# Review of the Quantum Tetrahedron – part 1

Over the last few posts the level of mathematics as been rather high  so I’ve decided to review our basic understanding of the Quantum Tetrahedron in this post. This review is based on the work of Carlo Rovelli and Francesca Vidotto. Other parts of this review will look at the graviton propagator and also coherent states.

If we pick a simple geometrical object ,an elementary portion of space, such as a small tetrahedron t, not necessarily regular.

The geometry of a tetrahedron is characterized by the length of its sides, the area of its faces, its volume, the dihedral angles at its edges, the angles at the vertices of its faces, and so on. These are all local functions of the gravitational field, because geometry is the same thing as the gravitational field. These geometrical quantities are related to one another. A set of independent quantities is provided for instance by the six lengths of the sides, but these are not appropriate for studying quantization, because they are constrained by inequalities. The length of the three sides of a triangle, for instance, cannot be chosen arbitrarily: they must satisfy the triangle inequalities. Non-trivial inequalities between dynamical variables, like all global features of phase space, are generally difficult to implement in quantum theory.

Instead, we choose the four vectors

defined for each triangle a as 1/2 of the outward oriented vector product of two edges bounding the triangle.

These four vectors have several nice properties. Elementary geometry shows that they can be equivalently defined in one of the two following ways:

•  The vectors La are outgoing normals to the faces of the tetrahedron and their norm is equal to the area of the face.
• The matrix of the components

where M is the matrix formed by the components of three edges of the tetrahedron that emanate from a common vertex.

The vectors La have the following properties:

•  They satisfy the closure relation:

• The quantities La determine all other geometrical quantities such as areas, volume, angles between edges and dihedral angles between faces.
• All these quantities, that is, the geometry of the tetrahedron, are invariant under a common SO(3) rotation of the four La. Therefore a tetrahedron is determined by an equivalence class under rotations of a quadruplet of vectors La’s satisfying

•  The area Aa of the face a is |La|.
• The volume V is determined by the properly oriented triple product of any three faces:

We can describe the gravitational field in terms of triads and tetrads. If the tetrahedron is small compared to the scale of the local curvature, so that the metric can be assumed to be locally flat andLa can be identified with the flux of the triad field

across the face a, then

Since the triad is the gravitational field, this gives the explicit relation between La and the gravitational field. Here the triad is defined in the 3d hyperplane determined by the tetrahedron.

The above give all the ingredients for jumping to quantum gravity. The geometry of a real physical tetrahedron is determined by the gravitational field, which is a quantum field. Therefore the normals La can be described by quantum operators, if we take the quantum nature of gravity into account. These will obey commutation relations. The commutation relation can be obtained from the hamiltonian analysis of GR, by promoting Poisson brackets to operators,  but ultimately they are quantization postulates. The simplest possibility -see post Quantum tetrahedra and simplicial spin networks, is:

where lo is a constant proportional h and with the dimension of an area. These commutation relations are realizations of the algebra of SU(2), reflecting again the rotational symmetry in the description of the tetrahedron. This is useful, for instance we see that C as defined  is precisely the generator of common rotations and therefore the closure condition  is an immediate condition of rotational invariance, which is what we want: the geometry is determined by the La up to rotations, which here are gauge.

The constant lo must be related to the Planck scale , which is the only dimensional constant in quantum gravity. Setting,

where γ is a dimensionless parameter of the order of unity that fixes the precise scale of the theory.

One consequence is immediate -see post Discreteness of area and volume in quantum gravity, the quantity Aa = |La| behaves as total angular momentum. As this quantity is the area, it follows immediately that the area of the triangles bounding any tetrahedron is quantized with eigenvalues:

This is the gist of loop quantum gravity. The result extends to any surface, not just the area of the triangles bounding a tetrahedron.

Say that the quantum geometry is in a state with area eigenvalues    j1, …, j4. The four vector operators La act on the tensor product H of four representations of SU(2), with respective spins j1, …, j4. That is, the Hilbert space of the quantum states of the geometry of the tetrahedron at fixed values of the area of its faces is

Have to take into account the closure equation, which is a condition the states must satisfy, if they are to describe a tetrahedron. But C is nothing else than the generator of the global diagonal action of SU(2) on the four representation spaces. The states that solve the closure equation, namely

are the states that are invariant under this action, namely the states in the subspace:

The volume operator V is well defined in K because it commutes with C, namely it is rotationally invariant. Therefore we have a well-posed eigenvalue problem for the self-adjoint volume operator on the Hilbert space K. As this space is finite dimensional, it follows that its eigenvalues are discrete. Therefore we have the result that the volume has discrete eigenvalues as well. In other words, there are ‘quanta of volume’ or ‘quanta of space’: the volume of our tetrahedron can grow only in discrete steps.

Eigenvalues of the volume

Computing the volume eigenvalues for a quantum of space whose sides have minimal non vanishing area. Recall that the volume operator V is determined by:

where the operators La satisfy the commutation relations:

If the faces of the quantum of space have minimal area, the Casimir of the corresponding representations have minimal non-vanishing value. Therefore the four operators La act on the fundamental representations j1 = j2 = j3 = j4 = 1/2 . Therefore they are proportionalto the self-adjoint generators of SU(2), which in the fundamental representation are Pauli matrices. That is:

The proportionality constant has the dimension of length square, is of Planck scale and is fixed by comparing with the commutation relations of the Pauli matrices. This gives

The Hilbert space on which these operators act is therefore

H = H½ x H½xH½xH½

This is the space of objects with 4 spinor indices A, B = 0, 1, each being in the ½- representation of SU(2).

The operator La acts on the a-th index. Therefore the volume operator acts as:

Now implementing the closure condition . Let

We only have to look  for subspaces that are invariant under a common rotation for each space Hji , namely we should look for a quantity with four spinor indices that are invariant under rotations. What is the dimension of this space? Remembering that for SU(2) representations:

implies that:

Since the trivial representation appears twice, the dimension of is two. Therefore there must be two independent invariant tensors with four indices. These are easy to guess, because the only invariant objects available are ε(AB) and σ(AB), obtained raising the indices of the Pauli matrices σi:

Therefore two states that spanare:

These form a non orthogonal basis in. These two states span the physical SU(2)-invariant part of the Hilbert space, that gives all the shapes of our quantum of space with a given area. To find the eigenvalues of the volume it suffices to diagonalize the 2×2 matrix V²nm:

An straightforward calculation with Pauli matrices gives:

so that,

and the diagonalization gives the eigenvalues

The sign depends on the fact that this is the oriented volume square, which depends on the relative orientation of the triad of normal chosen. Inserting the value for α, we have finally the eigenvalue of the non oriented volume

About 10¹°º quanta of volume of this size fit into a cm³.

# Numerical work with sagemath 16: Moduli Space of Shapes of a Tetrahedron with Faces of Equal Area

One of the things  I’ve been  working on this week  is : Moduli Space of Shapes of a Tetrahedron with Faces of Equal Area.

This follows on the work done in the posts:

The space of shapes of a tetrahedron with fixed face areas is naturally a symplectic manifold of real dimension two. This symplectic manifold turns out to be a Kahler manifold and can be parametrized by a single complex coordinate Z given by the cross ratio of four complex numbers obtained by stereographically projecting the unit face normals onto the complex plane.

This post shows how this works in the simplest case of a tetrahedron T whose four face areas are equal. For convenience, the cross-ratio coordinate Z is shifted and rescaled to z=(2Z-1)/Sqrt[3] so that the regular tetrahedron corresponds to z=i, in which case the upper half-plane is mapped conformally into the unit disc w=(i-z)/(i+z).The equi-area tetrahedron T is then drawn as a function of the unit disc coordinate w.

# 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.

Introduction
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.

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:

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

where,

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

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:

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

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

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

The boundary state is taken to be:

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:

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.

The volume term is the volume of a spherical tetrahedron.  This is approximated  as a flat tetrahedron and expand the volume in δj1
and  δj2:

The factor in 2π,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) ˜ 2joj1.  This gives an asymptotic formula for the propagator:

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:

where

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:

where A is the matrix of coefficients of δjiδjk:

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 :

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

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.

Conclusions

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.

# Numerical work with sagemath 15: Holomorphic factorization

This week I have been  reviewing the new spinfoam vertex in 4d models of quantum gravity. This was discussed in the recent posts:

In this post I explore the large spins asymptotic properties of the overlap coefficients:

characterizing the holomorphic intertwiners in the usual real basis. This consists of the normalization coefficient times the shifted Jacobi polynomial.

In the case  of n = 4. I can study the asymptotics of the shifted Jacobi polynomials in the limit ji → λji, λ → ∞.  A  convenient integral representation for the shifted Jacobi polynomials is given by a contour integral:

This leads to the result that:

This formula relates the two very different descriptions of the phase space of shapes of a classical tetrahedron – the real one in terms of the k, φ parameters and the complex one in terms of the cross-ratio
coordinate Z. As is clear from this formula, the relation between the two descriptions is non-trivial.

In this post I have only worked with the simplest case of this relation when all areas are equal. In this ‘equi-area‘ case where all four representations are equal ji = j, ∀i = 1, 2, 3, 4, as described in the post: Holomorphic Factorization for a Quantum Tetrahedron the overlap function is;

Using sagemath I am able to evaluate the overlap coefficients for various values of j and the cross-ratios z.

Here I plot the modulus of the equi-area case state Ck, for j = 20, as a function of the spin label k, for the value of the cross-ratio Z = exp(iπ/3) that corresponds to the equilateral tetrahedron. It is obvious that the distribution looks Gaussian. We also see that the maximum is reached for kc = 2j/√3 ∼ 23, which agrees with an asymptotic analysis.

Here I plot the modulus of the equi-area case state Ck for various j values as a function of the spin label k, for the value of the cross-ratio Z = exp(iπ/3) that corresponds to the equilateral tetrahedron.

Here I have  have plotted the modulus of the j = 20 equi-area state Ck for increasing cross-ratios Z = 0.1i, 0.8i, 1.8i. The Gaussian distribution progressively moving its peak from 0 to 2j. This illustrates how changing the value of Z affects the semi-classical geometry of the tetrahedron.

Conclusions

In this post I we have studied a holomorphic basis for the Hilbert space Hj1,…,jn of SU(2) intertwiners. In particular I have looked at the case of 4-valent intertwiners that can be interpreted as quantum states of a quantum tetrahedron. The formula

gives the inner product in Hj1,…,jn in terms of a holomorphic integral over the space of ‘shapes’ parametrized by the cross-ratio coordinates Zi. In the tetrahedral n = 4 case there is a single cross-ratio Z. The n=4 holomorphic intertwiners parametrized by a single cross-ratio variable Z are coherent states in that they form an over-complete basis of the Hilbert space of intertwiners and are semi-classical states peaked on the geometry of a classical tetrahedron as shown by my numerical studies. The new holomorphic intertwiners are related to the standard spin basis of intertwiners that are usually used in loop quantum gravity and spin foam models, and the change of basis coefficients are given by Jacobi polynomials.

In the canonical framework of loop quantum gravity, spin network states of quantum geometry are labeled by a graph as well as by SU(2) representations on the graph’s edges e and intertwiners on its vertices v. It is now possible to put holomorphic intertwiners at the vertices of the graph, which introduces the new spin networks labeled by representations je and cross-ratios Zv. Since each holomorphic intertwiner can be associated to a classical tetrahedron, we can interpret these new spin network states as discrete geometries. In particular, geometrical observables such as the volume can be expected to be peaked on their classical values as shown in my numerical studies for j=20. This should be of great help when looking at the dynamics of the spin network states and when studying how they are coarse-grained and refined.