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December 2010 The Hilbert space of 3d gravity: quantum group symmetries and observables
Catherine Meusburger, Karim Noui
Adv. Theor. Math. Phys. 14(6): 1651-1715 (December 2010).


We relate three-dimensional loop quantum gravity to the combinatorial quantization 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 quantization approaches and clarifies the role of quantum group symmetries in three-dimensional gravity.


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Catherine Meusburger. Karim Noui. "The Hilbert space of 3d gravity: quantum group symmetries and observables." Adv. Theor. Math. Phys. 14 (6) 1651 - 1715, December 2010.


Published: December 2010
First available in Project Euclid: 24 April 2012

zbMATH: 1241.83036
MathSciNet: MR2872468

Rights: Copyright © 2010 International Press of Boston

Vol.14 • No. 6 • December 2010
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