Temporal Lorentzian Spectral Triples

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.

25 pages, a proposition has been added (Prop. 11) concerning the recovering of the Lorentzian signature, final version

References in corpus (5)

• Algebras of distributions suitable for phase-space quantum mechanics. I
• Algebras of distributions suitable for phase-space quantum mechanics. II. Topologies on the Moyal algebra
• An algebraic formulation of causality for noncommutative geometry
• Pseudo-Riemannian spectral triples and the harmonic oscillator
• Equivariant Lorentzian Spectral Triples

Cited by in corpus (19)

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• The Lorentzian distance formula in noncommutative geometry
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• Exploring the Causal Structures of Almost Commutative Geometries
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• The noncommutative geometry of Zitterbewegung
• Metrics and causality on Moyal planes
• Spectral Noncommutative Geometry, Standard Model and all that
• Noncommutative geometry, Lorentzian structures and causality
• Noncommutative geometry, the Lorentzian Standard Model and its B-L extension
• Lorentzian fermionic action by twisting euclidean spectral triples
• Linear hyperbolic PDEs with non-commutative time
• Spectral Distance on Lorentzian Moyal Plane
• Families of spectral triples and foliations of space(time)
• Non-commutative coordinates from quantum gravity
• Quantum causality constraints on kappa-Minkowski space-time
• Remarks on the spectrum of the Dirac operator of pseudo-Riemannian spin manifolds
• Physical models from noncommutative causality
• Twisting Noncommutative Geometries with Applications to High Energy Physics