In this paper Speziale looks at the extraction of the 2-point function of linearised quantum gravity, within the spinfoam formalism. The author that this process relies on the use of a boundary state, which introduces a semi–classical flat geometry on the boundary.

In this paper, Speziale investigates this proposal by considering a toy model in the Riemannian 3d case, where the semi–classical limit is understood. The author shows that in this the semi-classical limit the propagation kernel of the model is that for the for the harmonic oscillator – which leads to expected 1/*l* behaviour of the 2-point function.

**The toy model**

The toy model considered in this paper is a tetrahedron with dynamics described by the Regge action, whose fundamental variables are the edge lengths *l*e. Since there is only a single tetrahedron, all edges are boundary edges, and the action consists only of the

boundary term, namely it coincides with the Hamilton function of the system:

Here the *θe* are the dihedral angles of the tetrahedron, namely the angles between the outward normals to the triangles. They represent a discrete version of the extrinsic curvature, they satisfy the non–trivial relation

In this discrete setting, assigning the six edge lengths is equivalent to the assignment of

the boundary gravitational field.

The quantum dynamics is described by the Ponzano–Regge (PR) model . In the model, the lengths are promoted to operators whose spectrum is labelled by the half–integer j which labels SU(2) irreducible representations and the Casimir operator C^2 = j(j+1). In the model, each tetrahedron has an amplitude given by Wigner’s {6j} symbol for the recoupling theory of SU(2).

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