Ideal quantum metrics from fractional Laplacians

TL;DR

This paper introduces a new framework for quantum metrics based on fractional Laplacians and Schatten ideals, providing explicit spectral formulas and extending to dynamical systems within noncommutative geometry.

Contribution

It develops a novel approach to quantum metrics using fractional Laplacians and spectral analysis, with applications to fractal geometry and dynamical systems.

Findings

Derived closed spectral formulas for quantum metrics

Developed new noncommutative geometric techniques including a Weyl law

Extended fractional analysis to hyperbolic dynamical systems

Abstract

We develop a novel framework for Monge--Kantorovi\v{c} metrics using Schatten ideals and commutators of fractional Laplacians on Ahlfors regular spaces. Notably, for those metrics we derive closed formulas in terms of spectra of higher-order fractional Laplacians. For our proofs we develop new techniques in noncommutative geometry, in particular a Weyl law and Schatten-class commutators, yielding refined quantum metrics on the space of Borel probability measures. Lastly, our fractional analysis extends to dynamical systems. We showcase this in the setting of expansive algebraic $\mathbb Z^m$-actions and homoclinic $C^*$-algebras of certain hyperbolic dynamical systems. These findings illustrate the versatility of fractional analysis in fractal geometry, dynamical systems and noncommutative geometry.