Quantum states of elementary three-geometry

Abstract

We introduce a quantum volume operator $K$ in three--dimensional Quantum Gravity by taking into account a symmetrical coupling scheme of three SU(2) angular momenta. The spectrum of $K$ is discrete and defines a complete set of eigenvectors which is alternative with respect to the complete sets employed when the usual binary coupling schemes of angular momenta are considered. Each of these states, that we call quantum bubbles, represents an interference of extended configurations which provides a rigorous meaning to the heuristic notion of quantum tetrahedron. We study the generalized recoupling coefficients connecting the symmetrical and the binary basis vectors, and provide an explicit recursive solution for such coefficients by analyzing also its asymptotic limit.

15 pages, LaTex

References in corpus (2)

• Discrete structures in gravity
• Implementing holographic projections in Ponzano--Regge gravity

Cited by in corpus (13)

• Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe
• Discreteness of the volume of space from Bohr-Sommerfeld quantization
• Semiclassical Mechanics of the Wigner 6j-Symbol
• Properties of the Volume Operator in Loop Quantum Gravity I: Results
• Bohr-Sommerfeld Quantization of Space
• A semiclassical tetrahedron
• Properties of the Volume Operator in Loop Quantum Gravity II: Detailed Presentation
• Hamiltonian dynamics of a quantum of space: hidden symmetries and spectrum of the volume operator, and discrete orthogonal polynomials
• Symmetric angular momentum coupling, the quantum volume operator and the 7-spin network: a computational perspective
• 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