We study a noncommutative analog of a spacetime foliated by spacelike hypersurfaces, in both Riemannian and Lorentzian signatures. First, in the classical commutative case, we show that the canonical Dirac operator on the total spacetime can be reconstructed from the family of Dirac operators on the hypersurfaces. Second, in the noncommutative case, the same construction continues to make sense for an abstract family of spectral triples. In the case of Riemannian signature, we prove that the construction yields in fact a spectral triple, which we call a product spectral triple. In the case of Lorentzian signature, we correspondingly obtain a “Lorentzian spectral triple,” which can also be viewed as the “reverse Wick rotation” of a product spectral triple. This construction of “Lorentzian spectral triples” fits well into the Krein space approach to noncommutative Lorentzian geometry.
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Families of spectral triples and foliations of space(time)
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June 2018
Research Article|
June 19 2018
Families of spectral triples and foliations of space(time)

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Koen van den Dungen
Koen van den Dungen
a)
SISSA (Scuola Internazionale Superiore di Studi Avanzati)
, Via Bonomea, 265, 34136 Trieste, Italy
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Koen van den Dungen
a)
SISSA (Scuola Internazionale Superiore di Studi Avanzati)
, Via Bonomea, 265, 34136 Trieste, Italy
a)
Present address: Mathematisches Institut der Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany. Email: [email protected].
J. Math. Phys. 59, 063507 (2018)
Article history
Received:
January 03 2018
Accepted:
May 25 2018
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A companion article has been published:
Generalizing noncommutative geometry moves toward integrating quantum mechanics with gravity
Citation
Koen van den Dungen; Families of spectral triples and foliations of space(time). J. Math. Phys. 1 June 2018; 59 (6): 063507. https://doi.org/10.1063/1.5021305
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