manifolds.

Provides a rigorous mathematical foundation for real-time quantization formalism.

Abstract

The Berezin--Simon (BS) quantization is a rigorous version of the ``operator formalism'' of quantization procedure. The goal of the paper is to present a rigorous real-time (not imaginary-time) path-integral formalism corresponding to the BS operator formalism of quantization; Here we consider the classical systems whose phase space $M$ is a (possibly non-compact) K\"ahler manifold which satisfies some conditions, with a Hamiltonian $H:M\rightarrow\mathbb{R}$. For technical reasons, we consider only the cases where $H$ is smooth and bounded. We use G\"uneysu's extended version of the Feynman--Kac theorem to formulate the path-integral formula.