Quantum gravity and non-commutative spacetimes in three dimensions: a unified approach

Abstract

These notes summarise a talk surveying the combinatorial or Hamiltonian quantisation of three dimensional gravity in the Chern-Simons formulation, with an emphasis on the role of quantum groups and on the way the various physical constants (c,G,Λ,\hbar) enter as deformation parameters. The classical situation is summarised, where solutions can be characterised in terms of model spacetimes (which depend on c and Λ), together with global identifications via elements of the corresponding isometry groups. The quantum theory may be viewed as a deformation of this picture, with quantum groups replacing the local isometry groups, and non-commutative spacetimes replacing the classical model spacetimes. This point of view is explained, and open issues are sketched.

Talk given at Geometry and Physics in Cracow, September 2010; 22 pages, 2 figures

References in corpus (13)

• Symplectic structures associated to Lie-Poisson groups
• Combinatorial quantization of the Hamiltonian Chern-Simons theory I
• Symplectic structure of the moduli space of flat connections on a Riemann surface
• Hamiltonian Quantization of Chern-Simons theory with SL(2,C) Group
• Noncommutative geometry of angular momentum space U(su(2))
• Generalised Chern-Simons actions for 3d gravity and kappa-Poincare symmetry
• Three Dimensional Quantum Geometry and Deformed Poincare Symmetry
• Galilean quantum gravity with cosmological constant and the extended q-Heisenberg algebra
• Quantum double of a (locally) compact group
• Phase space structure of Chern-Simons theory with a non-standard puncture
• Boundary conditions and symplectic structure in the Chern-Simons formulation of (2+1)-dimensional gravity
• Lessons from (2+1)-dimensional quantum gravity
• Notes on a paper of Mess

Cited by in corpus (6)

• Observables in Loop Quantum Gravity with a cosmological constant
• Discretization of 3d gravity in different polarizations
• On the semiduals of local isometry groups in 3d gravity
• Classical r-matrices for the generalised Chern-Simons formulation of 3d gravity
• Towards Non-Commutative Deformations of Relativistic Wave Equations in 2+1 Dimensions
• Minimal Factorization of Chern-Simons Theory -- Gravitational Anyonic Edge Modes