Simple Spin Networks as Feynman Graphs
Laurent Freidel, Kirill Krasnov
TL;DR
This work reframes simple ${\\rm SO}(D)$ spin networks as Feynman graphs on a homogeneous space using class-1 representations, enabling a propagator-based evaluation $G^{(\\rho)}(x,y)$ with $G_N^{(D)}(x,y) = {D+2N-2\\over D-2} C_N^{(D-2)/2}(x\\cdot y)$. It shows how to apply this framework to compute the large-$N$ asymptotics of a $D$-simplex amplitude by stationary phase, yielding an oscillatory term governed by the Regge action in higher dimensions, e.g. $\\phi_{(\\Gamma,\\rho)} \\sim \\cos\\left( \\sum_{k<l} (N_{kl}+p)\\theta_{kl} + \\kappa \\frac{\\pi}{4} \\right)$ with $p=(D-2)/2$ and $\\kappa={ (D+1)D \\over 2 }(4-D)$. The approach generalizes Barrett’s 4D results to arbitrary $D$, provides explicit Gegenbauer-based expressions for simple ${\\rm SO}(D)$ intertwiners and graphs (e.g. $\\Theta^{(D)}(N_1,N_2,N_3)$), and offers a versatile tool for analyzing simplex amplitudes in loop quantum gravity. This contributes a general, diagrammatic, and tractable method for studying spin-network amplitudes across dimensions, linking representation theory, harmonic analysis on spheres, and discrete gravity actions.
Abstract
We show how spin networks can be described and evaluated as Feynman integrals over an internal space. This description can, in particular, be applied to the so-called simple SO(D) spin networks that are of importance for higher-dimensional generalizations of loop quantum gravity. As an illustration of the power of the new formalism, we use it to obtain the asymptotics of an amplitude for the D-simplex and show that its oscillatory part is given by the Regge action.