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The EPRL amplitude is supported on flat connections

Carlos E. Beltrán, José A. Zapata

Abstract

For the version of the EPRL model based on the original vertex amplitude and the face amplitude selected by its gluing properties, we prove that the EPRL amplitude of any region with the topology of a 4-ball is supported on flat connections. We state immediate consequences of this result, comment on some applications, and discuss physical implications. The results hold in general; they do not rely on a semiclassical analysis.

The EPRL amplitude is supported on flat connections

Abstract

For the version of the EPRL model based on the original vertex amplitude and the face amplitude selected by its gluing properties, we prove that the EPRL amplitude of any region with the topology of a 4-ball is supported on flat connections. We state immediate consequences of this result, comment on some applications, and discuss physical implications. The results hold in general; they do not rely on a semiclassical analysis.
Paper Structure (7 sections, 3 theorems, 9 equations)

This paper contains 7 sections, 3 theorems, 9 equations.

Table of Contents

  1. Introduction: The EPRL spin foam model
  2. Local factorization of spin foam amplitudes
  3. The support of EPRL's amplitude
  4. Immediate consequences
  5. General form of atomic amplitudes supported on flat connections
  6. Outlook and discussion
  7. Acknowledgements

Key Result

Theorem 1

The support of EPRL's atomic amplitude $W_\nu$ is the set of flat connections ${\cal A}_{\partial \nu}^{\rm Flat} \subset {\cal A}_{\partial \nu}$.

Theorems & Definitions (3)

  • Theorem 1
  • Corollary 1
  • Corollary 2