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The Dwelling of the Spectral Action

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Spectral Action in Noncommutative Geometry

Part of the book series: SpringerBriefs in Mathematical Physics ((BRIEFSMAPHY,volume 27))

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Abstract

The natural habitat of the spectral action is Connes’ noncommutative geometry. Therefore, it is indispensable to lay out its rudiments encoded in the notion of a spectral triple. We will, however, exclusively focus on the aspects of the structure, which are relevant for the spectral action computations. These include i.a. the abstract pseudodifferential calculus, the dimension spectrum and noncommutative integrals, based on both the Wodzicki residue and the Dixmier trace.

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Notes

  1. 1.

    One could in principle also allow for essential singularities of \(\zeta _{T,D}\), as long as they are isolated.

  2. 2.

    Using the notation of [37] we have for \(k \ge 1\).

  3. 3.

    For any \(s \in \mathbb {C}\), \(n \in \mathbb {N}\), with the convention . See [1, Sect. 24.1.3].

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Authors and Affiliations

  1. Copernicus Center for Interdisciplinary Studies, Kraków, Poland

    Michał Eckstein & Bruno Iochum

  2. Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Kraków, Poland

    Michał Eckstein

  3. National Quantum Information Centre, Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, Gdańsk, Poland

    Michał Eckstein

  4. Aix-Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

    Bruno Iochum

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  1. Michał Eckstein
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Eckstein, M., Iochum, B. (2018). The Dwelling of the Spectral Action. In: Spectral Action in Noncommutative Geometry. SpringerBriefs in Mathematical Physics, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-319-94788-4_1

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