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On the semiclassical limit of 4d spin foam models

Florian Conrady, Laurent Freidel

Abstract

We study the semiclassical properties of the Riemannian spin foam models with Immirzi parameter that are constructed via coherent states. We show that in the semiclassical limit the quantum spin foam amplitudes of an arbitrary triangulation are exponentially suppressed, if the face spins do not correspond to a discrete geometry. When they do arise from a geometry, the amplitudes reduce to the exponential of i times the Regge action. Remarkably, the dependence on the Immirzi parameter disappears in this limit.

On the semiclassical limit of 4d spin foam models

Abstract

We study the semiclassical properties of the Riemannian spin foam models with Immirzi parameter that are constructed via coherent states. We show that in the semiclassical limit the quantum spin foam amplitudes of an arbitrary triangulation are exponentially suppressed, if the face spins do not correspond to a discrete geometry. When they do arise from a geometry, the amplitudes reduce to the exponential of i times the Regge action. Remarkably, the dependence on the Immirzi parameter disappears in this limit.

Paper Structure

This paper contains 22 sections, 8 theorems, 99 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Path integral representation of spin foam models
  3. Semiclassical limit
  4. Classical equations
  5. Variation on interior and exterior edges
  6. Variation of $u_e$ and $n_{ef}$ and maximality
  7. Rewriting the equations
  8. Parallel transport to vertices
  9. Projection to SO(4)
  10. Discrete geometry
  11. Co--tetrads and tetrads on a simplicial complex
  12. Solutions
  13. Determination of $h$
  14. Semiclassical approximation of effective amplitude
  15. Evaluation of action
  16. ...and 7 more sections

Key Result

Proposition 6.2

Given any non--degenerate co--tetrad $E$, there is a unique SO(4) connection $\Omega$ on $\Delta^*$ that satisfies the condition We call this connection $\Omega$ the spin connection associated to $E$.

Figures (4)

  • Figure 1: (a) Face $f$ of dual complex $\Delta^*$. (b) Subdivision of face $f$ into wedges. The arrows indicate starting point and orientation for wedge holonomies.
  • Figure 2: Variation of the group variable $\mathbf{h}_{fe}$ on the edge $fe$ in the interior of the face $f$.
  • Figure 3: Variation of the group variable $\mathbf{g}_{ev}$ on the segment $ev$ between faces.
  • Figure 5: Choice of labelling at neighbouring vertices $v$ and $v'$.

Theorems & Definitions (10)

  • Definition 6.1
  • Proposition 6.2
  • Definition 6.3
  • Proposition 6.4
  • Proposition 7.1
  • Proposition 7.2
  • Proposition 8.1
  • Proposition A.1
  • Proposition B.1
  • Proposition B.2