Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model

TL;DR

This work develops a background-independent perturbative framework for 3d quantum gravity within the spinfoam (Ponzano–Regge) setting by expressing the graviton propagator on a single tetrahedron as a pair of $SU(2)$ group integrals. Using exact group-integral representations for the isosceles $6j$-symbol and carefully chosen boundary states, the authors perform a complete saddle-point expansion in the boundary length scale $d_{j_t}$, obtaining explicit leading, next-to-leading, and next-to-next-to-leading order corrections to the two-point function $W_{1122}$. In parallel, they derive the corresponding perturbative expansion of the isosceles $6j$-symbol itself, uncovering a clear isosceles-specific refinement of the Ponzano–Regge asymptotics in terms of alternating cosine and sine phases of the Regge action $S_R$, with a computable polynomial coefficient $P_1(k_1,k_2)$. The results corroborate with numerical simulations and illustrate how a boundary-state–driven, group-integral formulation can yield the full perturbative graviton propagator in this non-perturbative quantum gravity setting, offering a blueprint for extending to 4d spinfoams and more complex triangulations.

Abstract

We consider an exact expression for the 6j-symbol for the isosceles tetrahedron, involving integrals over SU(2), and use it to write the two-point function of 3d gravity on a single tetrahedron as a group integral. The perturbative expansion of this expression is then performed with respect to the boundary geometry using a simple saddle-point analysis. We derive the complete expansion in inverse powers of the length scale and evaluate explicitly the quantum corrections up to second order. Finally, we use the same method to provide the complete expansion of the isosceles 6j-symbol with the explicit phases at all orders and the next-to-leading correction to the Ponzano-Regge asymptotics.

Figures (3)

• Figure 1: Physical setting to compute the 2-point function. The two edges whose correlations of length fluctuations will be computed are in fat lines, and have length $j_1+\frac{1}{2}$ and $j_2+\frac{1}{2}$. These data are encoded in the boundary state of the tetrahedron. In the time-gauge setting, the four bulk edges have imposed lengths $j_t+\frac{1}{2}$ interpreted as the proper time of a particle propagating along one of these edges. Equivalently, the time between two planes containing $e_1$ and $e_2$ has been measured to be $T=(j_t+\frac{1}{2})/\sqrt{2}$.
• Figure 2: Log-log plots comparing numerical simulations with analytical results. Left plot: a numerical simulation of \ref{['exact propagator']} (diamond symbol) compared with the leading order of \ref{['equilateral NLO']} (star symbol). Middle plot: the next to leading order of \ref{['equilateral NLO']}, in star shape, with the numerics, in diamond shape. Right plot: the next to next to leading order \ref{['equilateral NNLO']}.
• Figure 3: Differences between the 6j-symbol and the analytical result \ref{['6j NLO']} for three pairs $(k_1,k_2)=(k,k)$: from left to right, $k=1/2$, $k=3/14$, $k=3/42$. The X axis stands for $d_{j_{t}}=Nd_{j_{0}}$, for $d_{j_{0}}$ respectively fixed to 1, 7, and 21, while $N$ goes from 200 to 800.