Towards the graviton from spinfoams: the 3d toy model by Simone Speziale

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.

The dynamical tetrahedron

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 le. 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:

toy model hamiltonian

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

toy model dihedral angle

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).

propagation kernal toy model

 

 

 

 

 

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