A finiteness proof for the Lorentzian state sum spinfoam model for quantum general relativity

TL;DR

The paper proves that the normalized Lorentzian spinfoam state sum for four-dimensional quantum gravity is finite on any nondegenerate, finite triangulation when using the Perez:2001ec normalization and balanced Lorentz representations. It expresses amplitudes as traces of relativistic spin networks realized through hyperbolic-space integrals and establishes rigorous decay bounds for Θ4 and I10 amplitudes, enabling a convergent partition function via careful power counting. This finiteness supports the view of a perturbatively finite quantum theory of gravity in Lorentzian signature, while leaving open questions about degenerate triangulations and the continuum limit. The result highlights a delicate interplay between representation theory, hyperbolic geometry, and spinfoam summations, suggesting promising connections to field-theoretic and TQFT-like formulations.

Abstract

We show that the normalized Lorentzian state sum is finite on any triangulation. It thus provides a candidate for a perturbatively finite quantum theory of general relativity in four dimensions with Lorentzian signature.

Figures (3)

• Figure 1: The $\Theta_{4}$ spin net.
• Figure 2: The $I_{10}$ spin net.
• Figure 3: The $I_{6}$ spin net.

Theorems & Definitions (15)

• Lemma 1
• Lemma 2
• Lemma 3
• proof
• Corollary 1
• proof
• Lemma 4
• proof
• Lemma 5
• proof