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Towards the graviton from spinfoams: the 3d toy model

Simone Speziale

TL;DR

This work examines extracting the linearised graviton 2-point function from spinfoam amplitudes using a 3D toy model based on a single tetrahedron with Ponzano–Regge dynamics. By employing a Gaussian boundary state peaked on flat geometry, and analyzing the semi-classical limit $j_0 o ty$ with $ ext{ell}= ext{ell}_{ m P} j$, the boundary-projected propagator reduces to a harmonic-oscillator kernel with frequency $oldsymbol{ ext{omega}}$, reproducing the expected large-scale scaling $W( ext{ell}) o 3/(2 ext{ell})$. The study also reveals short-distance quantum corrections that manifest as inverse powers of $ ext{ell}$ and are gauge-dependent, underscoring the need to extend the model to multiple tetrahedra to access the full tensor structure and diffeomorphism symmetry. These results provide a clearer link between boundary data in spinfoams and propagation, offering guidance for extending to the 4D case where a true graviton propagator would be sought.

Abstract

Recently, a proposal has appeared for the extraction of the 2-point function of linearised quantum gravity, within the spinfoam formalism. This relies on the use of a boundary state, which introduces a semi-classical flat geometry on the boundary. In this paper, we investigate this proposal considering a toy model in the (Riemannian) 3d case, where the semi-classical limit is better understood. We show that in this limit the propagation kernel of the model is the one for the harmonic oscillator. This is at the origin of the expected 1/L behaviour of the 2-point function. Furthermore, we numerically study the short scales regime, where deviations from this behaviour occur.

Towards the graviton from spinfoams: the 3d toy model

TL;DR

This work examines extracting the linearised graviton 2-point function from spinfoam amplitudes using a 3D toy model based on a single tetrahedron with Ponzano–Regge dynamics. By employing a Gaussian boundary state peaked on flat geometry, and analyzing the semi-classical limit with , the boundary-projected propagator reduces to a harmonic-oscillator kernel with frequency , reproducing the expected large-scale scaling . The study also reveals short-distance quantum corrections that manifest as inverse powers of and are gauge-dependent, underscoring the need to extend the model to multiple tetrahedra to access the full tensor structure and diffeomorphism symmetry. These results provide a clearer link between boundary data in spinfoams and propagation, offering guidance for extending to the 4D case where a true graviton propagator would be sought.

Abstract

Recently, a proposal has appeared for the extraction of the 2-point function of linearised quantum gravity, within the spinfoam formalism. This relies on the use of a boundary state, which introduces a semi-classical flat geometry on the boundary. In this paper, we investigate this proposal considering a toy model in the (Riemannian) 3d case, where the semi-classical limit is better understood. We show that in this limit the propagation kernel of the model is the one for the harmonic oscillator. This is at the origin of the expected 1/L behaviour of the 2-point function. Furthermore, we numerically study the short scales regime, where deviations from this behaviour occur.

Paper Structure

This paper contains 8 sections, 32 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The toy model
  3. The boundary state
  4. The field insertions
  5. Graviton propagator
  6. Dependence on the distance
  7. Conclusions
  8. 2-point function from the kernel for the harmonic oscillator

Figures (2)

  • Figure 1: The dynamical tetrahedron as evolution between two hyperplanes. The labels give the physical lengths as $a=\ell_{\rm P} C(j_1)$, $b=\ell_{\rm P} C(j_2)$, $c=\ell_{\rm P} C(j_0)$, and $T=t_2-t_1=c/\sqrt 2$.
  • Figure 2: The crosses are the numerical evaluations of (\ref{['Wexact']}), plotted against the asymptotic behaviour $\frac{3}{2j_0}$ on a bi--logarithmic scale. The agreement is very good for $j_0\geq 50$, while deviations appear for smaller values.