Propagator with Positive Cosmological Constant in the 3D Euclidian Quantum Gravity Toy Model by Bunting and Carlo Rovelli

This week I have been doing some more numerical work on holomorphic factorization and  coherent states  – which will be posted later.

This post looks at a paper which adds to the literature on the 3d toy model reviewed in posts:

In this paper the authors look at  the propagator on a single tetrahedron in a three dimensional toy model of quantum gravity  with positive cosmological constant. The cosmological constant is included in the model via q-deformation  of the spatial symmetry algebra, i.e. using the Tuarev-Viro amplitude. The expected repulsive effect of dark energy is recovered in numerical and analytic calculations of the propagator at large scales comparable to the infrared cutoff.

This paper extends work the work on the 3d toy model by evaluating quantum gravity two-point functions on a single tetrahedron in three euclidean dimensions to the case with positive cosmological constant. At long distance scales the two point function for a quantum field theory gives the Newton force associated to that theory’s gauge boson. In this toy model, there is only one tetrahedron with a state peaked around an equilateral configuration of that tetrahedron, therefore it will not reproduce the exact Newton limit of 3D gravity with a cosmological constant. However it still shows an asymptotically repulsive force associated to the dark energy.

The inclusion of the cosmological constant corresponds to a deformation of the SU(2) spatial rotation symmetry. This quantum gravity toy model is known as the q-deformed Ponzano-Regge model or the Tuarev-Viro model. There are two key differences in the propagator from the Ponzano-Regge model.

First, in the case where the tetrahedra is large compared with the infrared cutoff imposed by the cosmological constant, the sums will be cut off by the deformation and not the triangular inequalities. This feature of the quantum algebra is essential in four dimensions where so-called “bubble” configurations of 2-complexes, cannot be escaped without an infrared cutoff, without this cutoff these transition amplitudes would diverge.

Second, in the case where the size of the tetrahedron is smaller than the cutoff, the modification will simply affect the asymptotics of the two point function via the addition of a volume term to the Regge action.

In this paper the authors first compute the Tuarev-Viro propagator numerically. Then they compare the numerical computations with an analytic calculation of the propagator asymptotics. These two calculations are found to match strongly in an easily computable regime, that is for an unrealistically large cosmological constant. However, the agreement is expected to persist as the cosmological constant is taken smaller.

The paper studies the correlator between two edges on a single tetrahedron. The fix four edges around the tetrahedron to jo which can be thought of as a time, thus j1 and j2  are lengths on two different time slices of the spacetime.

positive fig1

It studies the modulus of the propagator [P] as a function of the distance j and the infrared cutoff jmax. Because in the 3D case all of the calculations can be done in a gauge where they all give zero, following Speziale a Coulomb-like gauge where the field operators have non-trivial projections along a bone (edge of a triangle in the triangulation. This choice produces a calculation similar to the one that can be done in the 4D spinfoam model. The operator notations used in this paper are different to those of Speziale, but in line with more recent work in quantum gravity correlation functions. The two point function is:

positive equ1

where instead of the usual metric field insertions the operators perturbed around flat space.

positive equ2


positive fig1a

are the Penrose operators acting on links that go between nodes           a and n, and b and n in the spin network state |s>, Cq is the SUq(2) casimir defined as

positive fig1b

W is the Tuarev-Viro transition amplitude and S is a boundary state peaked on an equilateral  tetrahedron. For a tetrahedron the spin network state is also a tetrahedron dual to the original  one where there are nodes on all the faces. Choose the node labels such that the correlator is between the two edges labeled by j1 and j2. Then interested in P1122. This calculated by evaluating with the q-deformed state and transition amplitude:

positive equ2a

where  from the Tuarev-Viro model that the transition amplitude for a single tetrahedron is just a quantum 6j-symbol:

positive equ2bThe deformation parameter q is taken to be   a root of unity;

positive equ2c

where Λ  is the cosmological constant that fixes the infrared cutoff jmax as:

positive equ2d

The boundary state is taken to be:

positive equ2e

The factor with Λ in this state is included so that the asymptotics do not have oscillatory terms in the limit as jo →∞ that come from the addition of the cosmological constant term to the Regge action. Numerical evaluation of equation above gives the main result of the paper, shown below:

positive fig2

Here can see that at small scales jo≈  1  this gives the same quantum deviations as Speziale from the Newtonian limit. In the range where we are not too close to the cutoff and not too small lengths ie.

1 «jo « jmax/2

The behavior is  similar to the Newtonian limit with cosmological constant [P]  ≈3/2jo. Lastly, when the spins get close to the infrared cutoff there is  a repulsion representative of the repulsive force of dark energy.

Discussion and  Analysis

The authors next analyze analytically the asymptotics of this toy model to see if they match the numerical results. The asymptotics of {6j}q symbols are given by a cosine of the Regge action with a volume term.

positive equ3a

The volume term is the volume of a spherical tetrahedron.  This is approximated  as a flat tetrahedron and expand the volume in δj1
and  δj2:

positive equ3b

The factor in 2π,r, G is approximately independent of j1 and j2 therefore it will cancel with the same factor in the normalization. In the large spin limit the difference of the Casimir operators will go like C²(j1) – C²(j0) ˜ 2joj1.  This gives an asymptotic formula for the propagator:

positive equ3c

omitting the other exponential term that is rapidly oscillating in j1 and j2.

Expanding the action and the volume terms around the equilateral configuration, and canceling factors independent of j1 and j2 with the same ones in the normalization gives:

positive equ3d


positive equ3e

is the matrix of derivatives of the dihedral angles with respect to the
edge lengths evaluated for an equilateral tetrahedron. Passing to continuous variables z = j1 and dz = dj1 gives the gaussian integral:

positive equ3f

where A is the matrix of coefficients of δjiδjk:

positive equ3Aa

where α  = 4/3jo which is compatible with the requirement that the state be peaked on the intrinsic and extrinsic geometry . From this the full expression for the asymptotics of the propagator modulus is :

positive equ3h

The plot of this analytic expression versus the numerical calculation shows the agreement is seen to be very good. Expanding this expression about λ= 0 it becomes more clear that a repulsion is the dominating correction to the Speziale result

positive equ3i

This expression has a term proportional to the cosmological constant times the volume of the tetrahedron. While this differs from what we might expect from the Newton law, this is a simplified model where there is only one simplex that spans the cosmological distance, and it is peaked on an equilateral configuration.


The paper shows that the inclusion of a cosmological constant in a 3D euclidean toy model of quantum gravity on a single tetrahedron reproduces the expected qualitative behavior near the infrared cutoff –  an additional repulsive force on large distance scales. The effect is reproduced in both numerical calculations and an analytic evaluation of the propagator asymptotics. This work also strengthens the argument for the interpretation of the cosmological constant as a deformation parameter in the theory.

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