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A spin foam model without bubble divergences

Alejandro Perez, Carlo Rovelli

TL;DR

The paper addresses the problem of bubble divergences in spin foam quantum gravity by proposing a spin foam model that modifies BF theory with a Barrett-Crane–type constraint implemented in the interaction term. The authors develop both group-field-theory (coordinate) and momentum-space formalisms, establishing finiteness of the fundamental bubble amplitudes (notably the 5-bubble and 1-bubble) through explicit bounds on $6j$ and Barrett-Crane intertwiners. They provide detailed combinatorial and integral analyses (including gauge fixing and lemma-based bounds) to show that the vacuum bubble amplitudes are finite, and argue that all-order finiteness is plausible. The work suggests a promising route toward a finite, background-independent quantum gravity model that preserves the BF-to-Einstein-Hilbert constraint structure, with potential implications for the classical limit and broader spin foam divergences.

Abstract

We present a spin foam model in which the fundamental ``bubble amplitudes'' (the analog of the one-loop corrections in quantum field theory) are finite as the cutoff is removed. The model is a natural variant of the field theoretical formulation of the Barrett-Crane model. As the last, the model is a quantum BF theory plus an implementation of the constraint that reduces BF theory to general relativity. We prove that the fundamental bubble amplitudes are finite by constructing an upper bound, using certain inequalities satisfied by the Wigner (3n)j-symbols, which we derive in the paper. Finally, we present arguments in support of the conjecture that the bubble diagrams of the model are finite at all orders.

A spin foam model without bubble divergences

TL;DR

The paper addresses the problem of bubble divergences in spin foam quantum gravity by proposing a spin foam model that modifies BF theory with a Barrett-Crane–type constraint implemented in the interaction term. The authors develop both group-field-theory (coordinate) and momentum-space formalisms, establishing finiteness of the fundamental bubble amplitudes (notably the 5-bubble and 1-bubble) through explicit bounds on and Barrett-Crane intertwiners. They provide detailed combinatorial and integral analyses (including gauge fixing and lemma-based bounds) to show that the vacuum bubble amplitudes are finite, and argue that all-order finiteness is plausible. The work suggests a promising route toward a finite, background-independent quantum gravity model that preserves the BF-to-Einstein-Hilbert constraint structure, with potential implications for the classical limit and broader spin foam divergences.

Abstract

We present a spin foam model in which the fundamental ``bubble amplitudes'' (the analog of the one-loop corrections in quantum field theory) are finite as the cutoff is removed. The model is a natural variant of the field theoretical formulation of the Barrett-Crane model. As the last, the model is a quantum BF theory plus an implementation of the constraint that reduces BF theory to general relativity. We prove that the fundamental bubble amplitudes are finite by constructing an upper bound, using certain inequalities satisfied by the Wigner (3n)j-symbols, which we derive in the paper. Finally, we present arguments in support of the conjecture that the bubble diagrams of the model are finite at all orders.

Paper Structure

This paper contains 15 sections, 89 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Definition of the model
  3. TOCY model as a QFT over a group manifold
  4. The Barrett-Crane model in coordinate space
  5. The new model
  6. Bubble amplitudes
  7. Fundamental bubbles and Pachner moves
  8. The 5-bubble amplitude
  9. The 1-bubble amplitude
  10. Spin foam formulation
  11. Bubbles in the spin foam formulation
  12. Conclusion
  13. Acknowledgments
  14. Irreducible representations of $SO(4)$
  15. Some lemmas

Figures (5)

  • Figure 1: The structure of the propagator.
  • Figure 2: The structure of the interaction vertex.
  • Figure 3: The 1-5 Pachner move, on the right the fundamental 5-bubble Feynman diagram.
  • Figure 4: The 2-4 Pachner move, on the right the fundamental 1-bubble Feynman diagram.
  • Figure 5: Two possible terms contributing to the ${\em A}_5$ amplitude