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Asymptotic analysis of the EPRL four-simplex amplitude

John W. Barrett, R. J. Dowdall, Winston J. Fairbairn, Henrique Gomes, Frank Hellmann

Abstract

The semiclassical limit of a 4-simplex amplitude for a spin foam quantum gravity model with an Immirzi parameter is studied. If the boundary state represents a non-degenerate 4-simplex geometry, the asymptotic formula contains the Regge action for general relativity. A canonical choice of phase for the boundary state is introduced and is shown to be necessary to obtain the results.

Asymptotic analysis of the EPRL four-simplex amplitude

Abstract

The semiclassical limit of a 4-simplex amplitude for a spin foam quantum gravity model with an Immirzi parameter is studied. If the boundary state represents a non-degenerate 4-simplex geometry, the asymptotic formula contains the Regge action for general relativity. A canonical choice of phase for the boundary state is introduced and is shown to be necessary to obtain the results.

Paper Structure

This paper contains 34 sections, 9 theorems, 142 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The EPRL four-simplex amplitude
  3. Posing the asymptotic problem
  4. Coherent states
  5. Coherent tetrahedra
  6. Coherent boundary 3D manifolds
  7. Boundary data
  8. Regge-like boundary data
  9. Regge states
  10. Exponentiated expression for the four-simplex amplitude
  11. $\gamma < 1$ case.
  12. $\gamma > 1$ case.
  13. Symmetries of the action.
  14. Asymptotic formula
  15. Proof of the asymptotic formula
  16. ...and 19 more sections

Key Result

Theorem 1

(Asymptotic formula) Given a set $\mathcal{B}=\{ \mathbf n_{ab} , k_{ab} \}_{a \neq b}$ of boundary data, then in the limit $\lambda \rightarrow \infty$

Figures (4)

  • Figure 1: The $\mathrm{Spin}(4)$ intertwiner $\iota$.
  • Figure 2: The propagator $\mathcal{P}_{ab}$ for a single edge.
  • Figure 3: The three-valent intertwiner $C^{j_{ab}^- j_{ab}^+}_{k_{ab}}$ for the case $\gamma <1$ showing how the projection to the highest spin subspace $k_{ab}$ makes two of the symmetrizers redundant.
  • Figure 4: The three-valent intertwiner $C^{j_{ab}^- j_{ab}^+}_{k_{ab}}$ for the case $\gamma >1$ projecting to the lowest weight state $k_{ab} = j^+_{ab} - j^+_{ab}$.

Theorems & Definitions (9)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Lemma 2
  • Theorem 3
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6