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Three Dimensional Loop Quantum Gravity: Particles and the Quantum Double

Karim Noui

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 $Σ$ and the Hilbert structure is defined by the Ashtekar-Lewandowski measure. In the case where $Σ$ 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 $Σ$.

Three Dimensional Loop Quantum Gravity: Particles and the Quantum Double

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 and the Hilbert structure is defined by the Ashtekar-Lewandowski measure. In the case where is the sphere , 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 .

Paper Structure

This paper contains 13 sections, 6 theorems, 100 equations, 8 figures.

Table of Contents

  1. 1. Motivations
  2. 2. Space of Physical States
  3. 3. Transition amplitudes and the Quantum double
  4. 4. Conclusion and Perspectives

Key Result

Proposition 1

The Hilbert space of partial kinematical states ${\cal H}_{Pkin}$ is isomorphic to the space $Fun(G^{2g}\otimes (S^2)^{n-1})$ endowed with the measure $d\mu_G^{\otimes 2g} \otimes d\mu_{S^2}^{\otimes (n-1)}$. Indeed, a function $\psi \in {\cal H}(\gamma;G)$ is a partial kinematical states if and onl where $\tilde{\Lambda}_e = \int dh \; \Lambda_e h \in S^2$ and $dh$ is the measure on the Cartan to

Figures (8)

  • Figure 1: The picture on the left is a spin-network state whose edges are labelled with irreps of $SU(2)$ and vertices with normalized $SU(2)$ intertwiners. On the right side, we have drawn the spin-foam picture illustrating the transition amplitude between the no-state and the state on the left. The amplitude associated to this spin-foam is a $(6j)$ symbol.
  • Figure 2: On the l.h.s. chunk of spin-foam amplitude between two spin-network states represented in thick lines: the boundary crosses the face labelled with the irrep $j_4$. On the r.h.s. the dual picture: the circle around the edge $j_4$ reminds the presence of the boundary.
  • Figure 3: Example of transition between spin-network states involving explicitely boundaries degrees of freedom. New faces can emerge from the boundaries.
  • Figure 4: A minimal graph on a genus 2 surface with two particles. On the r.h.s., the "flat projection" of the minimal graph with the labellings of each edge or loop.
  • Figure 5: Pictorial illustration of the expression (\ref{['vev']}). The dot colored with the mass $m$ denotes insertion of the element $h(m)$; any trivalent vertex denotes $(3j)$ coefficients of $SU(2)$; each edge is colored with an irrep of $SU(2)$ and can end up with a magnetic number like $a_i$ or $0$.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Proposition 1: Partial kinematical Hilbert space
  • Proposition 2: Partial physical Hilbert space
  • Theorem 1: The Drinfeld double in LQG
  • Proposition 3: Reduction of the genus
  • Theorem 2: Reduction to the case of the sphere
  • Proposition 4: Inclusions