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

TL;DR

This paper completes the calculation of the graviton two-point function in loop quantum gravity using the Barrett-Crane vertex and finds it does not produce the correct long-distance limit, highlighting fundamental issues with the vertex's structure.

Contribution

It identifies the limitations of the Barrett-Crane vertex in reproducing correct semiclassical behavior and links this to its intertwiner-independence, proposing a direction for improved models.

Findings

Barrett-Crane vertex fails to produce correct long-distance limit

Intertwiner-independence causes mismatch with canonical spin networks

Suggests alternative vertex amplitudes can address this issue

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.