Tag Archives: Sage (mathematics software)

Calculations on Quantum Cuboids and the EPRL-FK path integral for quantum gravity

This week I have been studying a really great paper looking at Quantum Cuboids and path-integral calculations for the EPRL vertex in LQG and also beginning to write some calculational software tools for performing these calculations using Sagemath.

In this work the authors investigate the 4d path integral for Euclidean quantum gravity on a hypercubic lattice, as given by the EPRL-FK model. To tackle the problem, they restrict the path to a set of quantum geometries that reflects the lattice symmetries. In particular, the sum over intertwiners is restricted to quantum cuboids, that is,  coherent intertwiners which describe a cuboidal
geometry in the large-j limit.

Using asymptotic expressions for the vertex amplitude, several interesting properties of the state sum are found.

  • The value of coupling constants in the amplitude functions determines whether geometric or non-geometric configurations dominate the path integral.
  • There is a critical value of the coupling constant α, which separates two phases.  In one phase the main contribution
    comes from very irregular and crumpled states. In the other phase, the dominant contribution comes from a highly regular configuration, which can be interpreted as flat Euclidean space, with small non-geometric perturbations around it.
  • States which describe boundary geometry with high
    torsion have exponentially suppressed physical norm.

The symmetry-restricted state sum

Will work on a regular hypercubic lattice in 4d. On this lattice consider only states which conform to the lattice symmetry. This is a condition on the intertwiners, which  corresponds to cuboids.
A cuboid is completely determined by its three edge lengths, or equivalently by its three areas.


All internal angles are π/2 , and the condition of regular cuboids on all dual edges of the lattice result in a high degree of symmetries on the labels: The area and hence the spin on each two parallel squares of the lattice which are translations perpendicular to the squares, have to be equal.

The high degree of symmetry will make all quantum geometries flat. The analysis carried out here is therefore not suited for describing local curvature.


The plan of the paper is as follows:

  • Review of the EPRL-FK spin foam model
  • Semiclassical regime of the path integral
  • Construction of the quantum cuboid intertwiner
  • Full vertex amplitude, in particular describe its asymptotic expression for large spins
  • Numerical investigation of the quantum path integral

The spin foam state sum  employed is the Euclidean EPRL-FK model with Barbero-Immirzi parameter γ < 1. The EPRL-FK model is defined on an arbitrary 2-complexes. A 2-complex 􀀀 is determined by its vertices v, its edges e connecting two vertices, and faces f which are bounded by the edges.

The path integral is formulated as a sum over states. A state in this context is given by a collection of spins –  irreducible representations
jf ∈ 1/2 N of SU(2) to the faces, as well as a collection of intertwiners ιe on edges.

The actual sum is given by


where Af , Ae and Av are the face-, edge- and vertex- amplitude functions, depending on the state. The sum has to be carried out over all spins, and over an orthonormal orthonormal basis in the intertwiner space at each edge.

The allowed spins jf in the EPRL-FK model are such
that jf are both also half-integer spins.

The face amplitudes are either


The edge amplitudes Ae are usually taken to be equal to 1.

In Sagemath code this looks like:


Coherent intertwiners

In this paper, the space-time manifold used is  M∼ T³×[0, 1] is the product of the 3-torus T3 and a closed interval. The space is compactified toroidally. M is covered by 4d hypercubes, which
form a regular hypercubic lattice H.There is a vertex for each hypercube, and two vertices are connected by an edge whenever two hypercubes intersect ina 3d cube. The faces of 􀀀 are dual to squares in H, on which four hypercubes meet.The geometry will be encoded in the state, by specification of spins jf
and intertwiners ιe.


Intertwiners ιe can be given a geometric interpretation in terms of polyhedra in R³. Given a collection of spins j1, . . . jn and vectors n1, . . . nn which close closure. Can define the coherent polyhedron


The geometric interpretation is that of a polyhedron, with face areas jf and face normals ni. The closure condition ensures that such a polyhedron exists.

We are interested in the large j-regime of the quantum cuboids. In this limit, these become classical cuboids  which are completely specified by their three areas. Therefore, a
semiclassical configuration is given by an assignment of
areas a = lp² to the squares of the hypercubic lattice.

Denote the four directions in the lattice by x, y, z, t. The areas satisfy


The two constraints which reduce the twisted geometric
configurations to geometric configurations are given by:


For a non-geometric configuration, define the 4-volume of a hypercube as:


Define the four diameters to be:


then we have, V4 = dxdydzdt

We also define the non- geometricity as:


as a measure of the deviation from the constraints.

In sagemath code this looks like:


Quantum Cuboids

We let’s look at  the quantum theory. In the 2-complex, every edge has six faces attached to it, corresponding to the six faces of the cubes. So any intertwiner in the state-sum will be six-valent, and therefore can be described by a coherent polyhedron with six faces. In our setup, we restrict the state-sum to coherent cuboids, or quantum cuboids. A cuboid is characterized by areas on opposite sides of the cuboid being equal, and the respective normals being negatives of one another


The state ιj1,j2,j3 is given by:


The vertex amplitude for a Barbero-Immirzi parameter γ < 1 factorizes as Av = A+vAv with


with the complex action


where, a is the source node of the link l, while b is its target node.

Large j asymptotics
The amplitudes A±v possess an asymptotic expression for large jl. There are two distinct stationary and critical points, satisfying the equations.

cuboidequ15for all links ab . Using the convention shown below

cuboidfig4having fixed g0 = 1, the two solutions Σ1 and Σ2 are


The amplitudes A±satisfy, in the large j limit,


In the large j-limit, the norm squared of the quantum cuboid states is given by:


For the state sum, in the large-j limit on a regular hypercubic lattice:


In sagemath code this looks like:






Related articles


Exact Computation and Asymptotic Approximations of 6j Symbols: Illustration of Their Semiclassical Limits by Mirco Ragni et al

In recent posts I have been doing some numerical work based on a series of papers on the ‘Exact Computation and Asymptotic Approximations of 6j Symbols’.

and a review of their use in spin networks

In this posts I’ll be looking at a paper which looks and their semiclassical properties.

In this paper the authors describe a direct method for the exact computation of 3nj symbols from the defining series. The properties and asymptotic formulas of the 6j symbols or Racah coefficients are discussed. Relationships with families of hypergeometric orthogonal polynomials are presented and the asymptotic behaviour is studied both from the viewpoints of the basic geometrical significance and as a source of accurate approximation formulas.

Numerical aspects are specifically investigated in detail, regarding the relationship between functions of discrete and of continuous variables, exhibiting the transition in the limit of large angular momenta toward both Wigner’s reduced rotation matrices or Jacobi
polynomials and harmonic oscillators or Hermite polynomials.

This paper contains a presentation of properties, useful formulas, and  illustrations for a basic mathematical tool, the 6j-symbol, also known in quantum mechanical angular momentum theory as Racah coefficient and the building block of 3nj-symbols and spin networks.

There are basic connections among the 6j symbols of angular momentum theory with both the theory of superposition coefficients of hyperspherical harmonics and the theory of discrete orthogonal polynomials. There is a connection between the Askey
scheme of orthogonal polynomials and the tools of angular momentum theory  such as 6j, 3j, rotation d-matrices. Going down the Askey scheme corresponds in quantum mechanics to the semiclassical limit, while going up provides discretization algorithms for quantum mechanical calculations for example the hyperquantization algorithm.


Explicit expressions for the 6j coefficients can be written according to the series expressions of the Racah type, or as generalized hypergeometric series, or in connection with the so-called Racah polynomials. Orthogonal polynomials of a discrete variable are important tools of numerical analysis for the representation of functions on grids.

Computation of Mathematical Functions and Angular Momentum

We can calculate the 3nj-symbols and Wigner d functions
by directly summing the defining series using multiprecision arithmetic. The multiple precision arithmetic allows convenient calculation of hypergeometric functions, pFq, of small
and large argument by their series definition.



d functions with 2F1:


Clebsch–Gordan coefficients and 3j-symbols with  3F2


6j-symbols with 4F3


 Semiclassical Limits and Schulten–Gordon Approach

In the Askey scheme for orthogonal polynomials
of hyperspherical family and its counterpart for the
tools of angular momentum theory shown arrows pointing out downwards are asymptotic connections.

A basic role is played by the relationship which relates
three 6j symbols with an argument differing by one


In this formula, one can introduce a quantity R such
that either and m1=j23−j, m2=j3−j23, m3 = j −j3.

We have


So that when R goes to infinity, we obtain a three terms
recurrence relationship for the 3j symbols as a function of j1.



when R goes to infinity, we obtain another three term recurrence relationship for the 3j symbols as function of m2.

Wigner Reduced Rotation Matrix Elements as Limits of 6j Symbols

The post Numerical work with sagemath 23: Wigner Reduced Rotation Matrix Elements as Limits of 6j Symbols deals with this section.

Geometrical interpretation

The equation:


has an interesting geometrical interpretation, based on the vector model visualization of quantum angular momentum coupling by
the triangle of vectors that we would draw in classical mechanics.


In view of this when we consider 6j properties as correlated to those of the tetrahedron,


we use the substitution

Jx =jx

which greatly improves all asymptotic formulas down to surprisingly low values of the entries.

The square of the area of each triangular face is given by the formula:


where a, b, c are the sides of the face. Similarly, the square of the volume of an irregular tetrahedron,can be written as the Cayley-Menger determinant:


When the values of J1, J2, J12, J3, and J are fixed, the maximum value for the volume as a function of J23 is given at:


The corresponding volume is


Therefore, the two values of J23 for which the volume is zero are:

hyperequ18They mark the boundaries between classical and nonclassical regions.

Introduce a parameter λ indicating the growth of the angular momentum. Consider the Schulten-Gordon relationships:

For λ =1




where F(a, b, c) is area of abc triangle from


The coefficients in


are connected to the geometry of the tetrahedron:


In terms of the finite difference operator:

hyperequ22We have


From these formulas, and from that of the volume,
we have that

  • V=0 implies cosθ1 =±1 and establishes the classical domain between J1min and J1max
  • F(J1, J2, J3)=0 or F(J1, L2, L3)=0 establish the definition limits j1min and j1max

For a Schrodinger type equation


its discrete analog in a grid having one as a step


and so comparing this with hyperequ22a

we have


allowing the identification


Plots corresponding to the three cases are given below:


On the closed loop, we can enforce Bohr–Sommerfeld phase space


where the role of q is played by j12. This is an eigenvalue equation for allowed L1. The illustration of these formulas is below:

hyperfig6Illustration of phase space for semiclassical quantization with  j1=92, j2 =47, j3 =80, j =121 for j12 =139 (dots), j12 =134 (triangles) ,j12 =129 (plus signs)


Values for the integral hyperequ29 for different
number of nodes n.

  • j12 = 139 ⇒ n = 0,
  • j12 =134 ⇒n =5,
  • j12 =129 ⇒n =10.

The values of the integrals are connected by the line while the dots are evaluated with p given by hyperequ29 – see table below:


 The 6j symbol and the oscillator wavefuncions
The Askey scheme and its counterpart point out at the connection in the angular momentum case between the top, the 6j symbol, and
the bottom, the harmonic oscillator. The geometrical insight of the Ponzano–Regge theory and its implementation in the Schulten–Gordon asymptotic formulas consistently lead to the expected Airy function behaviour astride of the transitions between classical and nonclassical regions of the ranges of elongations of the oscillator.

There is a  connection between the harmonic oscillator wavefunctions and 6j coefficients for large angular momentum arguments. Ponzano and Regge have approximated 6j-coefficients with sine and cosine functions as well as Airy functions. The formulas and extensions by Schulten and Gordon are excellent for the uniform semi-classical approximation for 6j-coefficients.

These semi-classical approximations  rely on the volume, surface areas, and angles that characterize the tetrahedron that corresponds to each 6j-coefficient that is required. A simple method connects a large set of special 6j-coefficients to harmonic oscillator wavefunctions by using only three parameters that are uniquely given from a simple algebraic analysis of the volumes of some tetrahedra related to the desired set of 6j-coefficients.

Consider the approximation of the 6j-coefficients:


as a function of j12 and j23. These discrete functions are orthonormal with relations:


Compare the 6j-coefficients with one dimensional quantum mechanical harmonic oscillator wavefunctions which belong to
an orthogonal set. Consider weighted 6j-coefficients:

hyperequ32Draw the connection with the harmonic oscillator wave-functions by noting that for given j1, j2, j12, j3, j there will be a value of j23max that will yield the maximum volume for the corresponding tetrahedron. The volume is given by


The maximum V² is obtained by finding the appropriate value of j23
that gives d (V²)/dj23=0. Consider the particular 6j-symbol:


where j2 = j1 and j= j3. This symbol has n nodes as j23 is varied. Set up the approximation using harmonic oscillator wave functions:



Looking at the 6j-symbol:


which gives the values for j23max and α. The figures
show n = 0, n =2, and n =7.


Representation of the 6j symbols by the harmonic oscillator wavefuntion for the case j1=1000, j2=1000, j12 =200, j3 =100,
j =100.


Representation of the 6j symbols by the harmonic oscillator wavefunction for the case j1 = 1000, j2 = 1000, j12 =198, j3 = 100, j =100.


Representation of the 6j symbols by the harmonic oscillator wavefunction for the case j1 = 1000, j2 =1000, j12 =193, j3 = 100, j =100.



Representation of the 6j symbols by the harmonic oscillator wavefunction for the case for j1 = 4000, j2 = 4000, j12 = 200, j3 =100, j =100


Representation of the 6j symbols by the harmonic oscillator wavefunction for the case for j1 = 8000, j2 =8000, j12 = 200, j3 =100, j = 100.

The harmonic oscillator parameters obtained from the two parameters:  j23max  and α provide a  representation of the behaviour of specific 6j-symbols by harmonic oscillator wavefunctions. The present state of the theory shows that the agreement should get better with increasing j.

 Related articles

Sagemath21: Knot theory calculations

This week I have been working on a variety of topics, but I’d like to post about some sagemath knot theory investigations Ive been doing.  This is connected to one of my longer term aims, which  is to  perform SU(2) recoupling calculations based on  Temperley-Lieb algebras using Sage Mathematics Software. Essentially I would like to to write a set of tools for doing SU(2) recoupling theory, Penrose binor calculus and  Temperley—Lieb algebra calculations using sagemath. 

Lets see what sagemath can do:

knots code

knot output1




animate knot

animated knot