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Boundary terms in the Barrett-Crane spin foam model and consistent gluing

Daniele Oriti

TL;DR

The work investigates how the Barrett-Crane spin foam amplitudes for quantum gravity emerge from a discretized BF theory with explicit boundary terms. By analyzing boundary conditions that fix either the connection or the B field (metric) on the boundary and performing the gluing of 4-simplices, the authors demonstrate that the Perez-Rovelli version of the Barrett-Crane model is the consistent outcome, with boundary terms correctly accounted for. They also explore several alternatives—including projections on exposed edges and variations in how the simplicity and gauge invariance constraints are imposed—and show that many lead to gauge-invariance breaking or inconsistent gluing, while non-minimal multi-projection constructions are theoretically possible. Overall, the paper clarifies the role of boundary data in spin foam amplitudes and confirms the robustness of the Perez-Rovelli construction within the explored framework, while outlining directions for non-minimal extensions and further consistency checks in both Euclidean and Lorentzian settings.

Abstract

We extend the lattice gauge theory-type derivation of the Barrett-Crane spin foam model for quantum gravity to other choices of boundary conditions, resulting in different boundary terms, and re-analyze the gluing of 4-simplices in this context. This provides a consistency check of the previous derivation. Moreover we study and discuss some possible alternatives and variations that can be made to it and the resulting models.

Boundary terms in the Barrett-Crane spin foam model and consistent gluing

TL;DR

The work investigates how the Barrett-Crane spin foam amplitudes for quantum gravity emerge from a discretized BF theory with explicit boundary terms. By analyzing boundary conditions that fix either the connection or the B field (metric) on the boundary and performing the gluing of 4-simplices, the authors demonstrate that the Perez-Rovelli version of the Barrett-Crane model is the consistent outcome, with boundary terms correctly accounted for. They also explore several alternatives—including projections on exposed edges and variations in how the simplicity and gauge invariance constraints are imposed—and show that many lead to gauge-invariance breaking or inconsistent gluing, while non-minimal multi-projection constructions are theoretically possible. Overall, the paper clarifies the role of boundary data in spin foam amplitudes and confirms the robustness of the Perez-Rovelli construction within the explored framework, while outlining directions for non-minimal extensions and further consistency checks in both Euclidean and Lorentzian settings.

Abstract

We extend the lattice gauge theory-type derivation of the Barrett-Crane spin foam model for quantum gravity to other choices of boundary conditions, resulting in different boundary terms, and re-analyze the gluing of 4-simplices in this context. This provides a consistency check of the previous derivation. Moreover we study and discuss some possible alternatives and variations that can be made to it and the resulting models.

Paper Structure

This paper contains 7 sections, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Derivation of the Barrett-Crane spin foam model: constraining and gluing
  3. Fixing the boundary metric
  4. Exploring alternatives
  5. Projections on the exposed edges
  6. Imposing the projections differently
  7. Conclusions

Figures (2)

  • Figure 1: Fig.1 - A wedge (the part of a dual face belonging to a single 4-simplex) with the D-functions for the group elements assigned to its boundary edges.
  • Figure 2: Structure of the boundary term corresponding to a single tetrahedron, i.e. a single dual edge with the 4 wedges incident to it, and the corresponding 4 exposed edges.