Temporal Lorentzian spectral triples

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    Abstract

    We present the notion of temporal Lorentzian spectral triple which is an extension of the notion of pseudo-Riemannian spectral triple with a way to ensure that the signature of the metric is Lorentzian. A temporal Lorentzian spectral triple corresponds to a specific 3 + 1 decomposition of a possibly noncommutative Lorentzian space. This structure introduces a notion of global time in noncommutative geometry. As an example, we construct a temporal Lorentzian spectral triple over a Moyal-Minkowski spacetime. We show that, when time is commutative, the algebra can be extended to unbounded elements. Using such an extension, it is possible to define a Lorentzian distance formula between pure states with a well-defined noncommutative formulation.

    Original languageEnglish
    Article number1430007
    Number of pages23
    JournalReviews in Mathematical Physics
    Volume26
    Issue number8
    DOIs
    Publication statusPublished - 22 Sept 2014

    Keywords

    • Lorentzian geometry
    • Noncommutative geometry
    • Spectral triples

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