A noncommutative integral on spectrally truncated spectral triples, and a link with quantum ergodicity
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866916874274471936 |
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| author | Hekkelman, Eva-Maria McDonald, Edward A. |
| author_facet | Hekkelman, Eva-Maria McDonald, Edward A. |
| contents | 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 Szegő 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00628 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A noncommutative integral on spectrally truncated spectral triples, and a link with quantum ergodicity Hekkelman, Eva-Maria McDonald, Edward A. Operator Algebras Functional Analysis Spectral Theory 58B34, 58J51, 46L55 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 Szegő 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. |
| title | A noncommutative integral on spectrally truncated spectral triples, and a link with quantum ergodicity |
| topic | Operator Algebras Functional Analysis Spectral Theory 58B34, 58J51, 46L55 |
| url | https://arxiv.org/abs/2412.00628 |