Noncommutative geometry, Lorentzian structures and causality

Abstract

The theory of noncommutative geometry provides an interesting mathematical background for developing new physical models. In particular, it allows one to describe the classical Standard Model coupled to Euclidean gravity. However, noncommutative geometry has mainly been developed using the Euclidean signature, and the typical Lorentzian aspects of space-time, the causal structure in particular, are not taken into account. We present an extension of noncommutative geometry à la Connes suitable the for accommodation of Lorentzian structures. In this context, we show that it is possible to recover the notion of causality from purely algebraic data. We explore the causal structure of a simple toy model based on an almost commutative geometry and we show that the coupling between the space-time and an internal noncommutative space establishes a new `speed of light constraint'.

24 pages, review article. in `Mathematical Structures of the Universe', eds. M. Eckstein, M. Heller, S.J. Szybka, CCPress 2014

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• Causality in noncommutative two-sheeted space-times
• Finite Pseudo-Riemannian Spectral Triples and The Standard Model
• Metrics and causality on Moyal planes
• Noncommutative geometrical origin of the energy-momentum dispersion relation
• Linear hyperbolic PDEs with non-commutative time
• Spectral Distance on Lorentzian Moyal Plane
• Quantum causality in $\kappa$-Minkowski and related constraints