Ideal quantum metrics from fractional Laplacians
Dimitris Michail Gerontogiannis, Bram Mesland
TL;DR
The paper develops a uniform construction of Monge–Kantorovich metrics on probability measures on Ahlfors regular spaces by using Schatten-class commutators with the fractional Dirichlet Laplacian $\Delta_{\alpha}$. It establishes a Weyl law and domain properties for $\Delta_{\alpha}$ and shows that commutators with Hölder functions lie in Schatten classes, yielding explicit metric expressions in terms of the spectrum of $\Delta_{\beta(\alpha)}$ in the low-spectral-dimension regime. The framework is extended to dynamical systems via crossed products, enabling CQMS structures for expansive $\mathbb{Z}^m$-actions and Smale/homoclinic $C^*$-algebras, and to Pontryagin duals in the abelian setting. A key result is an explicit Connes distance formula expressed as a spectral sum, facilitating practical computations on Cantor-type and related spaces. Overall, the work unifies fractional analysis, fractal geometry, noncommutative geometry, and dynamics to produce versatile quantum metric space tools with concrete spectral data.
Abstract
We develop a novel framework for Monge--Kantorovič 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.