Hamiltonian dynamics of a quantum of space: hidden symmetries and spectrum of the volume operator, and discrete orthogonal polynomials

TL;DR

This paper explores the Hamiltonian dynamics of a quantum space volume operator, revealing hidden symmetries, spectrum characteristics, and introducing new discrete orthogonal polynomials related to quantum gravity models.

Contribution

It uncovers hidden symmetries in the volume operator, analyzes its spectrum and wavefunctions, and constructs novel discrete orthogonal polynomials linked to quantum gravity.

Findings

Hidden symmetries influence the spectrum of the volume operator.

A discrete Schrödinger-like equation describes the operator's action.

New orthogonal polynomials characterize oscillatory modes.

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 Schr\"odinger-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.