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Observables in Loop Quantum Gravity with a cosmological constant

Maïté Dupuis, Florian Girelli

TL;DR

The paper shows how a nonzero cosmological constant can be incorporated into loop quantum gravity by inserting the real quantum group U_q({\mathfrak su}(2)) into spin networks. It develops a tensor-operator framework that yields a full set of intertwiner observables, and demonstrates that these observables generate a q-deformed U_q({\mathfrak u}(n)) structure acting on intertwiners, thereby encoding curved (hyperbolic) discrete geometries. Explicit realizations of rank-1/2 and rank-1 tensor operators are given via q-harmonic oscillators, enabling construction of geometric operators such as length, area, and angle, which realize a quantum hyperbolic triangle in 3d Euclidean LQG with Λ<0 (and informing the 4d Lorentzian case Λ>0). The results provide evidence that quantum-group methods offer a consistent mechanism to incorporate Λ into LQG and bridge to spinfoam models like Turaev-Viro through a q-deformed geometric observable framework.

Abstract

An open issue in loop quantum gravity (LQG) is the introduction of a non-vanishing cosmological constant $Λ$. In 3d, Chern-Simons theory provides some guiding lines: $Λ$ appears in the quantum deformation of the gauge group. The Turaev-Viro model, which is an example of spin foam model is also defined in terms of a quantum group. By extension, it is believed that in 4d, a quantum group structure could encode the presence of $Λ\neq0$. In this article, we introduce by hand the quantum group $\mathcal{U}_{q}(\mathfrak{su}(2))$ into the LQG framework, that is we deal with $\mathcal{U}_{q}(\mathfrak{su}(2))$-spin networks. We explore some of the consequences, focusing in particular on the structure of the observables. Our fundamental tools are tensor operators for $\mathcal{U}_{q}(\mathfrak{su}(2))$. We review their properties and give an explicit realization of the spinorial and vectorial ones. We construct the generalization of the U($n$) formalism in this deformed case, which is given by the quantum group $\mathcal{U}_{q}(\mathfrak{u}(n))$. We are then able to build geometrical observables, such as the length, area or angle operators ... We show that these operators characterize a quantum discrete hyperbolic geometry in the 3d LQG case. Our results confirm that the use of quantum group in LQG can be a tool to introduce a non-zero cosmological constant into the theory.

Observables in Loop Quantum Gravity with a cosmological constant

TL;DR

The paper shows how a nonzero cosmological constant can be incorporated into loop quantum gravity by inserting the real quantum group U_q({\mathfrak su}(2)) into spin networks. It develops a tensor-operator framework that yields a full set of intertwiner observables, and demonstrates that these observables generate a q-deformed U_q({\mathfrak u}(n)) structure acting on intertwiners, thereby encoding curved (hyperbolic) discrete geometries. Explicit realizations of rank-1/2 and rank-1 tensor operators are given via q-harmonic oscillators, enabling construction of geometric operators such as length, area, and angle, which realize a quantum hyperbolic triangle in 3d Euclidean LQG with Λ<0 (and informing the 4d Lorentzian case Λ>0). The results provide evidence that quantum-group methods offer a consistent mechanism to incorporate Λ into LQG and bridge to spinfoam models like Turaev-Viro through a q-deformed geometric observable framework.

Abstract

An open issue in loop quantum gravity (LQG) is the introduction of a non-vanishing cosmological constant . In 3d, Chern-Simons theory provides some guiding lines: appears in the quantum deformation of the gauge group. The Turaev-Viro model, which is an example of spin foam model is also defined in terms of a quantum group. By extension, it is believed that in 4d, a quantum group structure could encode the presence of . In this article, we introduce by hand the quantum group into the LQG framework, that is we deal with -spin networks. We explore some of the consequences, focusing in particular on the structure of the observables. Our fundamental tools are tensor operators for . We review their properties and give an explicit realization of the spinorial and vectorial ones. We construct the generalization of the U() formalism in this deformed case, which is given by the quantum group . We are then able to build geometrical observables, such as the length, area or angle operators ... We show that these operators characterize a quantum discrete hyperbolic geometry in the 3d LQG case. Our results confirm that the use of quantum group in LQG can be a tool to introduce a non-zero cosmological constant into the theory.

Paper Structure

This paper contains 21 sections, 5 theorems, 118 equations, 1 figure.

Table of Contents

  1. ${{\mathcal{U}}_{q}({\mathfrak su}(2))}$ in a nutshell
  2. Definition of ${{\mathcal{U}}_{q}({\mathfrak su}(2))}$
  3. $q$-harmonic oscillators and the Schwinger-Jordan trick
  4. Tensor operators for ${{\mathcal{U}}_{q}({\mathfrak su}(2))}$
  5. Definition and Wigner-Eckart theorem
  6. Product of tensor operators: scalar product, vector product and triple product
  7. Scalar product
  8. Vector product
  9. Tensor products of tensor operators
  10. Realization of tensor operators of rank $1/2$ and $1$ for ${{\mathcal{U}}_{q}({\mathfrak su}(2))}$
  11. Rank $1/2$ tensor operators
  12. Rank $1$ tensor operators
  13. Observables for the intertwiner space
  14. General construction and properties of intertwiner observables
  15. ${{\mathcal{U}}_{q}({\mathfrak u}(n))}$ formalism for LQG defined over ${{\mathcal{U}}_{q}({\mathfrak su}(2))}$
  16. ...and 6 more sections

Key Result

Lemma 1.1

Let ${\mathcal{C}}\in {{\mathcal{U}}_{q}({\mathfrak su}(2))}$ invariant under the adjoint action, then ${\mathcal{C}}$ commutes with the generators $J_{\sigma}$, $\sigma=+,-,z$. Conversely, if ${\mathcal{C}}\in {{\mathcal{U}}_{q}({\mathfrak su}(2))}$ commutes with $J_{\sigma}$, then it is invariant

Figures (1)

  • Figure 1: The hyperbolic triangle is represented in the Poincaré disc. The (outgoing) normals $\hat{n}_i$ are defined in the tangent plane at the vertex of the triangle, as the orthogonal vectors to the tangent vectors $\hat{u}_i$.

Theorems & Definitions (6)

  • Lemma 1.1
  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5