# Canonical quantization and the spectral action, a nice example

Abstract : We study the canonical quantization of the theory given by Chamseddine-Connes spectral action on a particular finite spectral triple with algebra $M_2(\Cset)\oplus\Cset$. We define a quantization of the natural distance associated with this noncommutative space and show that the quantum distance operator has a discrete spectrum. We also show that it would be the same for any other geometric quantity. Finally we propose a physical Hilbert space for the quantum theory. This spectral triple had been previously considered by Rovelli as a toy model, but with a different action which was not gauge-invariant. The results are similar in both cases, but the gauge-invariance of the spectral action manifests itself by the presence of a non-trivial degeneracy structure for our distance operator.
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Journal of Geometry and Physics, Elsevier, 2007, 57 (9), pp.1757-1770
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https://hal.archives-ouvertes.fr/hal-00129590
Contributeur : Fabien Besnard <>
Soumis le : dimanche 17 juin 2007 - 11:40:23
Dernière modification le : lundi 18 juin 2007 - 09:02:31
Document(s) archivé(s) le : mardi 21 septembre 2010 - 12:43:06

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Fabien Besnard. Canonical quantization and the spectral action, a nice example. Journal of Geometry and Physics, Elsevier, 2007, 57 (9), pp.1757-1770. <hal-00129590v2>

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