Tag Archives: Angular momentum

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




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.

Related articles

Special properties of spin network functions

My work on quantum geometric operators makes heavy use of special properties of the spin network functions in particular the fact that the action of the flux operators on spin network basis states can be mapped to the problem of evaluating angular momentum operators on angular momentum eigenstates, which is familiar from ordinary quantum mechanics. This results in a great simplification and provides a convenient way to work in the SU(2)gauge-invariant regime using powerful techniques from the recoupling theory of angular momenta.

This post provide details on the conventions used in the construction of recoupling schemes. The post is organized as follows:

  • Basic properties of matrix representations of SU(2), whose matrix elements are used for the definition of spin network functions.
  • The theory of angular momentum from quantum mechanics and the notion of recoupling of an arbitrary number of angular momenta in terms of recoupling schemes.
  • The definition of 6j-symbols andtheir basic properties. This provides an explicit notion of recoupling schemes in terms of polynomials of quantum numbers.

Representations of SU(2)
Irreducible matrix representations of SU(2) can be constructed in (2j+1)-dimensional linear vector spaces, where j ≥ 0 is a half integer number; j = 0 denotes the trivial representation.

General Conventions — Defining Representation j = 1/2
Generators of SU(2) use the τ-matrices given by τk := − iσk, with σk being the Pauli-matrices:

Additionally use


SU(2) Representations
In the defining representation of SU(2), for a group element h ∈ SU(2)





and we have the additional properties that


Use the following convention for the matrix elements


For the τk’s we additionally have


General Conventions for (2j + 1)-dimensional SU(2) Representation Matrices
General Formula for SU(2) Matrix Element

The (2j +1)-dimensional representation matrix of h ∈ SU(2), given in terms of the parameters of the defining representation can be written as


where ℓ takes all integer values such that none of the factorials in the denominator gets a negative argument. By
construction every representation of SU(2) consists of special unitary matrices and


Generators and ε-Metric
Applying the representation matrix element formula and the ansatz,


where in the defining two dimensional representation the exponential can be explicitly evaluated, one obtains

and finds that


Angular Momentum Theory
Basic Definitions
The angular momentum orthonormal basis u(j,m; n) =|j m ; n〉 of a general (2j + 1) dimensional representation of SU(2). The index n stands for additional quantum numbers, not affected by the action of the angular momentum operators J fulfilling the commutation


Formulate ladder operators as


The |j m ; n〉  simultaneously diagonalize the two operators: the squared total angular momentum (J)² and the
magnetic quantum number J³ :


That is, |j m ; n〉 is a maximal set of simultaneous eigenvectors of (J)² and J³ find the following commutation relations


such that for the (2j + 1)-dimensional matrix representation with arbitrary weight j



Fundamental Recoupling

Clebsch -Gordan theorem on tensorized representations of SU(2):
Theorem b1

If we couple two angular momenta j1, j2, we can get resulting angular momenta j12 varying in the range |j1 − j2| ≤ j12 ≤ j1 + j2. The tensor product space of two representations of SU(2) decomposes into a direct sum of representation spaces, with one space for every possible value of recoupling j12 with the according dimension
2j12 + 1.


Recoupling of n Angular Momenta — 3nj-Symbols
The successive coupling of three angular momenta to a resulting j can be generalized to an n-fold tensor product of representations            πj1 ⊗ πj2 ⊗ . . . ⊗ πjn  by reducing out step by step every pair of representations.

This procedure is carried out until all tensor products are reduced out. One then ends up with a direct sum of representations, each of which has a weight corresponding to an allowed value of the total angular momentum to which the n single angular momenta j1, j2, . . . , jn can couple. However, there is an arbitrariness in how one couples the n angular momenta together, that is the order in which πj1 ⊗πj2 ⊗. . .⊗πjn is reduced out matters.

Consider a system of n angular momenta. First we fix a labelling of these momenta, such that we have j1, j2, . . . , jn. Again the first choice would be a tensor basis |j m〉 of all single angular momentum states |jk mk〉, k = 1 . . . n defined by:


Recoupling Scheme
recoupling b1

Standard Basis

Theorem b2



Properties of Recoupling Schemes

A general standard recoupling scheme is defined as follows:

For the scalar product of two recoupling schemes we have:


Properties of the 6j-Symbols
6j-symbols are the basic structure in recoupling calculations, as every coupling of n angular momenta can be expressed in terms of them.
The 6j-symbol is defined in


The factors in the summation are Clebsch-Gordon coefficients.

Explicit Evaluation of the 6j-Symbols
A general formula for the numerical value of the 6j-symbols has been derived by Racah


Symmetry Properties
The 6j-symbols are invariant under:

  • any permutation of the columns:


• simultaneous interchange of the upper and lower arguments of two columns

Orthogonality and Sum Rules
Orthogonality Relations


Composition Relation


Sum Rule of Elliot and Biedenharn


Quantum states of elementary three–geometry by Carbone, Carfora , and Marzuoli

This is a relatively old paper but its so clearly written that it is well worth reviewing and understanding the material.

In this paper the authors introduce a quantum volume operator K in three–dimensional Quantum Gravity by taking into account a symmetrical coupling scheme of three SU(2) angular momenta. The spectrum of K is discrete and defines a complete set of eigenvectors
which is an alternative with respect to the complete sets employed when the usual binary coupling schemes of angular momenta are considered. Each of these states,  called quantum bubbles, represents an interference of extended configurations which provides a rigorous meaning to the heuristic notion of quantum tetrahedron. They study the generalized recoupling coefficients connecting the symmetrical and the binary basis vectors, and provide an explicit recursive solution for such coefficients by analyzing also its asymptotic limit.


Ponzano–Regge gravity is based on an asymptotic formula for theSU(2) 6j symbols,


The physical interpretation follows if we recognize that the exponential includes the classical Regge action – the discretized version of the Hilbert– Einstein action for the tetrahedron T. The
presence of the slowly varying volume term in front of the phase factor means that {6j}as can be interpreted as a probability amplitude in the approach to the classical limit. The probability amplitude for an elementary block of Euclidean three–geometry to emerge from the recoupling of quantum angular momenta or from a spin network.

Physical reality can only be ascribed to the tetrahedron T if and only if its volume –written in terms of the squares of the edges through the Cayley determinant– satisfies the condition (V (T))² > 0. The triangle inequalities on the four triples of spin variables  – associated with the four faces of the tetrahedron – which ensure the existence of the 6j symbol are weaker than the condition that T exists as a realizable solid.

The authors try to answer the following questions:

  • Why does a classical three–geometry emerge from a recoupling of angular momenta?
  • What does it mean to give the quantum state of an elementary block of geometry?
  • What are the degrees of freedom of a quantum tetrahedron?

Quantum bubbles

The theory of the coupling of states of three SU(2) angular momenta operators J₁, J₂,  J₃ to states of sharp total angular momentum J is usually developed in the framework of binary couplings. Starting from the ordered triple,  denote symbolically the admissible schemes according to



The corresponding state vectors may be written as


In this framework the Wigner 6j symbol is just the recoupling coefficient relating the two sets  and , namely


where Φ ≡ J₁+J₂+J₃

In spite of the tetrahedral symmetry of the 6j symbol, at the quantum level we cannot recognize something like a quantum tetrahedron: it is only in the approach to the classical limit  that both the coupling schemes coexist and the tetrahedron may take shape.

This situation changes  if we bring into play the symmetrical coupling
scheme for the addition of three SU(2) angular momenta to give a fourth definite angular momentum J with projection J₀ symbolically

3gequ7 such a coupling is characterized by the simultaneous diagonalization of the six Hermitian operators

J²₁,J²₂,J²₃,  K, J² and J₀, where K is a scalar operator built from the irreducible tensor operators J₁, J₂,  J₃ and defined according to


Since K is the mixed product of three angular momentum vectors we can call it a volume.

Each of the eigenvectors of this new set of operators is denoted by 3gequ8

the set of eigenvalues in each subspace with (jm) fixed is discrete and
consists of pairs (k,−k), with at most one zero eigenvalue.

From a geometrical point of view a state |kjm >, for k ≠ 0, represents a quantum volume of size kħ³ and k2 > 0 is the quantum counterpart of the condition (V (T))² > 0.

According to the Correspondence Principle in the region of large quantum numbers the states state |kjm >characterize angular momentum vectors confined to narrow ranges
around specific values: the narrower are the ranges, the closer we approach the classical regime. This implies that the classical limit of the operator coincides with the expression of the volume of the classical tetrahedron T spanned by the counterparts of the operators  J₁, J₂,  J₃.

The state |kjm > for fixed (jm) provides a proper description of the quantum state of an elementary block of Euclidean three–geometry. However |kjm > cannot be directly interpreted as a quantum tetrahedron, but rather as a quantum interference of extendend configurations from which information about all significant geometrical quantities can be extracted.

 Generalized recoupling theory

The 6j symbol can be interpreted as a propagator between two states belonging to alternative binary bases. The asymptotic expression:  3gequ1is actually the generating functional of a path–sum evaluated at the semiclassical level and the physical information about the underlying classical theory is encoded in the form of the action.

For the quantitative analysis of the relations between one of the sets of eigenvectors arising from a binary coupling parametrized by the eigenvalue of the intermediate momentum J₁₂ and the symmetrical basis  we introduce a unitary transformation defined as


The generalized recoupling coefficient


and its inverse defined as


A three–term recursion relation for the generalized recoupling coefficient is


The functions α (l =  J₁₂, J₁₂ + 1) can be cast in the form


Since 3gequ21the conditions at the extrema fixed by  are


The formal solution of 3gequ19



Asymptotic limit

To explore the generalized recoupling coefficient we use a semiclassical (WKB) approximation to the recursion relation


Which  gives an ordinary second order difference equation
of the type


we search for a solution of this for each k, in the form


with ρ and A to be determined. By substitution we get a pair of differential equations corresponding respectively to the imaginary and real parts;


Finally, since the classical counterpart of the operator K is the volume V ≡ V (T) of the Euclidean tetrahedron T we see that the generalized recoupling coefficient in the asymptotic limit behaves as



where θ₁₂ is the dihedral angle between the faces of the tetrahedron T which share J₁₂.

 Related articles

The quantum tetrahedron as a quantum harmonic oscillator

It is known that the large-volume limit of a quantum tetrahedron is a quantum  harmonic oscillator. In particular the volume operator of a quantum tetrahedron is, in the sector of large eigenvalues, accurately described by a quantum harmonic oscillator.

Using  Vpython, it is possible to visualise the quantum tetrahedron as a  semi classical quantum simple harmonic oscillator.  The implementation of this is shown below;

Quantum simple harmonic oscillator

The quantum tetrahedron as a  Quantum  harmonic oscillator .

quantum tetrahedron as quantum SHM

Click to view animated gif

Review of the basic mathematical structure of a quantum tetrahedron

A quantum tetrahedron consists of four angular momenta Ji, where i = {1,2,3, 4} representing its faces and coupling to a vanishing total angular momentum – this means that the Hilbert space consists of all states ful filing:

j1 +j2 +j3 +j4 = 0

In the coupling scheme both pairs j1;j2 and j3;j4 couple frst to two irreducible SU(2)representations of dimension 2k+1 each, which are then added to give a singlet state.

The quantum number k ranges from kmin to kmax in integer steps with:
kmin = max(|j1-j2|,|j3 -j4|)
kmax = min(j1+j2,j3 +j4)

The total dimension of the Hilbert space is given by:
d = kmax -kmin + 1.

The volume operator can be formulated as:

large vol equ 1

where the operators

large vol equ 2

represent the faces of the tetrahedron with

large vol equ 3and being the Immirzi parameter.

It is useful to use the operator

large vol equ 4

which, in the basis of the states can be represented as

large vol equ 5


large vol equ 6

and where

large vol equ 7

is the area of a triangle with edges a; b; c expressed via Herons formula,

These choices mean that the matrix representation of Q is real and antisymmetric and  the spectrum of Q consists for even d of pairs of eigenvalues q and -q differing in sign. This makes it much easier to find the eigenvalues as I found in  Numerical work with sage 7.

The large volume limt of Q is found to be given by:

large vol equ 8

The eigenstates of Q are just the well-known wave functions diagonalizing the harmonic oscillator in real-space representation:

large vol equ 9

For a monochromatic quantum tetrahedron we can obtain closed analytical expressions:


Below is shown a sagemath program which displays the wavefunction  of a Quantum simple harmonic oscillator using Hermite polynomials Hn:


This an really important step in the development of Loop Quantum Gravity, because now we have  a  quantum field theory of geometry. This leads from combinatorial or simplicial description of geometry, through spin foams and the  quantum tetrahedron, to many particle states of quantum tetrahedra and the emergence of spacetime as a Bose-Einstein condensate.

quantum tetrahedron concepts

Angle and Volume Studies in Quantized Space by Michael Seifert

In this paper seifert reviews the angle and volume operators in Loop Quantum Gravity. I’m particularly interested at the moment in his treatment of the angle operator.

In LQG, quantum states are represented by spin networks –  graphs with weighted edges. Spatial observables such as area, volume, and angle are given by the eigenvalues of Hermitian operators on the spin network states. The author presents results on  the angle and volume operators which act on the vertices of spin networks. He finds that the minimum observable angle is inversely proportional to the square root of the total spin of the vertex, with a fairly slow decrease to  zero. He also presents numerical results indicating that the angle operator can reproduce the classical angle distribution. Noting that the volume operator is significantly harder to investigate analytically the author presents analytical and  numerical results indicating that the volume of a region scales as the 3/2 power of its bounding surface, a result which corresponds to the classical model of space.

Angular Momentum Operators

One of the aims of this paper is to describe the angles associated with a given vertex.  To define angles, need to be able to associate a direction with an edge or a group of edges. In Penrose’s original formulation of spin networks, the labels on the edges refer to angular momenta carried by the edges  so one of the natural directions to associate with an edge is its angular momentum vector. The angular momenta can be measured by the angular momentum operators, which act upon the edges. Angular momentum operators are expressed in terms of the Pauli spin matrices σi:

seirfeit pauli matrices

Using this, we can define angular momentum operators Ji = 1/2hσi which act on the edges. Can also construct the Jˆ2 operator this is equal to the sum of the squares of the three Ji operators.

The eigenvalues simplify to the  form:

Jˆ2 = 1/2hj(j +1)

The Area Operatorseirfeit areas

The total area of  is then given by

Discreteness of Area and Volume fig 3

where the summation of i is over all edges that intersect , and Ji is the edge label of the edge I.

The Angle Operator

seirfeit angle partition

The angle operator will measure the angle between the edges that traverse S1 and those that traverse S2. The  direction that is associated with the edges traversing S1, S2, and Sr is the angular momentum vectors associated with their internal edges J1,J2, and Jr.

Classically it is expected that

cos θ = J1.J2 / |J1||J2|

The angle operator is then defined as

seirfeit angle op

The eigenvalues of the angle operator:

seirfeit angle eigenvalue

where ji = ni/2.

From the discreteness of the angle operator Seifert derives three  results: will be referred to as

Small angle  property for angle e

seirfeit small angle

where the total value of core spin nT = n1 + n2 + nr

Angular resolution property

seirfeit angle resolution

where n is valence of the vertex.

Angular distribution property

Angular momentum: An approach to combinatorial space-time by Roger Penrose

This week I have also been reading about Penrose’s “spin networks“. In this paper Penrose attempts to build a purely combinatorial description of spacetime starting from the mathematics of spin-1/2 particles. The appraoch he describes in this paper leads into twistor theory. The spin networks he describe give an interesting theory of space. Penrose’s spin networks are purely combinatorial structures: graphs with edges labelled by numbers j = 0, 1/2, 1, 3/2,… These numbers represent the total angular momentum or “spin”. In the spin networks described in this paper it is required that three edges meet at each vertex, with the corresponding spins j1, j2, j3 adding up to an integer and satisfying the triangle inequalities

|j1 – j2| ≤ j3 ≤ j1 + j2

These rules are motivated by the quantum mechanics of angular momentum in that if we combine a system with spin j1 and a system with spin j2, the spin j3 of the combined system satisfies exactly these constraints. In this paper a spin network represents a quantum state of the geometry of space and this interpretation is justified by computations using a special rule for computing the norm of any spin network.


A spin network