Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2008-06-16
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
49 pages, one pdf figure; revised version (several small changes, improved figure) similar to one which will appear in J.Phys.
Scientific paper
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{-\Lambda}/m_p}, where m_p is the Planck mass and \Lambda is 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{|\Lambda|}. 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.
Majid Shahn
Schroers Bernd J.
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