Structure preserving discretization: A Berezin-Toeplitz Quantization viewpoint
Damien Tageddine, Jean-Christophe Nave
TL;DR
This work proposes a unified axiomatic framework for structure-preserving discretization based on commutative diagrams, arguing that discretization induces a noncommutative differential structure realized by a self-adjoint $D$. It demonstrates, via Berezin-Toeplitz quantization, that continuous Poisson algebras can be faithfully discretized into finite-dimensional matrix algebras with convergent Lie and differential structures, and provides a precise limit theorem relating discrete and continuous Laplacians. The authors also show the framework encompasses key discretization paradigms—FEEC, diffeomorphism discretization, and BT quantization—culminating in a noncommutative Laplacian that converges to the classical Laplacian. The sphere example ties the theory to a concrete, well-studied setting, highlighting Zeitlin-type models as natural structure-preserving discretizations with potential broad impact across numerical PDEs and noncommutative geometry.
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.