A new construction of strict deformation quantization for Lagrangian fiber bundles

TL;DR

This paper introduces a novel method for strict deformation quantization of symplectic manifolds with Lagrangian fiber bundles, linking geometric and Berezin-Toeplitz quantizations.

Contribution

It presents a new construction that approximates the correspondence between differential operators and principal symbols, connecting geometric and Berezin-Toeplitz quantizations.

Findings

Provides a lattice approximation framework for deformation quantization.

Analyzes the formal deformation quantization associated with the new construction.

Explores relations between geometric and Berezin-Toeplitz quantizations.

Abstract

We give a new construction of strict deformation quantization of symplectic manifolds equipped with a proper Lagrangian fiber bundle structure, whose representation spaces are the quantum Hilbert spaces obtained by geometric quantization. The construction can be regarded as a "lattice approximation of the correspondence between differential operators and principal symbols". We analyze the corresponding formal deformation quantization. We also investigate into relations between our construction and Berezin-Toeplitz deformation quantization.