2012/10/31 by Nicolas Franco · 1 citation
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Noncommutative and Quantum Gravity Theories #gr-qc #math-ph #math.MP #msc:46C20 #msc:46L52 #msc:53C50 #msc:58B34
paper · pdf · doi:10.1142/s0129055x14300076
published as Rev. Math. Phys., Vol. 26, No. 8, 1430007 (2014) · 25 pages, a proposition has been added (Prop. 11) concerning the recovering of the Lorentzian signature, final version
openalex publication_date 2014/08/20 · arxiv created 2014/08/25 · arxiv updated 2014/09/11 · openalex created_date 2020/11/23 · openalex updated_date 2026/07/28
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.