Advances in Theoretical and Mathematical Physics

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

Catherine Meusburger and Karim Noui

Full-text: Open access

Abstract

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.

Article information

Source
Adv. Theor. Math. Phys., Volume 14, Number 6 (2010), 1651-1715.

Dates
First available in Project Euclid: 24 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.atmp/1335273531

Mathematical Reviews number (MathSciNet)
MR2872468

Zentralblatt MATH identifier
1241.83036

Citation

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


Export citation