1. Micha³ Eckstein, Nicolas Franco
Causal structure for noncommutative geometry
Frontiers of Fundamental Physics, vol. 14, pp. 138-xxx (2015).
[abstract] [journal] | Abstract:
Noncommutative Geometry à la Connes offers a promising framework for models of fundamental interactions. To guarantee the correct signature, the theory of Lorentzian spectral triples has been developed. We will briefly summarise its main elements and show that it can accommodate a sensible notion of causality understood as a partial order relation on the space of states on an algebra. For almost-commutative algebras of the form $C^\infty \otimes \A_F$, with $\A_F$ being finite-dimensional, the space of (pure) states is a simple product of space-time $\M$ and an internal space. The exploration of causal structures in this context leads to a surprising conclusion: The motion in both space-time and internal space is restricted by a "finite speed of light" constraint. We will present this phenomena on 2 simple toy-models. | 2. Edited by James Ladyman, Stuart Presnell, Gordon McCabe, Micha³ Eckstein, Sebastian J. Szybka
Road to Reality with Roger Penrose
CCPress [abstract] [preprint] [journal] | Abstract:
Where does the road to reality lie? This fundamental question is addressed in this collection of essays by physicists and philosophers, inspired by the original ideas of Sir Roger Penrose. The topics range from black holes and quantum information to the very nature of mathematical cognition itself. | 3. Editors: Micha³ Eckstein, Michael Heller, Sebastian J. Szybka
Mathematical Structures of the Universe
Copernicus Center Press (2014) [abstract] [journal] | Abstract:
The book contains a collection of essays on mathematical structures that serve us to model the Universe. The authors discuss such topics as: the interplay between mathematics and physics, geometrical structures in physical models, observational and conceptual aspects of cosmology. The reader can also contemplate the scientific method on the verge of its limits. | 4. Nicolas Franco, Micha³ Eckstein
Noncommutative geometry, Lorentzian structures and causality
in “Mathematical Structures of the Universe”, eds. M. Eckstein, M. Heller, S.J. Szybka, Copernicus Center Press (2014), pp. 315-340 [abstract] [preprint] [journal] | 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 \`a 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'. | |
