SIGMA 10 (2014), 010, 23 pages      arXiv:1310.8225      https://doi.org/10.3842/SIGMA.2014.010
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

Exploring the Causal Structures of Almost Commutative Geometries

Nicolas Franco ^a and Michał Eckstein ^b
a) Copernicus Center for Interdisciplinary Studies, ul. Sławkowska 17, 31-016 Kraków, Poland
^b) Faculty of Mathematics and Computer Science, Jagellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland

Received October 31, 2013, in final form January 20, 2014; Published online January 28, 2014

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

Key words: noncommutative geometry; causal structures; Lorentzian spectral triples.

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