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Temporal Lorentzian spectral triples

Nicolas Franco

Copernicus Center for Interdisciplinary Studies, ul. Sławkowska 17, PL-31-016 Kraków, Poland

University of Namur, Department of Mathematics, Rempart de la Vierge 8, B-5000 Namur, Belgium

https://doi.org/10.1142/S0129055X14300076Cited by:15 (Source: Crossref)

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.

Keywords:

AMSC: 58B34, 53C50, 46C20, 46L52

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