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 three-body problem and recently recognized to play a fundamental role in discretized quantum gravity models, can be given as a second order difference equation which, by a complex phase change, we turn into a discrete Schrödinger-like equation. The introduction of discrete potential-like 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 emerges of the semiclassical and classical regimes. The definition of coordinates adapted to Regge symmetry is exploited for the construction of a novel set of discrete orthogonal polynomials, characterizing the oscillatory components of torsion-like modes.

13 pages, 5 figures

References in corpus (4)

• Spectral Analysis of the Volume Operator in Loop Quantum Gravity
• Discreteness of the volume of space from Bohr-Sommerfeld quantization
• Bohr-Sommerfeld Quantization of Space
• Ladder Operators for Lamé Spheroconal Harmonic Polynomials

Cited by in corpus (10)

• Holonomy Operator and Quantization Ambiguities on Spinor Space
• The Screen representation of spin networks: 2D recurrence, eigenvalue equation for 6j symbols, geometric interpretation and Hamiltonian dynamics
• The Screen representation of spin networks. Images of 6j symbols and semiclassical features
• Symmetric angular momentum coupling, the quantum volume operator and the 7-spin network: a computational perspective
• Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages
• The Large-Volume Limit of a Quantum Tetrahedron is a Quantum Harmonic Oscillator
• Symmetric coupling of angular momenta, quadratic algebras and discrete polynomials
• Classical and Quantum Polyhedra
• Projective Ponzano-Regge spin networks and their symmetries
• The Ponzano-Regge asymptotic of the supersymmetric 6jS symbols