A semiclassical tetrahedron

TL;DR

This paper addresses constructing semiclassical states for a quantum tetrahedron in loop quantum gravity, where geometry is quantised by SU(2) spins and intertwiners and many classical variables do not commute. The authors explicitly build semiclassical states inside the intertwiner space ${\cal I}_{j_1\ldots j_4}$ by forming Gaussian superpositions peaked at classical edge areas and dihedral angles, and they introduce phases derived from an auxiliary tetrahedron to fix the second angle. Using the asymptotics of the $\{6j\}$ symbol and Regge action, they show that the resulting states have expectation values for area, dihedral angles, and the volume operator $U$ that reproduce the classical geometry with vanishing relative uncertainties in the large-spin limit, though the states are not minimum-uncertainty coherent states. This construction provides concrete semiclassical tetrahedron states, facilitating the study of the semiclassical limit and correlations in loop quantum gravity.

Abstract

We construct a macroscopic semiclassical state state for a quantum tetrahedron. The expectation values of the geometrical operators representing the volume, areas and dihedral angles are peaked around assigned classical values, with vanishing relative uncertainties.