Three Dimensional Loop Quantum Gravity: Particles and the Quantum Double
arXiv:gr-qc/0612144 · doi:10.1063/1.2352860
Abstract
It is well known that the quantum double structure plays an important role in three dimensional quantum gravity coupled to matter field. In this paper, we show how this algebraic structure emerges in the context of three dimensional Riemannian loop quantum gravity (LQG) coupled to a finite number of massive spinless point particles. In LQG, physical states are usually constructed from the notion of SU(2) cylindrical functions on a Riemann surface and the Hilbert structure is defined by the Ashtekar-Lewandowski measure. In the case where is the sphere , we show that the physical Hilbert space is in fact isomorphic to a tensor product of simple unitary representations of the Drinfeld double DSU(2): the masses of the particles label the simple representations, the physical states are tensor products of vectors of simple representations and the physical scalar product is given by intertwining coefficients between simple representations. This result is generalized to the case of any Riemann surface .
36 pages, published in Jour. Math. Physics
References in corpus (24)
- Background Independent Quantum Gravity: A Status Report
- Spin Foam Models for Quantum Gravity
- Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity
- A Lorentzian Signature Model for Quantum General Relativity
- Ponzano-Regge model revisited III: Feynman diagrams and Effective field theory
- Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity
- Three dimensional loop quantum gravity: physical scalar product and spin foam models
- Combinatorial quantization of the Hamiltonian Chern-Simons theory I
- Ponzano-Regge model revisited I: Gauge fixing, observables and interacting spinning particles
- Testing the Master Constraint Programme for Loop Quantum Gravity I. General Framework
- Hamiltonian Quantization of Chern-Simons theory with SL(2,C) Group
- Combinatorial Quantization of the Hamiltonian Chern-Simons Theory II
- Two dimensional lattice gauge theory based on a quantum group
- On the relation between 2+1 Einstein gravity and Chern Simons theory
- On relativistic spin network vertices
- Spin Networks for Non-Compact Groups
- Testing the Master Constraint Programme for Loop Quantum Gravity IV. Free Field Theories
- Ponzano-Regge model revisited II: Equivalence with Chern-Simons
- A generalized Hamiltonian Constraint Operator in Loop Quantum Gravity and its simplest Euclidean Matrix Elements
- Three dimensional loop quantum gravity: coupling to point particles
- Testing the Master Constraint Programme for Loop Quantum Gravity III. SL(2,R) Models
- Quantum double of a (locally) compact group
- Tensor product representations of the quantum double of a compact group
- Unfashionable observations about three dimensional gravity
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- Three Dimensional Quantum Geometry and Deformed Poincare Symmetry
- Observables in Loop Quantum Gravity with a cosmological constant
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- Quasi-local holographic dualities in non-perturbative 3d quantum gravity I - Convergence of multiple approaches and examples of Ponzano-Regge statistical duals
- Quantum gravity kinematics from extended TQFTs
- On entanglement entropy in non-Abelian lattice gauge theory and 3D quantum gravity
- Flux formulation of loop quantum gravity: Classical framework
- A model for the motion of a particle in a quantum background
- Most general theory of 3d gravity: Covariant phase space, dual diffeomorphisms, and more
- Black hole spectroscopy from Loop Quantum Gravity models
- A Note on B-observables in Ponzano-Regge 3d Quantum Gravity
- Discretization of 3d gravity in different polarizations
- Three dimensional Loop Quantum Gravity: towards a self-gravitating Quantum Field Theory
- Two-dimensional topological field theories coupled to four-dimensional BF theory
- On a self-dual phase space for 3+1 lattice Yang-Mills theory
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- A bound on Planck-scale modifications of the energy-momentum composition rule from atomic interferometry
- Gauge fixing in (2+1)-gravity: Dirac bracket and spacetime geometry
- (2-)Drinfel'd Double and (2-)BF Theory
- Motion in Quantum Gravity