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Asymptotics of 6j and 10j symbols

Laurent Freidel, David Louapre

Abstract

It is well known that the building blocks for state sum models of quantum gravity are given by 6j and 10j symbols. In this work we study the asymptotics of these symbols by using their expressions as group integrals. We carefully describe the measure involved in terms of invariant variables and develop new technics in order to study their asymptotics. Using these technics we compute the asymptotics of the various Euclidean and Lorentzian 6j-symbols. Finally we compute the asymptotic expansion of the 10j symbol which is shown to be non-oscillating in agreement with a recent result of Baez et al. We discuss the physical origin of this behavior and a way to modify the Barrett-Crane model in order to cure this disease.

Asymptotics of 6j and 10j symbols

Abstract

It is well known that the building blocks for state sum models of quantum gravity are given by 6j and 10j symbols. In this work we study the asymptotics of these symbols by using their expressions as group integrals. We carefully describe the measure involved in terms of invariant variables and develop new technics in order to study their asymptotics. Using these technics we compute the asymptotics of the various Euclidean and Lorentzian 6j-symbols. Finally we compute the asymptotic expansion of the 10j symbol which is shown to be non-oscillating in agreement with a recent result of Baez et al. We discuss the physical origin of this behavior and a way to modify the Barrett-Crane model in order to cure this disease.

Paper Structure

This paper contains 16 sections, 11 theorems, 152 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Asymptotics of the 6j-symbol
  3. Integral expression for the square of the 6j-symbol
  4. Analysis of the integral
  5. Asymptotic of $I^>(Nl_{IJ})$
  6. Asymptotics of $I^<(Nl_{IJ})$
  7. Lorentzian 6j symbol
  8. Spacelike tetrahedra
  9. Timelike tetrahedra
  10. General method for the stationarity equations
  11. 10j symbol
  12. Integral expression for the 10j-symbol
  13. Asymptotics for the 10j symbol
  14. Discussion
  15. Geometry of the Euclidean and Lorentzian flat tetrahedron
  16. ...and 1 more sections

Key Result

Theorem 1

where ${\mathcal{D}}_{\pi}$ is the subset of $[0;\pi]^6$ of angles satisfying the relations : for any triple $(I,J,K)$ of distinct elements. Geometrically this domain is the set of all possible spherical tetrahedra.

Figures (1)

  • Figure 1: Dual angles in a spherical tetrahedron. The dual $\,{\tilde{\!\tilde{\theta}}}_{23}$ of $\tt_{23}$ in the dashed spherical triangle appears as the dihedral angle of the edge 01.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Theorem 3
  • Proposition 6
  • Proposition 7
  • ...and 1 more