Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2008-09-17
Adv. Theor. Math. Phys. Vol. 14, Issue 6 (2010), 1651-1716
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
52 pages, 13 figures
Scientific paper
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.
Meusburger Catherine
Noui Karim
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