Selected publications

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1. Nicolas Franco, Micha³ Eckstein
Causality in noncommutative two-sheeted space-times
, vol. xxx, pp. xxx-xxx (2015).
[abstract] [preprint]

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.

2. Nicolas Franco
Temporal Lorentzian Spectral Triples
Rev. Math. Phys., vol. 26, 8, p. 14300076 (2014).
[abstract] [preprint] [journal]

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.

3. Nicolas Franco, Micha³ Eckstein
Exploring the Causal Structures of Almost Commutative Geometries
SIGMA, vol. 10, 010 (2014).
[abstract] [preprint] [journal] [download]

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 $\mathcal{S}(\mathbb{R}^{1,1}) \otimes M_2(\mathbb{C})$, 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.

4. Nicolas Franco, Micha³ Eckstein
An algebraic formulation of causality for noncommutative geometry
Class. Quantum Grav., vol. 30, p. 135007 (2013).
[abstract] [preprint] [journal]

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.

5. Nicolas Franco
Global Eikonal Condition for Lorentzian Distance Function in Noncommutative Geometry
SIGMA, vol. 6, 064 (2010).
[abstract] [preprint] [journal]

Connes' noncommutative Riemannian distance formula is constructed in two steps, the first one being the construction of a path-independent geometrical functional using a global constraint on continuous functions. This paper generalizes this first step to Lorentzian geometry. We show that, in a globally hyperbolic spacetime, a single global timelike eikonal condition is sufficient to construct a path-independent Lorentzian distance function.


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