Construction of Self-Adjoint Berezin-Toeplitz Operators on Kahler Manifolds and a Probabilistic Representation of the Associated Semigroups

TL;DR

This paper develops criteria for the self-adjointness of Berezin-Toeplitz operators on Kahler manifolds and introduces a probabilistic path integral representation of their semigroups using Brownian motion, linking geometric quantization with stochastic analysis.

Contribution

It provides new self-adjointness conditions for Berezin-Toeplitz operators and establishes a Wiener-regularized path integral representation of their semigroups, connecting geometric quantization with probabilistic methods.

Findings

Self-adjointness criteria for Berezin-Toeplitz operators

Probabilistic Wiener-regularized path integral representation of semigroups

Relation between Berezin-Toeplitz and Schrödinger operators

Abstract

We investigate a class of operators resulting from a quantization scheme attributed to Berezin. These so-called Berezin-Toeplitz operators are defined on a Hilbert space of square-integrable holomorphic sections in a line bundle over the classical phase space. As a first goal we develop self-adjointness criteria for Berezin-Toeplitz operators defined via quadratic forms. Then, following a concept of Daubechies and Klauder, the semigroups generated by these operators may under certain conditions be represented in the form of Wiener-regularized path integrals. More explicitly, the integration is taken over Brownian-motion paths in phase space in the ultra-diffusive limit. All results are the consequence of a relation between Berezin-Toeplitz operators and Schrodinger operators defined via certain quadratic forms. The probabilistic representation is derived in conjunction with a version of the Feynman-Kac formula.