q-Deformation and Semidualisation in 3d Quantum Gravity

Abstract

We explore in detail the role in euclidean 3d quantum gravity of quantum Born reciprocity or `semidualisation'. The latter is an algebraic operation defined using quantum group methods that interchanges position and momentum. Using this we are able to clarify the structural relationships between the effective non-commutative geometries that have been discussed in the context of 3d gravity. We show that the spin model based on D(U(su_2)) for quantum gravity without cosmological constant is the semidual of a quantum particle on a three-sphere, while the bicrossproduct (DSR) model is the semidual of a quantum particle on hyperbolic space. We show further how the different models are all specific limits of q-deformed models with q=e^{-\hbar \sqrt{-Λ}/m_p}, where m_p is the Planck mass and Λis the cosmological constant, and argue that semidualisation interchanges m_p and l_c, where l_c is the cosmological length scale l_c=1/\sqrt{|Λ|}. We investigate the physics of semidualisation by studying representation theory. In both the spin model and its semidual we show that irreducible representations have a physical picture as solutions of a respectively non-commutative/curved wave equation. We explain, moreover, that the q-deformed model, at a certain algebraic level, is self-dual under semidualisation.

49 pages, one pdf figure; revised version (several small changes, improved figure) similar to one which will appear in J.Phys.A

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Cited by in corpus (6)

• Generalised Chern-Simons actions for 3d gravity and kappa-Poincare symmetry
• Galilean quantum gravity with cosmological constant and the extended q-Heisenberg algebra
• Group Momentum Space and Hopf Algebra Symmetries of Point Particles Coupled to 2+1 Gravity
• The Poincaré group as a Drinfel'd double
• Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data
• Towards Non-Commutative Deformations of Relativistic Wave Equations in 2+1 Dimensions