Three Dimensional Loop Quantum Gravity: Particles and the Quantum Double

TL;DR

This paper demonstrates how the quantum double algebraic structure naturally arises in three-dimensional Riemannian loop quantum gravity coupled with massive particles, revealing a deep connection between quantum gravity and algebraic structures.

Contribution

It shows that the physical Hilbert space in 3D LQG with particles is isomorphic to tensor products of simple representations of the Drinfeld double DSU(2), generalizing previous results.

Findings

Hilbert space is isomorphic to tensor products of simple DSU(2) representations

Particle masses label simple representations in the quantum double

Results extend from sphere to arbitrary Riemann surfaces

Abstract

It is well known that the quantum double structure plays an important role in three dimensional quantum gravity coupled to matter field. In this paper, we show how this algebraic structure emerges in the context of three dimensional Riemannian loop quantum gravity (LQG) coupled to a finite number of massive spinless point particles. In LQG, physical states are usually constructed from the notion of SU(2) cylindrical functions on a Riemann surface $\Sigma$ and the Hilbert structure is defined by the Ashtekar-Lewandowski measure. In the case where $\Sigma$ is the sphere $S^2$, we show that the physical Hilbert space is in fact isomorphic to a tensor product of simple unitary representations of the Drinfeld double DSU(2): the masses of the particles label the simple representations, the physical states are tensor products of vectors of simple representations and the physical scalar product is given by intertwining coefficients between simple representations. This result is generalized to the case of any Riemann surface $\Sigma$.