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

@article{Aquilanti2013HamiltonianDO, title={Hamiltonian dynamics of a quantum of space: hidden symmetries and spectrum of the volume operator, and discrete orthogonal polynomials}, author={Vincenzo Aquilanti and Dimitri Marinelli and Annalisa Marzuoli}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2013}, volume={46} }

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…

## 24 Citations

### Symmetric coupling of angular momenta, quadratic algebras and discrete polynomials

- 2014

Mathematics

Eigenvalues and eigenfunctions of the volume operator, associated with the symmetric coupling of three SU(2) angular momentum operators, can be analyzed on the basis of a discrete Schrödinger–like…

### Classical and quantum polyhedra

- 2014

Physics

Quantum polyhedra constructed from angular momentum operators are the building blocks of space in its quantum description as advocated by loop quantum gravity. Here we extend previous results on the…

### The Screen Representation of Spin Networks: 2D Recurrence, Eigenvalue Equation for 6j Symbols, Geometric Interpretation and Hamiltonian Dynamics

- 2013

Mathematics

ICCSA

2D and 1D recursion relations that are useful for the direct computation of the orthonormal 6j are presented and a convention for the order of the arguments of the 6j that is based on their classical and Regge symmetries are presented.

### The large-volume limit of a quantum tetrahedron is a quantum harmonic oscillator

- 2013

Physics

It is shown that the volume operator of a quantum tetrahedron is, in the sector of large eigenvalues, accurately described by a quantum harmonic oscillator. This result relies on the fact that (i)…

### Combinatorial and Geometrical Origins of Regge Symmetries: Their Manifestations from Spin-Networks to Classical Mechanisms, and Beyond

- 2017

Physics

ICCSA

Attention is dedicated to the recently pointed out connection between the quantum mechanics of spin recouplings and the Grashof analysis of four-bar linkages, with perspective implications at the molecular level.

### Spherical and Hyperbolic Spin Networks: The q-extensions of Wigner-Racah 6j Coefficients and General Orthogonal Discrete Basis Sets in Applied Quantum Mechanics

- 2017

Mathematics

ICCSA

A unified treatment of Racah recoupling coefficients or their closely related Wigner 6j symbols' q-extensions to non-Euclidean spaces: hyperbolic, for real q different from 1, and spherical, for \(q = r^{th}\) root of unity.

### Symmetric Angular Momentum Coupling, the Quantum Volume Operator and the 7-spin Network: A Computational Perspective

- 2014

Physics

ICCSA

The focus is on the quantum mechanical angular momentum theory of Wigner's 6j symbols and on the volume operator of the symmetric coupling in spin network approaches: here, crucial to this presentation are an appreciation of the role of the Racah sum rule and the simplification arising from the use of Regge symmetry.

### Projective Ponzano–Regge spin networks and their symmetries

- 2018

Mathematics

We present a novel hierarchical construction of projective spin networks of the Ponzano-Regge type from an assembling of five quadrangles up to the combinatorial 4-simplex compatible with a…

### Hypergeometric orthogonal polynomials as expansion basis sets for atomic and molecular orbitals: The Jacobi ladder

- 2019

Physics

State of The Art of Molecular Electronic Structure Computations: Correlation Methods, Basis Sets and More

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Physics

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