Tag Archives: Toy model

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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Numerical work with sagemath 13: 1+1 causal dynamical triangulations

This week I have been studying and working on several things. Firstly I have been reading a long paper on ‘Holomorphic Factorization for a Quantum Tetrahedron’ which I’ll review in my next post. This is quite a sophisticated paper with some quite high level mathematics so I’ve taken my time with it.

Then I been upgrading my work on the 3d toy model including the presentation of the equilateral tetrahedron and hyperplanes as seen in the posts:



The other thing I have been working on is 1 +1 dimensional causal dynamical triangulations, starting with two main references quantum gravity ( in addition to Loll, Amborn papers) on a laptop: 1+1 cdt by Normal S Israel and John F Linder and  Two Dimensional Causal Dynamical Triangulation by Normal S Israel. So far I have set up a rough framework including initial conditions, detailed balance for monte carlo analysis and initial graphical work.

cdt1+1 fig

cdt1+1 figb


What I want to be able to do is independently construct a working CDT code and build it up from 2d to 3d and finally 4d. For the 1+1 CDT code I would be aiming to be able to independently construct simulated universes which exhibit quantum fluctuations as shown below:

cdt1+1 fig1

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.

Related articles

Numerical work with sage 5 – Wigner {3j}, {6j}, {9j} symbol evaluation

I’ve been working on weighed digraphs, Dynkin diagrams and various other graph theory concepts over the last week. The reason for this is that as we saw in “Towards the graviton from spinfoams: the 3d toy model by Simone Speziale”  in  the Ponzano–Regge (PR) model  each tetrahedron has an amplitude given by Wigner’s {6j} symbol for the recoupling theory of SU(2).

propagation kernal toy model

This use of Wigner symbols is a common feature of Quantum Gravity models. Fortunately, Sagemaths provides a simple and straightforward way to evaluate Wigner symbols {3j}, {j6}, or {9j}. I am still working on evaluating {10j}, {15j} and {20j} Wigner symbols which are used in 3d and 4d Loop Quantum Gravity .

sage wigner symbols

The table below taken from Quantum Gravity By Rovelli shows how the Wigner symbols are used in various quantum gravity models.

Using Wigner symbol

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