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.
Abstract. We investigate the causal relations in the space of states of almost commutative Lorentzian geometries. We fully describe the causal structure of a simple model based on the algebra S(R 1,1 ) ⊗ M 2 (C), which has a non-trivial space of internal degrees of freedom. It turns out that the causality condition imposes restrictions on the motion in the internal space. Moreover, we show that the requirement of causality favours a unitary evolution in the internal space.
We investigate the causal structure of two-sheeted space-times using the tools of Lorentzian spectral triples. We show that the noncommutative geometry of these spaces allows for causal relations between the two sheets. The computation is given in details when the sheet is a 2-or 4-dimensional globally hyperbolic spin manifold. The conclusions are then generalised to a point-dependent distance between the two sheets resulting from the fluctuations of the Dirac operator.
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.a With our choice of signature, the flat Dirac matrix γ 0 is such that (γ 0 ) 2 = −1 and (γ 0 ) * = −γ 0 , so J = iγ 0 respects the conditions of a fundamental symmetry. The other flat Dirac matrices respect (γ a ) 2 = 1 and (γ a ) * = γ a for a = 1, 2, 3. 1430007-5N. Franco by J . We can notice that in the commutative case, with this definition of pseudo-Riemannian spectral triple as given in [12], the operator D corresponds to the usual Dirac operator times a factor i q .The compact resolvent condition is not present in this definition. This comes from the fact that, on a pseudo-Riemannian manifold, the Dirac operator D is not elliptic. Its principal symbol satisfies the relation σ D (ξ) 2 = c 2 (ξ) = g(ξ, ξ) and so it is not invertible any more. In order to recover a similar condition, we define. This operator is elliptic of order 1 since σ ∆J (ξ) 2 = g r (ξ, ξ), and is self-adjoint for the J -product. The compact condition is then required on ∆ −1 J = (1 + [D] 2 J ) − 1 2 and this is independent of the choice of the fundamental symmetry J [12].
For almost twenty years, a search for a Lorentzian version of the well-known Connes' distance formula has been undertaken. Several authors have contributed to this search, providing important milestones, and the time has now come to put those elements together in order to get a valid and functional formula. This paper presents a historical review of the construction and the proof of a Lorentzian distance formula suitable for noncommutative geometry.
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