Quantum states of elementary three-geometry

TL;DR

This paper introduces a new quantum volume operator in three-dimensional quantum gravity, defining quantum states called bubbles that represent quantum tetrahedra with a discrete spectrum, offering a novel basis for understanding quantum geometry.

Contribution

It develops a symmetric coupling scheme for SU(2) angular momenta, providing a complete set of quantum states and recoupling coefficients for quantum tetrahedra in 3D quantum gravity.

Findings

Spectrum of the quantum volume operator is discrete.

Quantum bubbles form a new basis for quantum tetrahedra.

Explicit recursive solutions for recoupling coefficients are provided.

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.