Hamiltonian dynamics of a quantum of space: hidden symmetries and spectrum of the volume operator, and discrete orthogonal polynomials
Abstract
The action of the quantum mechanical volume operator, introduced in connection with a symmetric representation of the threebody problem and recently recognized to play a fundamental role in discretized quantum gravity models, can be given as a secondorder difference equation which, by a complex phase change, we turn into a discrete Schrödingerlike equation. The introduction of discrete potentiallike functions reveals the surprising crucial role here of hidden symmetries, first discovered by Regge for the quantum mechanical 6j symbols; insight is provided into the underlying geometric features. The spectrum and wavefunctions of the volume operator are discussed from the viewpoint of the Hamiltonian evolution of an elementary ‘quantum of space’, and a transparent asymptotic picture of the semiclassical and classical regimes emerges. The definition of coordinates adapted to the Regge symmetry is exploited for the construction of a novel set of discrete orthogonal polynomials, characterizing the oscillatory components of torsionlike modes.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 May 2013
 DOI:
 10.1088/17518113/46/17/175303
 arXiv:
 arXiv:1301.1949
 Bibcode:
 2013JPhA...46q5303A
 Keywords:

 Quantum Physics;
 General Relativity and Quantum Cosmology;
 Mathematical Physics
 EPrint:
 13 pages, 5 figures