Structure preserving discretization: A Berezin-Toeplitz Quantization viewpoint

TL;DR

This paper develops a formal framework for structure-preserving discretization using commutative diagrams, and applies it to Berezin-Toeplitz quantization, demonstrating how continuous structures are approximated discretely while highlighting the emergence of noncommutative features.

Contribution

It introduces a rigorous axiomatization of structure-preserving discretization and applies it to Berezin-Toeplitz quantization, establishing a limit theorem for Laplacian approximation.

Findings

Discretization often leads to noncommutative structures.

Berezin-Toeplitz quantization satisfies the proposed discretization criteria.

A limit theorem for matrix approximation of the Laplacian is established.

Abstract

In this paper, we introduce a comprehensive axiomatization of structure-preserving discretization through the framework of commutative diagrams. By establishing a formal language that captures the essential properties of discretization processes, we provide a rigorous foundation for analyzing how various structures (such as algebraic, geometric, and topological features) are maintained during the transition from continuous to discrete settings. Specifically, we establish that the transition from continuous to discrete differential settings invariably leads to noncommutative structures, reinforcing previous observation on the interplay between discretization and noncommutativity. We demonstrate the applicability of our axiomatization by applying it to the Berezin-Toeplitz quantization, showing that this quantization method adheres to our proposed criteria for structure-preserving discretization. We establish in this setting a precise limit theorem for the approximation of the Laplacian by a sequence of matrix approximations. This work not only enriches the theoretical understanding of the nature of discretization but also sets the stage for further exploration of its applications across various discretization methods.