Berezin-Toeplitz quantization for eigenstates of the Bochner-Laplacian on symplectic manifolds

TL;DR

This paper investigates Berezin-Toeplitz quantization on symplectic manifolds using eigenstates of the Bochner Laplacian, demonstrating correct semiclassical behavior and constructing the associated star-product.

Contribution

It introduces a novel quantization scheme based on eigenstates of the Bochner Laplacian and constructs the corresponding star-product, advancing the mathematical understanding of quantization on symplectic manifolds.

Findings

The quantization exhibits correct semiclassical behavior.

A star-product consistent with the quantization is constructed.

Eigenstates near the origin effectively model quantum states.

Abstract

We study the Berezin-Toeplitz quantization using as quantum space the space of eigenstates of the renormalized Bochner Laplacian corresponding to eigenvalues localized near the origin on a symplectic manifold. We show that this quantization has the correct semiclassical behavior and construct the corresponding star-product.