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\nconnection with a symmetric representation of the three-body problem and\nrecently recognized to play a fundamental role in discretized quantum gravity\nmodels, can be given as a second order difference equation which, by a complex\nphase change, we turn into a discrete Schr\\"odinger-like equation. The\nintroduction of discrete potential-like functions reveals the surprising\ncrucial role here of hidden symmetries, first discovered by Regge for the\nquantum mechanical 6j symbols; insight is provided into the underlying\ngeometric features. The spectrum and wavefunctions of the volume operator are\ndiscussed from the viewpoint of the Hamiltonian evolution of an elementary\n"quantum of space", and a transparent asymptotic picture emerges of the\nsemiclassical and classical regimes. The definition of coordinates adapted to\nRegge symmetry is exploited for the construction of a novel set of discrete\northogonal polynomials, characterizing the oscillatory components of\ntorsion-like modes.\n