LQG propagator: III. The new vertex

TL;DR

The paper addresses the failure of the Barrett-Crane spinfoam vertex to reproduce the correct graviton propagator in the low-energy limit and argues that the corrected EPRL-type vertex, whose large-spin asymptotics have been recently analyzed, has the exact asymptotic structure needed to rectify this. By translating the Barrett-Fairbairn asymptotics into the intertwiner basis, the work shows the crucial phase factor $e^{i\frac{\pi}{2}k}$ arises, providing the necessary $\phi_n = -\frac{\pi}{2}$ offset that yields the correct semiclassical propagator when combined with an appropriate boundary state. The results support the viability of the new vertex for recovering the Regge-geometry semiclassical limit, and motivate redoing the graviton propagator calculation using Livine-Speziale coherent states. Future work includes computing higher-point functions and addressing remaining issues on gauge invariance and finiteness.

Abstract

In the first article of this series, we pointed out a difficulty in the attempt to derive the low-energy behavior of the graviton two-point function, from the loop-quantum-gravity dynamics defined by the Barrett-Crane vertex amplitude. Here we show that this difficulty disappears when using the corrected vertex amplitude recently introduced in the literature. In particular, we show that the asymptotic analysis of the new vertex amplitude recently performed by Barrett, Fairbairn and others, implies that the vertex has precisely the asymptotic structure that, in the second article of this series, was indicated as the key necessary condition for overcoming the difficulty.