Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model by Bonzom, Livine,Smerlak and Speziale

This week I have been doing further work on the Quantum Tetrahedron as an Quantum Harmonic Oscillator – which I’ll review in a later post and also looking at the 3d toy model in more detail. In particular I have been studying ‘Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model.’ In this paper the authors consider an exact expression for the 6j-symbol for the isosceles tetrahedron, involving SU(2) group integrals, 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 can then be performed with respect to the geometry of the boundary using a simple saddle-point analysis. They can then derive the complete expansion in inverse powers of the length scale and evaluate explicitly the quantum corrections up to second order. They also  use the same method to provide the complete expansion of the isosceles 6j-symbol with the next-to-leading correction to the Ponzano-Regge asymptotics.

Considering for simplicity the Riemannian case, the spinfoam amplitude for a single tetrahedron is the 6j-symbol of the Ponzano-Regge model. Its large spin asymptotics is dominated by exponentials of the Regge action for 3d general relativity. This is a key result, since the quantization of the Regge action is known to reproduce the correct free graviton propagator around flat spacetime.

This paper considers the simplest possible setting given by the 3d toy model introduced in the post Towards the graviton from spinfoams: the 3d toy model and studies analytically the full perturbative expansion of the 3d graviton. The results are based on a reformulation of the Wigner 6j symbol and the graviton propagator as group integrals. The authors compute explicitly
the leading order then both next-to-leading and next-to-next analytically. They also calculate  a formula for the next-to-leading order of the  Ponzano-Regge asymptotics of the 6j-symbol in the case of an isosceles tetrahedron.

Applying the same methods and tools to 4d spinfoam models would  allow a more thorough study of the full non-perturbative spinfoam graviton propagator and its correlations in 4d quantum gravity.

The boundary states and the kernel

Consider a triangulation consisting of a single tetrahedron. To define transition amplitudes in a background independent context for a certain region of spacetime, the main idea is to perform a perturbative expansion with respect to the geometry of the boundary. This classical geometry acts as a background for the perturbative expansion. To do so  – have to specify the values of the intrinsic and extrinsic curvatures of such a boundary, that is the edge lengths and the dihedral angles for a single tetrahedron in spinfoam variables. As in the post Towards the graviton from spinfoams: the 3d toy model, attention is restricted  to a situation in which the lengths of four edges have been measured, so that their values are fixed, say to a unique value jt + 1/2 . These constitute the time-like boundary and we are then interested in the correlations of length fluctuations between the two remaining and opposite edges which are the initial and final spatial slices. This setting is referred to as the time-gauge setting. The two opposite edges e1 and e2 have respectively lengths, j1 + 1/2 and j2 + 1/2 .

towards graviton fig1

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 j1 + 1/2 and j2 + 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 jt + 1/2 interpreted as the proper time of a particle propagating along one of these edges. Equivalently, the time between two planes containing e1 and e2 has been measured to be T = (jt + 1/2 )/sqrt(2).

In the spinfoam formalism, and in agreement with 3d LQG, lengths are quantized so that jt, j1 and j2 are half-integers.

The lengths and the dihedral angles are conjugated variables with regards to the boundary geometry, and have to satisfy the classical equations of motion. Here, it simply means that they must have admissible values to form a genuine flat tetrahedron. The dimension of the SU(2)-representation of spin j, dj ≡ 2j +1 is twice the edge length.

towards graviton equ 1

To assign a quantum state to the boundary, peaked on the classical geometry of the tetrahedron. Since jt is fixed, we only need such a state for e1, peaked on the length j1 + 1/2 , and for e2, peaked on        j2 + 1/2 .

Previous work have used a Gaussian ansatz for such states. However, it is more convenient to choose states which admit a well-defined Fourier transform on SU(2). In this perspective, the dihedral angles of the tetrahedron are interpreted as the class angles of SU(2) elements. So the Gaussian ansatz can be replaced for the edges e1 and e2 by the following Bessel state:

towards graviton equ 3a

The role of the cosine  is to peak the variable dual to j, i.e. the dihedral angle, on the value αe. Then the boundary state admits a well-defined Fourier transform, which is a Gaussian on the group SU(2). The SU(2) group elements are parameterized as:

towards graviton equ 4

These states carry the information about the boundary geometry necessary to induce a perturbative expansion around it. Interested in the following correlator,

towards graviton equ 6

W1122 measures the correlations between length fluctuations for the edges e1 and e2 of the tetrahedron, and it can be interpreted as
the 2-point function for gravity , contracted along the directions of e1 and e2. The 6j-symbol emerges from the usual spinfoam models for 3d gravity as the amplitude for a single tetrahedron.

towards graviton equ4043

Perturbative expansion of the isosceles 6j-symbol

The procedure described above can be applied directly to the isosceles 6j-symbol, obtaining the known Ponzano-Regge formula and its corrections.

This is interesting for a number of reasons. The corrections to the Ponzano-Regge formula are a key difference between the spinfoam perturbative expansion  and the one from quantum Regge calculus. The 6j-symbol is also the physical boundary state of 3d gravity
for a trivial topology and a one-tetrahedron triangulation. In 4d, it appears as a building block for the spin-foams amplitudes, such as the 15j-symbol. So for many aspects of spin-foams in 3d and 4d, in particular for the quantum corrections to the semiclassical limits, it good to have a better understanding of this object.

.The expansion of this isosceles 6j-symbol is;

towards graviton equ80

The leading order asymptotics, given by the original Ponzano-Regge formula is:

towards graviton equ80a

It is possible to compute analytically the two-point function – the graviton propagator – at all orders in the Planck length for the 3d toy model -the Ponzano-Regge model for a single isoceles tetrahedron as in Towards the graviton from spinfoams: the 3d toy model.

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