2014 in review

The WordPress.com stats helper monkeys prepared a 2014 annual report for this blog.

Here’s an excerpt:

The concert hall at the Sydney Opera House holds 2,700 people. This blog was viewed about 11,000 times in 2014. If it were a concert at Sydney Opera House, it would take about 4 sold-out performances for that many people to see it.

Click here to see the complete report.

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

Numerical work with sagemath 23: Wigner Reduced Rotation Matrix Elements as Limits of 6j Symbols

This work is based on the paper “Exact Computation and Asymptotic Approximations of 6j Symbols: Illustration of Their Semiclassical Limits by Mirco Ragni et al which I’ll be reviewing in my next post.

The 6j symbols tend asymptotically to Wigner dlnm functions when some angular momenta are large where θ assumes certain discrete values.







These formulas are illustrated below:




This can be modelled using sagemath.


The routine gives some great results:

For N=320, M=320, n=0, m=0, l=20, L=0,  Lmax=640

Wigner 6j vs cosθL


For N=320, M=320, n=0, m=0, l=10, L=0,  Lmax=640

Wigner 6j vs cosθL


For N=320, M=320, n=0, m=0, l=5, L=0,  Lmax=640

Wigner 6j vs cosθL



Numerical work with sagemath 22: The Hypergeometric function

In order to follow up some work on the the last two posts I have been looking at  the capabilities of Sagemath with regard to calculating hypergeometric functions:

Fortunately sagemath can implement at number of great codes for hypergeometric functions including:

These give excellent results:





The ability to use mpmath code is very useful since it enables me to calculate a wide range of hypergeometric functions as can be seen on the reference pages. I’ll be using this over the coming posts starting with :

Exact Computation and Asymptotic Approximations of 6j Symbols:
Illustration of Their Semiclassical Limits which is in preparation.

Related articles


Exact and asymptotic computations of elementary spin networks

This week I have been following up some work which I was introduced to in Dimitri Marinelli’s PhD thesis ‘Single and collective dynamics of discretized geometries’. Essentially this involves the analysis of the volume operator.  This is really exciting for me as it is in my specialist research area –  the numerical analysis of Quantum geometric operators and their spectra. I’ll be following up the literature survey with numerical work in sagemath.

The paper I’ll look at this week is ‘Exact and asymptotic computations of elementary spin networks: classification of the quantum–classical boundaries’ by  Bitencourt, Marzuoli,  Ragni, Anderson and and Aquilanti.

There has been increasing interest to the issues of exact computations and asymptotics of spin networks. The large–entries regimes – semiclassical limits, occur in many areas of physics and in particular in discretization algorithms of applied quantum mechanics.

The authors extend recent work on the basic building block of spin networks, namely the Wigner 6j symbol or Racah coefficient,  by exploiting its self–dual properties and studying it as a function of two discrete variables. This arises from its original definition as an orthogonal angular momentum recoupling matrix. Progress comes
from recognizing its role in the foundation of the modern theory of classical orthogonal polynomials, as extended to include discrete variables. Features of the imaging of various regimes of these orthonormal matrices are made explicit by computational
advances –based on traditional and new recurrence relations– which allow an interpretation of the observed behaviors in terms of an underlying Hamiltonian formulation.

The paper provides a contribution to the understanding of the transition between two extreme modes of the 6j, corresponding to the nearly classical and the fully quantum regimes, by studying the boundary lines – caustics in the plane of the two matrix labels. This analysis marks the evolution of the turning points of relevance for the semiclassical regimes and highlights the key role of the Regge symmetries of the 6j.


The diagrammatic tools for spin networks were developed by the Yutsis school and  in connection with applications to discretized models for quantum gravity after Penrose, Ponzano and Regge.

The basic building blocks of all spin networks are the Wigner 6j symbols or Racah coefficients, which are studied here by exploiting their self dual properties and looking at them as functions of two variables. This approach is natural in view of their origin as matrix elements describing recoupling between alternative angular momentum binary coupling schemes, or between alternative hyperspherical harmonics.

Semiclassical and asymptotic views are introduced to describe the dependence on parameters. They originated from the association due to Racah and Wigner to geometrical features, respectively a dihedral angle and the volume of an associated tetrahedron, which is the starting point of the seminal paper by Ponzano and Regge . Their results provided an impressive insight into the functional dependence of angular momentum functions showing a quantum mechanical picture in terms of formulas which describe classical and non–classical discrete wavelike regimes, as well as the transition between them.

The screen: mirror, Piero and Regge symmetries

The 6j symbol becomes the eigenfunction of the Schrodinger–like equation in the variable q, a continuous generalization of j12:


where Ψ(q) is related to


and p² is related with the square of the volume V of the associated tetrahedron.

exactequtetraThe Cayley–Menger determinant permits to calculate the square of the volume of a generic tetrahedron in terms of squares of its edge lengths according to:


The condition for the tetrahedron with fixed edge lengths to exist as a polyhedron in Euclidean 3-space amounts to require V²> 0, while the V²= 0 and V²< 0 cases were associated by Ponzano and Regge to “flat” and nonclassical tetrahedral configurations respectively.

Major insight is provided by plotting both 6js and geometrical functions -volumes, products of face areas – of the associated tetrahedra in a 2-dimensional j12 -􀀀 j23 plane , in whch the square “screen” of allowed ranges of j12 and j23 is used in all the pictures

  •  The mirror symmetry. The appearance of squares of tetrahedron edges entails that the invariance with respect to the exchange J ↔− 􀀀J implies formally j ↔ – 􀀀j 􀀀-1 with respect to the entries of the 6j symbol.
  • Piero line. In general, an exchange of opposite edges of a tetrahedron corresponds to different tetrahedra and different symbols. In Piero formula, there is a term due to this difference that vanishes when any pair of opposite edges are equal.
  • Regge symmetries. The these arises through connection with the projective geometry of the elementary quantum of space, which
    is associated to the polygonal inequalities -triangular and quadrilateral in the 6j case -, which have to be enforced in
    any spin networks.

The basic Regge symmetry can be written in the following form:


The range of both J12 and J23, namely the size of the screen, is given by 2min (J1, J2, J3, J, J1 +ρ , J2 +ρ ,J3 +ρ, J + ρ).

Features of the tetrahedron volume function

Looking at the volume V as a function of  x=J12 and  y=J23 we get the expressions for the xVmaxand yVmax that correspond to the maximum of the volume for a fixed value of x or y:


The plots of these are  called “ridge” curves on the x,y-screen. Each one marks configurations of the associated tetrahedron when two specific pairs of triangular faces are orthogonal. The corresponding values of the volume (xVmax,xand yVmax,y) are

exactequ8F is the area of the triangle with sides a, b and c.Curves corresponding to V = 0, the caustic curves, obey the equations:





Symmetric and limiting cases

When some or all the j’s are equal, interesting features appear in the screen. Similarly when some are larger than others.


Symmetric cases



exactequfig4Limiting cases

We can discuss the caustics of the 3j symbols as the limiting case of the corresponding 6j where three entries are larger than the other ones:






The extensive images of the exactly calculated 6j’s on the square screens illustrate how the caustic curves studied in this paper separate the classical and nonclassical regions, where they show wavelike and evanescent behaviour respectively. Limiting
cases, and in particular those referring to 3j and Wigner’s d matrix elements can be analogously depicted and discussed. Interesting also are the ridge lines, which separate the images in the screen tending to qualitatively different flattening of the quadrilateral,
namely convex in the upper right region, concave in the upper left and lower right ones, and crossed in the lower left region.

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