An algebraic formulation of causality for noncommutative geometry

Abstract

We propose an algebraic formulation of the notion of causality for spectral triples corresponding to globally hyperbolic manifolds with a well defined noncommutative generalization. The causality is given by a specific cone of Hermitian elements respecting an algebraic condition based on the Dirac operator and a fundamental symmetry. We prove that in the commutative case the usual notion of causality is recovered. We show that, when the dimension of the manifold is even, the result can be extended in order to have an algebraic constraint suitable for a Lorentzian distance formula.

24 pages, minor changes from v2, to appear in Classical and Quantum Gravity

Cited by in corpus (24)

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• The noncommutative geometry of Zitterbewegung
• Spectral Noncommutative Geometry, Standard Model and all that
• Clifford Structures in Noncommutative Geometry and the Extended Scalar Sector
• Two roads to noncommutative causality
• Generally covariant $N$-particle dynamics
• Noncommutative geometrical origin of the energy-momentum dispersion relation
• Spectral Distance on Lorentzian Moyal Plane
• Families of spectral triples and foliations of space(time)
• Doppler shift in semi-Riemannian signature and the non-uniqueness of the Krein space of spinors
• Quantum causality constraints on kappa-Minkowski space-time
• Time functions and $K$-causality between measures
• The representation of spacetime through time functions
• Quantum causality in $\kappa$-Minkowski and related constraints
• Destructuring Physics: A functional derivation of spacetime
• Physical models from noncommutative causality
• A critical survey of twisted spectral triples beyond the Standard Model