# Numerical work with sagemath 24: 6j Symbols and non-eucledean Tetrahedra

This week I have begun to look at Hyperbolic Tetrahedra and their geometry. In the paper ‘6j Symbols for Uq and non-eucledean Tetrahedra‘, Taylor and Woodward  relate the semiclassical asymptotics of the 6j symbols for the  quantized enveloping algebra Uq(sl2) to the geometry of spherical and  hyperbolic tetrahedra.

The quantum 6j symbol is a function of a 6-tuple jab, 1 ≤ a ≤ b ≤ 4. The 6j symbols

for q = 1 were introduced as a tool in atomic spectroscopy
by Racah, and then studied mathematically by Wigner. 6j symbols
for Uq(sl2) were introduced by Kirillov and Reshetikhin, who used them to generalize the Jones knot invariant. Turaev and Viro used them to define three manifold invariants or  quantum gravity with a cosmological constant.

I have started doing some preliminary work with sagemath on 3j symbols, 6j symbols, the quantum integer and on the gram matrix.

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# Generating Functionals for Spin Foam Amplitudes by Hnybida

This week I have been reading the PhD thesis ‘Generating Functionals for Spin Foam Amplitudes’ by Jeff Hnybida. This is a few useful topic because the exact evaluation could be useful to study
recursion relations, coarse graining moves, or to perform numerical calculations.

In the various approaches to Quantum Gravity such as Loop Quantum Gravity, Spin Foam Models and Tensor-Group Field theories use invariant tensors on a group, called intertwiners, as the basic building block of transition amplitudes. For the group SU(2) the contraction of these intertwiners in the pattern of a graph produces spin network amplitudes.

In this paper a generating functional for the exact evaluation of a coherent representation of these spin network amplitudes is constructed. This generating functional is defined for arbitrary graphs and depends only on a pair of spinors for each edge. The generating functional is a meromorphic polynomial in the spinor invariants which is determined by the cycle structure of the graph.
The expansion of the spin network generating function is given in terms of a basis of SU(2) intertwiners consisting of the monomials of the holomorphic spinor invariants. This basis, the discrete-coherent basis, is labelled by the degrees of the monomials and is  discrete. It also contains the precise amount of data needed to specify points in the classical space of closed polyhedra.

The focus the paper is on the 4-valent basis, which is the case of interest for Quantum Gravity. Simple relations between the discrete-coherent basis, the orthonormal basis, and the coherent basis are found.

The 4-simplex amplitude in this basis depends on 20 spins and is referred to as the 20j symbol. The 20j symbol is the exact evaluation of the coherent 4-simplex amplitude.

The asymptotic limit of the 20j symbol is found to give a generalization of the Regge action to Twisted Geometry.

3d quantum gravity

A triple of edge vectors meeting at a node must be invariant under the local rotational gauge transformations

There is only one invariant rank three tensor on SU(2) up to normalization: The Wigner 3j symbol or Clebsch-Gordan coefficient. The 3j symbol has the interpretation as a quantum triangle and its three spins correspond to the lengths of its three edges, which close to form a triangle due to the SU(2) invariance. Contracting four 3j symbols in the pattern of a tetrahedron gives the  well-known 6j symbol which is the amplitude for each tetrahedron.

Coherent BF Theory
The coherent intertwiners are  a coherent state representation of the space of invariant tensors on SU(2). The exact evaluations computed later are a result of a special exponentiating property of
coherent states. Each SU(2) coherent state is labelled by a spinor |z 〉,  |z] denotes its contragradient version. Using a bra-ket notation for the spinors

such that given two spinors z and w the two invariants which can be formed by contracting with either epsilon or delta are denoted

The exponentiating property of the coherent states corresponds to the fact that the spin j representation is simply the tensor product of 2j copies of the spinor |z〉⊗2j   . A coherent rank n tensor on SU(2) is therefore the tensor product of n exponentiated spinors.

To make the coherent tensor invariant we group average using the Haar measure

which is the denition of the Livine-Speziale intertwiner.

The coherent 6j symbol is constructed by contracting 4 coherent intertwiners in the pattern of a tetrahedron. Labeling each vertex by i = 1,..,4 and edges by pairs (ij) this amplitude depends on 6 spins jij = jji and 12 spinors |zij 〉≠ |zji〉 where the upper index denotes the vertex and the lower index the connected vertex. The coherent
amplitude in 3d is given by

The asymptotics of the coherent amplitude have been studied extensively, however the actual evaluation of these amplitudes was not known. While the asymptotic analysis is important to check the semi-classical limit, the exact evaluation could be useful to study
recursion relations, coarse graining moves, or to perform numerical calculations.

To obtain the exact evaluation we use a special property of the Haar measure on SU(2) to express the group integrals above as Gaussian integrals. The generating functional is defined as

we are able to compute the Gaussian integrals in above, not just for the tetrahedral graph but for any arbitrary graph. Performing the Gaussian integrals produces a determinant depending purely on the spinors. The determinant can be evaluated in general and can be expressed in terms of loops of the spin network graph.

For example, after integration and evaluating the determinant, the generating functional of the 3-simplex takes the form:

4d quantum gravity

General Relativity in four dimensions is not topological, but it can  be formulated by a constrained four dimensional BF theory. That is if B is constrained to be of the form

for a real tetrad 1-form e then the BF action becomes the Hilbert Palatini action for General Relativity. The aim of the spin foam program is to formulate a discretized version of these constraints that can break the topological invariance of BF theory and give rise
to the local degrees of freedom of gravity.

The advantage of formulating GR as a constrained BF theory is that, instead of quantizing Plebanski’s action, we can instead use the topological nature of BF theory to quantize  the discretized BF action and impose the  discretized constraints at the quantum level.
The first model of this type was proposed by Barret and Crane.
While this is not a quantization of a constrained system in the sense of Dirac it is a quantization of the Gupta-Bleuler type which was realised by Livine and Speziale  and led to corrected versions of the Barret-Crane model by Engle, Livine, Pereira, Rovelli  and by Freidel, Krasnov.

The behaviour of our spin network generating functional under
general coarse graining moves is a simple transformation of the coarse grained action in terms of lattice paths. For a square lattice, the generating functional expressed as sums over loops similar to gives precisely the partition function for the 2d Ising model.

Since the Ising model and its renormalization are very well understood this example could provide a toy model for which one could base a study of the more complicated spin foam renormalization.

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# The tetrahedron and its Regge conjugate

This week I have been reading the PhD thesis ‘Single and collective dynamics of discretized geometries’ by Dimitri Marinelli. In this post I’ll look at a small portion about Regge calculus, the  tetrahedron and its Regge conjugate.

Regge Calculus is a dynamical theory of space-time introduced in 1961 by Regge as a discrete approximation for the Einstein theory of gravity. The basic idea is to replace a smooth space-time with a collection of simplices. The collective dynamics of these geometric objects is driven by the Regge action and the dynamical variables are their edge lengths – which play the role of the metric tensor of General Relativity. Simplices are the n-dimensional generalization of triangles and tetrahedra. Regge Calculus inspired and is at the base of almost all the present discretized models for a quantum theory of gravity for at least two reasons:

• It is a discretized model, so it represents a possible atomistic system typical of quantum systems
• There is a deep connection between the Regge action, the asymptotic of the 6j symbol and a path integral formulation of gravity.

Let’s see  how the Regge transformation acts on a tetrahedral shape. The formulas

and the association between 6j symbol and an Euclidean tetrahedron tell us that any Regge transformation acts on four edges of a tetrahedron keeping a pair of opposite edges unchanged. The Regge-transformed tetrahedra is called `conjugate’.

Using the Ponzano-Regge formula for the 6j,

we can immediately say that the volume of a tetrahedron and that of a Regge transformed one must coincide.

The volume of a tetrahedron is also invariant under the Regge transformation of four consecutive edges.

The volume of a tetrahedron, being a function of six parameters, can be expressed in several ways. For the tetrahedron below:

where AABC and AACD are respectively the areas of the triangles ABC and ACD, lAC is the length of the common edge and β is the dihedral angle between these two faces.

The importance of the Regge symmetry is that it constrains the shape dynamics of a single tetrahedron,  it relates different tetrahedra equating their quantum representations and it is the key tool to understand the classical motion of a four-bar linkage mechanical systems and its link to the the quantum dynamics of tetrahedra.

This thesis also contains a section on the Askey scheme which I’ll be following up in future posts:

# Asymptotic Analysis of Spin Networks with Applications to Quantum Gravity by Hal Haggard

This week I have been reading a couple of PhD thesis. The first is ‘The Chiral Structure of Loop Quantum Gravity‘ by Wolfgang  Wieland, the other is Hal Haggard’s PhD thesis, ‘Asymptotic Analysis of Spin Networks with Applications to Quantum Gravity‘ .

I’m going to focus in this post on a small section of  Haggard’s work about  the quantization of space.

In loop gravity quantum states are built upon a spin coloured graph. The nodes of this graph represent three dimensional grains
of space and the links of the graph encode the adjacency of these regions. In the semiclassical description of the grains of space – see the post:

these grains are  interpreted as giving rise to convex polyhedra in this limit. This geometrical picture suggests a natural proposal for the quantum volume operator of a grain of space: it should be an operator that corresponds to the volume of the associated classical polyhedron. This section provides a detailed analysis of such an operator, for a single 4-valent node of the graph. The valency of the node determines the number of faces of the polyhedron and so we will be focusing on classical and quantum tetrahedra.

The space of convex polyhedra with fixed face areas has a natural phase space and symplectic structure.  Using this kinematics we can study the classical volume operator and perform a Bohr-Sommerfeld quantization of its spectrum. This quantization
is in good agreement with previous studies of the volume operator spectrum and provides a simplified derivation.

This section of the thesis looks at:

• The node Hilbert Spaces Kn for general valency.
• The classical phase space and  its Poisson structure.
• The volume operator  in loop gravity.
•  The quantum tetrahedron
• Phase space for the classical tetrahedron and its Bohr-Sommerfeld quantization.
• Wavefunctions for the volume operator

Setup

Focus on a single node n and its Hilbert space
Kn. The space Kn is defined as the subspace of the tensor
product Hj1⊗ · · ·  ⊗HjF that is invariant under global SU(2) transformations

We call Kn the space of intertwiners. The diagonal action is generated by the operatorJ,

States of Kn are called intertwiners and can be expanded as:

The components i transform as a tensor under SU(2) transformations in such a way that the condition

is satisfied. These are precisely the invariant tensors captured graphically by spin networks.The finite dimensional intertwiner space Kn can be understood as the quantization of a classical phase space.

Begin with Minkowski’s theorem, which states the following: given F vectors Ar ∈ R³ (r = 1, . . . , F) whose sum is zero

then up to rotations and translations there exists a unique convex polyhedron with F faces associated to these vectors. These convex polyhedra are our semiclassical interpretation of the loop gravity grains of space.

The second step  is to associate a classical phase space to these
polyhedra

We call this the space of shapes and denote it P(A1, . . . ,AF ) or more briefly PF.  Interpret the partial sums

as generators of rotations about the μk = A1 + · · · + Ak+1 axis.This interpretation follows naturally from considering each of the Ar vectors to be a classical angular momentum, so that really Ar ∈ Λr ≅R³.

There is is a natural Poisson structure on each r and this extends to the bracket

With this Poisson bracket the μk do in fact generate rotations about the axis A1 + · · ·Ak+1. This geometrical interpretation suggests a natural conjugate coordinate, namely the angle φk of the rotation. Let k be the angle between the vectors

then

The pairs (μk,φk) are canonical coordinates for a classical phase space of dimension 2(F −3). Thisis called  the space of shapes and denote it P(A1, . . . ,AF ) or more briefly PF.

In the final step we  show that that quantization of PF is the Hilbert space Kn of an F-valent node n. The constraint

beomes

which is precisely the gauge invariance condition. This is where the classical vector model of angular momentum originates.

Volume operators in loop gravity

Most of the loop gravity research on the volume operator has been done in the context of the original canonical quantization of general relativity.  In Ashtekar’s formulation of classical general relativity the gravitational field is described in terms of the triad variables E. This triad corresponds to a 3-metric h and is called the electric field. The elementary quantum operator that measures the geometry of space
corresponds to the flux of the electric field through a surface S. When such a surface is punctured by a link of the spin network graph Γ the flux can be parallel transported, back along the link, to the node using the second of Ashtekar’s variables, the
Ashtekar-Barbero connection. This results in an SU(2) operator that acts on the intertwiner space Kn at the node n.The parallel transported flux operator at the node is proportional to the generator of SU(2) transformations

where γ is a free parameter of the theory called the Barbero-Immirzi parameter and Pl is the Planck length.

The volume of a region of space R is obtained by regularizing and quantizing the classical expression

using the operators Er. The total volume is obtained by summing the contributions from each node of the spin network graph Γcontained in the region R.

There are different proposals for the volume operator at a node. The operator originally proposed by Rovelli and Smolin is

A second operator introduced by Ashtekar and Lewandowski is

Both the Rovelli-Smolin and the Ashtekar-Lewandowski proposals have classical versions. This results in two distinct functions on phase space:

A third proposal for the volume operator at a node has emerged,
Dona, Bianchi and Speziale suggest the promotion of the classical volume of the polyhedron associated to {Ar} to an operator

In the case of a 4-valent node all three of these proposals agree.

The volume of a quantum tetrahedron

In the case of a node with four links, F = 4, all the proposals for the volume operator discussed above coincide and match the operator introduced by Barbieri for the volume of a quantum tetrahedron.

The Hilbert space K4 of a quantum tetrahedron is the intertwiner space of four representations of SU(2),

Introduce  basis into this Hilbert space using the recoupling channel Hj1 ⊗Hj2 and call these basis states |k>. The basis vectors are defined as

where the tensor ik is defined in terms of the Wigner 3j-symbols as

The index k ranges from kmin to kmax in integer steps with

The dimension d of the Hilbert space K4 is finite and given by

The states |k> form an orthonormal basis of eigenstates of the operator Er · Es. This operator measures the dihedral angle between the faces r and s of the quantum tetrahedron.
The operator The operator √Er · Er measures the area of the rth face of the quantum tetrahedron and states in K4 are area eigenstates with eigenvalues ,

The volume operator introduced by Barbieri is

and because of the closure relation

this operator coincides with the Rovelli-Smolin operator for                  α = 2√2/3. The volume operator introduce by Barbieri can be understood as a special case of the volume of a quantum polyhedron.

In order to compute the spectrum of the volume operator, it is useful to introduce the operator Q defined as

It represents the square of the oriented volume. The matrix elements of this operator are  computed in the post:

The eigenstates |q> of the operator Q,

are also eigenstates of the volume. The eigenvalues of the volume are simply given by the square-root of the modulus of q,

The matrix elements of the operator Q in the basis |k> are given by

The function Δ(a, b, c) returns the area of a triangle with sides of length (a, b, c) and is conveniently expressed in terms of Heron’s formula

This can be done numerically and we can  compare the eigenevalues calculated in this manner to the results of the Bohr-Sommerfeld quantization.

There are a number of properties of the spectrum of Q and therefore of V that can be determined analytically.

• The spectrum of Q is non-degenerate: it contains d distinct real eigenvalues. This is a consequence of the fact that the matrix elements of Q on the basis |k> determine a d × d Hermitian matrix of the form

with real coefficients ai.

• The non-vanishing eigenvalues of Q come in pairs ±q. A vanishing eigenvalue is present only when the dimension d of the intertwiner space is odd.
• For given spins j1, . . . , j4, the maximum volume eigenvalue can be estimated  using Gershgorin’s circle theorem and wefind that it scales as    where jmax is the largest of the four spins jr.
• The minimum non-vanishing eigenvalue (volume gap)
scales as

Tetrahedral volume on shape space

The starting point for the  Bohr-Sommerfeld analysis is the volume of a tetrahedron as a function on the shape phase space P(A1, . . . ,A4)≡  P4.

The Minkowski theorem guarantees the existence and uniqueness
of a tetrahedron associated to any four vectors Ar, (r = 1, . . . , 4) that satisfy A1 + · · · + A4 = 0. The magnitudes Ar ≡ |Ar|, (i = 1, . . . , 4) are interpreted as the face areas. A condition for the existence of a tetrahedron is that A1 +A2 +A3  ≥ A4, equality giving a flat -zero volume tetrahedron. The space of tetrahedra with four fixed face areas P(A1,A2,A3,A4) ≡ P4 is a sphere.

Consider the classical volume,

it will be more straightforward to work with the squared classical volume,

Writing the triple product of Q as the determinant of a matrix M = (A1,A2,A3) whose columns are the vectors A1,A2 and A3 and squaring yields,

Bohr-Sommerfeld quantization of tetrahedra

The Bohr-Sommerfeld quantization condition is expressed in
terms of the action I associated to the each of these orbits:

Wavefunctions

Just as the Bohr-Sommerfeld approximation can be used for finding the eigenvalues of the volume operator. Semiclassical techniques can
also be used to find the volume wavefunctions.

where,

and

Conclusions
At the Planck scale, a quantum behaviour of the geometry of space is expected. Loop gravity provides a specific realization of this expectation: it predicts a granularity of space with each grain having a quantum behaviour.

Based on semiclassical arguments applied to the simplest model for a grain of space, a Euclidean tetrahedron, and is closely related to Regge’s discretization of gravity and to more recent ideas about
general relativity and quantum geometry. The spectrum has been computed by applying Bohr-Sommerfeld quantization to the volume of a tetrahedron seen as an observable on the phase space of shapes.
There is quantitative agreement of the spectrum calculated here and
the spectrum of the volume in loop gravity. This result lends credibility to the intricate derivation of the volume spectrum in loop gravity, showing that it matches an elementary semiclassical approach.

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# Review of the Quantum Tetrahedron – part II

Intrinsic coherent states

A class of wave-packet states is given by the coherent states, which are  states labelled by classical variables (position and momenta) that minimize the spread of both. Coherent states are the basic tool for studying the classical limit in quantum gravity. They connect  quantum theory with classical general relativity. Coherent states in the Hilbert space of the theory can be used in proving the large distance behavior of the vertex amplitude and connecting it to the Einstein’s equations.

Given a classical tetrahedron, we can find a quantum state in Hγ such that all the dihedral angles are minimally spread around the classical values, these are the intrinsic coherent states

Tetrahedron geometry

Consider the geometry of a classical tetrahedron reviewed in Review of the Quantum Tetrahedron – part I . A tetrahedron in flat space can be determined by giving three vectors,, representing three of its sides emanating from a vertex P.

Forming a non-orthogonal coordinate system where the axes are along these vectors and the vectors determine the unit of coordinate length, then ei  is the triad and

is the metric in these coordinates. The three vectors

are normal to the three triangles adjacent to P and their length is the area of these faces. The products

define the matrix hab which is the inverse of the metric h =ea.eb. The volume of the tetrahedron is

Extending the range of the index a to 1, 2, 3, 4, and denote all the four normals, normalised to the area, as Ea. These satisfy the closure condition

The dihedral angle between two triangles is given by

Now we move to the quantum theory. Here, the quantities Ea are quantized as

in terms of the four operators La, which are the hermitian generators of the rotation group:

The commutator of two angles is:

From this commutation relation, the Heisenberg relation follows:

Now we want to look for states whose dispersion is small compared with their expectation value: semiclassical states where

SU(2) coherent states
Consider a single rotating particle. How do we write a state for which the dispersion of its angular momentum is minimized? If j is the quantum number of its total angular momentum, a basis of states is

since,

we have the Heisenberg relations

Every state satisfies this inequality. A state |j,j> that saturatesis one  for which.

In the large j limit we have

Therefore this state becomes sharp for large j.

The geometrical picture corresponding to this calculation is that the state |j, j> represents a spherical harmonic maximally concentrated on the North pole of the sphere, and the ratio between the spread and the radius decreases with the spin.

Other coherent states  are  obtained rotating the state |j, j> into an arbitrary direction n. Introducing Euler angles θ,Φ  to label rotations,

Then let and define the matrix R in SO(3) of the form  , With this, define:

The states |j,n> form a family of states, labelled by the continuous parameter n, which saturate the uncertainty relations for the angles. Some of their properties are the following.

For a generic direction n = (nx, ny, nz),therefore:

and

The expansion of these states in terms of Lz eigenstates is

The most important property of the coherent states is that they provide a resolution of the identity. That is

The left hand side is the identity in Hj. The integral is over all normalized vectors, therefore over a two sphere, with the standard R3 measure restricted to the unit sphere.

Observe that by taking tensor products of coherent states, we obtain coherent states. This follows from the properties of mean values and variance under tensor product.

Livine-Speziale coherent intertwiners

Now  introduce “coherent tetrahedra” states. A classical tetrahedron is defined by the four areas Aa and the four normalized normals na, up to rotations. These satisfy

Therefore consider the coherent state;

in

and project it down to its invariant part in the projection

The resulting state

is the element of Hγ that describes the semiclassical tetrahedron. The projection can be explicitly implemented by integrating over SO(3);

# 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.

# 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.