qdeformation and semidualization in 3D quantum gravity
Abstract
We explore in detail the role in euclidean 3D quantum gravity of quantum Born reciprocity or 'semidualization'. 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 noncommutative 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 3sphere, while the bicrossproduct (DSR) model based on \mathbb {C}[\mathbb {R}^2{\gt \!\!\!\triangleleft }\mathbb {R}]{\blacktriangleright \!\!\!\triangleleft } U(su_2) is the semidual of a quantum particle on hyperbolic space. We show further how the different models are all specific limits of qdeformed models with q=e^{\hbar \sqrt{\Lambda }/m_p}, where m_{p} is the Planck mass and Λ is the cosmological constant, and argue that semidualization interchanges m_{p} <> l_{c}, where l_{c} is the cosmological length scale l_c=1/\sqrt{\Lambda }. We investigate the physics of semidualization 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 noncommutative/curved wave equation. We explain, moreover, that the qdeformed model, at a certain algebraic level, is selfdual under semidualization.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 October 2009
 DOI:
 10.1088/17518113/42/42/425402
 Bibcode:
 2009JPhA...42P5402M