0809.2875 (C. Meusburger et al.)

The Hilbert space of 3d gravity: quantum group symmetries and
observables    [PDF]

C. Meusburger, K. Noui

We relate three-dimensional loop quantum gravity to the combinatorial
quantisation formalism based on the Chern-Simons formulation for
three-dimensional Lorentzian and Euclidean gravity with vanishing cosmological
constant. We compare the construction of the kinematical Hilbert space and the
implementation of the constraints. This leads to an explicit and very
interesting relation between the associated operators in the two approaches and
sheds light on their physical interpretation. We demonstrate that the quantum
group symmetries arising in the combinatorial formalism, the quantum double of
the three-dimensional Lorentz and rotation group, are also present in the loop
formalism. We derive explicit expressions for the action of these quantum
groups on the space of cylindrical functions associated with graphs. This
establishes a direct link between the two quantisation approaches and clarifies
the role of quantum group symmetries in three-dimensional gravity.

View original: http://arxiv.org/abs/0809.2875