An algebraic formulation of causality for noncommutative geometry

TL;DR

This paper introduces an algebraic approach to defining causality within noncommutative geometry, extending classical causality concepts to spectral triples and providing a foundation for Lorentzian distance formulations.

Contribution

It presents a novel algebraic formulation of causality for spectral triples, applicable to noncommutative geometries and compatible with classical causality in the commutative case.

Findings

Recovers classical causality in the commutative case

Extends to even-dimensional manifolds for Lorentzian distance

Defines causality via a cone of Hermitian elements respecting algebraic conditions

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.