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

  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},
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… 

Symmetric coupling of angular momenta, quadratic algebras and discrete polynomials

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

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

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

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

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

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

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

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

Discrete quantum mechanics

A comprehensive review of the discrete quantum mechanics with the pure imaginary shifts and the real shifts is presented in parallel with the corresponding results in the ordinary quantum mechanics.

Simplification of the spectral analysis of the volume operator in loop quantum gravity

The volume operator plays a crucial role in the definition of the quantum dynamics of loop quantum gravity (LQG). Efficient calculations for dynamical problems of LQG can therefore be performed only

Quantum states of elementary three-geometry

We introduce a quantum volume operator K—which could play a significant role in discretized quantum gravity models—by taking into account a symmetrical coupling scheme of three SU(2) angular momenta.

Properties of the volume operator in loop quantum gravity: I. Results

We analyze the spectral properties of the volume operator of Ashtekar and Lewandowski in loop quantum gravity, which is the quantum analog of the classical volume expression for regions in

Properties of the volume operator in loop quantum gravity: II. Detailed presentation

The properties of the volume operator in loop quantum gravity, as constructed by Ashtekar and Lewandowski, are analyzed for the first time at generic vertices of valence greater than four. We find

Quantum and semiclassical spin networks: from atomic and molecular physics to quantum computing and gravity

The mathematical apparatus of quantum-mechanical angular momentum (re)coupling, developed originally to describe spectroscopic phenomena in atomic, molecular, optical and nuclear physics, is embedded

Exact and Asymptotic Computations of Elementary Spin Networks: Classification of the Quantum-Classical Boundaries

This paper provides a contribution to the understanding of the transition between two extreme modes of the 6j, corresponding to the nearly classical and the fully quantum regimes, by studying the boundary lines (caustics) in the plane of the two matrix labels.

Combinatorics of angular momentum recoupling theory: spin networks, their asymptotics and applications

The quantum theory of angular momentum and the associated Racah–Wigner algebra of the Lie group SU(2) have been widely used in many branches of theoretical and applied physics, chemical physics, and

Volume operator in discretized quantum gravity.

    Physical review letters
  • 1995
The spectral properties of the volume operator in quantum gravity are investigated in the framework of a previously introduced lattice discretization and the presence of a well-defined scalar product permits to make definite statements about the hermiticity of quantum operators.