Temporal Lorentzian Spectral Triples

TL;DR

This paper introduces temporal Lorentzian spectral triples, extending pseudo-Riemannian spectral triples to include a global time concept, enabling Lorentzian distance measurement in noncommutative geometry, exemplified by Moyal--Minkowski spacetime.

Contribution

It defines temporal Lorentzian spectral triples with a 3+1 decomposition and constructs a noncommutative Lorentzian distance formula, advancing noncommutative Lorentzian geometry.

Findings

Constructed a temporal Lorentzian spectral triple over Moyal--Minkowski spacetime.

Extended algebra to unbounded elements for noncommutative Lorentzian distance.

Established a Lorentzian distance formula between pure states.

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.