Advances in Theoretical and Mathematical Physics
- Adv. Theor. Math. Phys.
- Volume 14, Number 6 (2010), 1651-1715.
The Hilbert space of 3d gravity: quantum group symmetries and observables
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.
Adv. Theor. Math. Phys., Volume 14, Number 6 (2010), 1651-1715.
First available in Project Euclid: 24 April 2012
Permanent link to this document
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Meusburger, Catherine; Noui, Karim. The Hilbert space of 3d gravity: quantum group symmetries and observables. Adv. Theor. Math. Phys. 14 (2010), no. 6, 1651--1715. https://projecteuclid.org/euclid.atmp/1335273531