2008/06/16 by Shahn Majid, S Majid, Majid, S +3 · 1 citation
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #gr-qc #hep-th #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.0806.2587
49 pages, one pdf figure; revised version (several small changes, improved figure) similar to one which will appear in J.Phys.A
openalex publication_date 2008/06/16 · arxiv created 2009/09/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(su2)) 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-ℏ √(-Λ)/mp, where mp is the Planck mass and Λis the cosmological constant, and argue that semidualisation interchanges mp and lc, where lc is the cosmological length scale lc=1/√(|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.