Unfashionable observations about three dimensional gravity
Résumé
It is commonly accepted that the study of 2+1 dimensional quantum gravity could teach us something about the 3+1 dimensional case. The non-perturbative methods developed in this case share, as basic ingredient, a reformulation of gravity as a gauge field theory. However, these methods suffer many problems. Firstly, this perspective abandon the non-degeneracy of the metric and causality as fundamental principles, hoping to recover them in a certain low-energy limit. Then, it is not clear how these combinatorial techniques could be used in the case where matter fields are added, which are however the essential ingredients in order to produce non trivial observables in a generally covariant approach. Endly, considering the status of the observer in these approaches, it is not clear at all if they really could produce a completely covariant description of quantum gravity. We propose to re-analyse carefully these points. This study leads us to a really covariant description of a set of self-gravitating point masses in a closed universe. This approach is based on a set of observables associated to the measurements accessible to a participant-observer, they manage to capture the whole dynamic in Chern-Simons gravity as well as in true gravity. The Dirac algebra of these observables can be explicitely computed, and exhibits interesting algebraic features related to Poisson-Lie groupoids theory.