The perturbative Regge-calculus regime of Loop Quantum Gravity

TL;DR

The paper demonstrates a precise semiclassical correspondence between Loop Quantum Gravity on boundary states peaked at large spins and perturbative area-Regge-calculus on a single 4-simplex. By deriving an integral representation of the Barrett-Crane vertex, performing a stationary-phase expansion, and computing two- and three-area correlations, it shows these LQG correlators match a perturbative Regge calculus with a well-defined action and measure, provided one identifies $l_0=A_0/(8\pi G_N)$ and $\delta l_{mn}=\delta A_{hkl}/(8\pi G_N)$. The matching extends to the second and third derivatives of the Regge action (tensors $K$ and $I$) and necessitates a specific perturbative measure $\mu_{l_0}$, linking spin and area fluctuations to gravitational dynamics. The results offer a concrete bridge between nonperturbative LQG dynamics (BC model) and perturbative Regge gravity, and they stress the importance of intertwiner dynamics and the measure for extending the correspondence beyond area correlations, with implications for evaluating new models of quantum gravity dynamics.

Abstract

The relation between Loop Quantum Gravity and Regge calculus has been pointed out many times in the literature. In particular the large spin asymptotics of the Barrett-Crane vertex amplitude is known to be related to the Regge action. In this paper we study a semiclassical regime of Loop Quantum Gravity and show that it admits an effective description in terms of perturbative area-Regge-calculus. The regime of interest is identified by a class of states given by superpositions of four-valent spin networks, peaked on large spins. As a probe of the dynamics in this regime, we compute explicitly two- and three-area correlation functions at the vertex amplitude level. We find that they match with the ones computed perturbatively in area-Regge-calculus with a single 4-simplex, once a specific perturbative action and measure have been chosen in the Regge-calculus path integral. Correlations of other geometric operators and the existence of this regime for other models for the dynamics are briefly discussed.