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Cosmological Deformation of Lorentzian Spin Foam Models

Karim Noui, Philippe Roche

Abstract

We study the quantum deformation of the Barrett-Crane Lorentzian spin foam model which is conjectured to be the discretization of Lorentzian Plebanski model with positive cosmological constant and includes therefore as a particular sector quantum gravity in de-Sitter space. This spin foam model is constructed using harmonic analysis on the quantum Lorentz group. The evaluation of simple spin networks are shown to be non commutative integrals over the quantum hyperboloid defined as a pile of fuzzy spheres. We show that the introduction of the cosmological constant removes all the infrared divergences: for any fixed triangulation, the integration over the area variables is finite for a large class of normalization of the amplitude of the edges and of the faces.

Cosmological Deformation of Lorentzian Spin Foam Models

Abstract

We study the quantum deformation of the Barrett-Crane Lorentzian spin foam model which is conjectured to be the discretization of Lorentzian Plebanski model with positive cosmological constant and includes therefore as a particular sector quantum gravity in de-Sitter space. This spin foam model is constructed using harmonic analysis on the quantum Lorentz group. The evaluation of simple spin networks are shown to be non commutative integrals over the quantum hyperboloid defined as a pile of fuzzy spheres. We show that the introduction of the cosmological constant removes all the infrared divergences: for any fixed triangulation, the integration over the area variables is finite for a large class of normalization of the amplitude of the edges and of the faces.

Paper Structure

This paper contains 18 sections, 4 theorems, 140 equations, 7 figures.

Table of Contents

  1. I. Introduction
  2. II. Spin foam models, quantum gravity and cosmological constant
  3. III. Quantum Hyperboloid and Quantum BC intertwiner.
  4. IV. Quantum Lorentzian Simple Spin Networks
  5. V. q-Lorentzian Spin Foam Models
  6. VI. Conclusion
  7. Appendix

Key Result

Proposition 1

Let $\alpha = (\alpha_1,\cdots, \alpha_n)$ a family of simple representations, let $A=(A_1,\cdots,A_n)$ a family of spins and $a=(a_1,\cdots,a_n)$ the magnetic numbers. We denote by $\stackrel{A}{e}_{a}[\alpha] = \bigotimes_{i=1}^n \stackrel{A_i}{e}_{a_i}\!\!(\alpha_i)$ and we have: where The series (sum) converges when $n\geq 3$ and in this case the q-BC Lorentzian intertwiner (QLI) is well de

Figures (7)

  • Figure 1: Diagramatic representation of the $15j$-symbol.
  • Figure 2: This graph represents the evaluation ${\cal Z}(ijklm)(\alpha \otimes \beta \otimes \gamma \otimes \delta \otimes \epsilon)$ where $(\alpha, \beta , \gamma, \delta, \epsilon)$ is a family of intertwiners.
  • Figure 3: The q-BC Lorentzian intertwiner and its generalization.
  • Figure 4: Invariance of the Quantum Lorentzian intertwiner by braidings.
  • Figure 5: The graph function associated to $\Gamma_2$.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Definition 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4