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Causal Structure for Generalized Spinfoams

Carlos E. Beltrán

Abstract

We investigate the role of causality in the generalized EPRL spinfoam model introduced by Kaminski, Kisielowski, and Lewandowski (EPRL-KKL). We first propose a definition of causal structure on an arbitrary 2-complex and analyze how causal orientations can be assigned to its 1-skeleton and 2-skeleton. We identify a criterion that determines when an orientation of the 2-skeleton induces a consistent causal structure on the 1-skeleton. This characterization naturally involves tools from graph theory and linear algebra over the Galois field $\mathbb{F}_2$. Using these results, we introduce a causal vertex amplitude that generalizes previous proposals by Bianchi--Dussaud and Bianchi--Chen--Gamonal. We study its asymptotic behavior and revisit the interpretation of the two critical points appearing in the semiclassical analysis of the EPRL and EPRL-KKL spinfoam models. Finally, we discuss several directions in which the causal amplitude introduced here may provide new insights for future developments, like the cosine problem and the presence of irregular light cone structures.

Causal Structure for Generalized Spinfoams

Abstract

We investigate the role of causality in the generalized EPRL spinfoam model introduced by Kaminski, Kisielowski, and Lewandowski (EPRL-KKL). We first propose a definition of causal structure on an arbitrary 2-complex and analyze how causal orientations can be assigned to its 1-skeleton and 2-skeleton. We identify a criterion that determines when an orientation of the 2-skeleton induces a consistent causal structure on the 1-skeleton. This characterization naturally involves tools from graph theory and linear algebra over the Galois field . Using these results, we introduce a causal vertex amplitude that generalizes previous proposals by Bianchi--Dussaud and Bianchi--Chen--Gamonal. We study its asymptotic behavior and revisit the interpretation of the two critical points appearing in the semiclassical analysis of the EPRL and EPRL-KKL spinfoam models. Finally, we discuss several directions in which the causal amplitude introduced here may provide new insights for future developments, like the cosine problem and the presence of irregular light cone structures.
Paper Structure (17 sections, 40 equations, 2 figures)

This paper contains 17 sections, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Discrete Causal Structure
  3. Discretization
  4. Discrete causality
  5. Causality in Minkowski spacetime
  6. Causal structure on $\mathcal{C}$
  7. Causal Structure in the 2-skeleton
  8. Example: The graph $K_{6}$ with three lines removed
  9. Causal structure of the boundary graph
  10. The EPRL-KKL Model and its Causal structure
  11. The EPRL-KKL Spinfoam Vertex
  12. Causal structure in the EPRL-KKL model
  13. The Causal EPRL-KKL Spinfoam Vertex
  14. Discussion
  15. Changes of signature
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Causal graph of a 5-valent vertex. To keep the picture clean, we we explicitly label only the external links.
  • Figure 2: Causal graph of a 6-valent vertex. As in Figure 1, we explicitly display only the labels of the external links.