Quantum Curved Tetrahedron, Quantum Group Intertwiner Space, and Coherent States

TL;DR

This paper develops a quantum model of curved tetrahedra with fixed areas, introducing a phase space with length and twist coordinates, and constructs coherent states relevant for Loop Quantum Gravity with a cosmological constant.

Contribution

It constructs the phase space and coherent states of quantum curved tetrahedra, linking quantum group intertwiners to Loop Quantum Gravity with a cosmological constant.

Findings

Quantum length operators form a fusion algebra.

Coherent states peak at classical phase space points.

Model relates to 3+1D Loop Quantum Gravity with cosmological constant.

Abstract

In this paper, we construct the phase space of a constantly curved tetrahedron with fixed triangle areas in terms of a pair of Darboux coordinates called the length and twist coordinates, which are in analogy to the Fenchel-Nielsen coordinates for flat connections, and their quantization. The curvature is identified to the value of the cosmological constant, either positive or negative. The physical Hilbert space is given by the $\mathcal{U}_q(\mathfrak{su}(2))$ intertwiner space. We show that the quantum trace of quantum monodromies, defining the quantum length operators, form a fusion algebra and describe their representation theory. We also construct the coherent states in the physical Hilbert space labeled by the length and twist coordinates. These coherent states describe quantum curved tetrahedra and peak at points of the tetrahedron phase space. This works is closely related to 3+1 dimensional Loop Quantum Gravity with a non-vanishing cosmological constant. The coherent states constructed herein serve as good candidates for the application to the spinfoam model with a cosmological constant.