A noncommutative integral on spectrally truncated spectral triples, and a link with quantum ergodicity

TL;DR

This paper introduces an approximation of the noncommutative integral in spectral triples, linking it to quantum ergodicity, Szegő limit formulas, and the ergodicity of geodesic flow, revealing deep connections in noncommutative geometry.

Contribution

It proposes a new approximation of the noncommutative integral and establishes its connection with quantum ergodicity and spectral properties of Dirac operators.

Findings

Derived a Szegő limit formula for noncommutative geometry.

Linked the approximation to the density of states.

Defined ergodicity of geodesic flow for spectral triples.

Abstract

We propose a simple approximation of the noncommutative integral in noncommutative geometry for the Connes--Van Suijlekom paradigm of spectrally truncated spectral triples. A close connection between this approximation and the field of quantum ergodicity and work by Widom in particular immediately provides a Szeg\H{o} limit formula for noncommutative geometry. We then make a connection to the density of states. Finally, we propose a definition for the ergodicity of geodesic flow for compact spectral triples. This definition is known in quantum ergodicity as uniqueness of the vacuum state for $C^*$-dynamical systems, and for spectral triples where local Weyl laws hold this implies that the Dirac operator of the spectral triple is quantum ergodic. This brings to light a close connection between quantum ergodicity and Connes' integral formula.