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arXiv · 2412.00628

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

Eva-Maria Hekkelman·Edward A. McDonald
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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.

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Eva-Maria Hekkelman, Edward A. McDonald. 2024-12-01. A noncommutative integral on spectrally truncated spectral triples, and a link with quantum ergodicity. https://doi.org/10.1016/j.jfa.2025.111154

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