Jena 2013 – scientific programme
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MP: Fachverband Theoretische und Mathematische Grundlagen der Physik
MP 7: Quanten+Gravitation 1
MP 7.2: Talk
Wednesday, February 27, 2013, 15:10–15:35, HS 8
Hamiltonian dynamics of a quantum of space:
hidden symmetries and spectrum of the volume operator,
and discrete orthogonal polynomials — Vincenzo Aquilanti1, •Dimitri Marinelli2, and Annalisa Marzuoli3 — 1Dipartimento di Chimica, Università di Perugia and I.M.I.P., C.N.R. Roma, Italy — 2Dipartimento di Fisica, Università degli Studi di Pavia and INFN, Sezione di Pavia, Italy — 3Dipartimento di Matematica ‘F. Casorati’, Università degli Studi di Pavia, and INFN, Sezione di Pavia, Italy
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 the Regge symmetries, first discovered 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 allows the construction of a novel set of discrete orthogonal polynomials.