The complete LQG propagator: I. Difficulties with the Barrett-Crane vertex

TL;DR

This work tests whether the Barrett-Crane vertex can reproduce the correct long-distance graviton propagator within loop quantum gravity by completing the non-diagonal tensor components and analyzing the intertwiner dependence. The authors construct a boundary state with explicit intertwiner structure and perform a detailed, large-$j_0$ asymptotic analysis of the propagator, finding that the BC vertex fails to produce the correct tensorial structure in the low-energy limit. They identify the root causes as intertwiner-independence and a fundamental mismatch between the Barrett-Crane framework and the canonical spin-network formalism, arguing that a different vertex is required for the correct semiclassical behavior. The results underscore that propagator calculations probe the dynamics beyond dimensional analysis and motivate alternative vertex constructions, which are explored in a companion paper II. Overall, the paper highlights the necessity of intertwiner-sensitive vertex amplitudes to recover classical general relativity at large scales and opens a path toward viable non-perturbative quantum gravity models.

Abstract

Some components of the graviton two-point function have been recently computed in the context of loop quantum gravity, using the spinfoam Barrett-Crane vertex. We complete the calculation of the remaining components. We find that, under our assumptions, the Barrett-Crane vertex does not yield the correct long distance limit. We argue that the problem is general and can be traced to the intertwiner-independence of the Barrett-Crane vertex, and therefore to the well-known mismatch between the Barrett-Crane formalism and the standard canonical spin networks. In a companion paper we illustrate the asymptotic behavior of a vertex amplitude that can correct this difficulty.