# The geometry of the tetrahedron and asymptotics of the 6j-symbol

This week I have been studying a useful PhD thesis ‘Asymptotics of quantum spin networks‘ by van der Veen. In this post I’ll look a the section dealing with the geometry of the tetrahedron.

Spin networks and their evaluations

In the special case of the tetrahedron graph the author presents a complete solution to the rationality property of the generating series of all evaluations with a fixed underlying graph using the combinatorics of the chromatic evaluation of a spin network, and a complete study of the asymptotics of the  6j-symbols in all cases: Euclidean, Plane or Minkowskian ,using the theory of Borel transform.

The three realizations of the dual metric tetrahedron depending on the sign of the Cayley-Menger determinant det(C). Also shown are the two exponential growth factors Λ ± of the 6j-symbol.

The author also computes the asymptotic expansions for all possible colorings,  including the degenerate and non-physical cases. They find that  quantities in the asymptotic expansion can be expressed as geometric properties of the dual tetrahedron graph interpreted as a metric polyhedron.

letting ( Γ,γ ) denote the tetrahedral spin network colored by an admissible coloring . Consider the planar dual graph which is also a tetrahedron with the same labelling on the edges. Regarding the labels of as edge lengths one may ask whether ( Γ,γ ) can be realized as a metric Euclidean tetrahedron with these edge lengths. The admissibility of ( Γ,γ ) implies that all faces of  the dual tetrahedron satisfy the triangle inequality. A well-known theorem of metric geometry implies that ( Γ,γ ) can be realized in exactly one of three flat geometries

(a) Euclidean 3-dimensional space R³
(b) Minkowskian space R²’¹
(c) Plane Euclidean R².

Which of the above applies  is decided by the sign of the Cayley-Menger determinant det(C) which is a degree six polynomial in the six edge labels.

Its assumed that ( Γ,γ ) is nondegenerate in the sense that all faces of are two dimensional. In the degenerate case, the evaluation of 6-symbol is a ratio of factorials, whose asymptotics are easily obtained from Stirling’s formula.

The geometry of the tetrahedron

The asymptotics of the 6j-symbols are related to the geometry of the planar dual tetrahedron. The 6j-symbol is a tetrahedral spin network ( Γ,γ ) admissibly labeled as shown below:

with  γ= (a, b, c, d, e, f). Its dual tetrahedron  ( Γ,γ ) is also labelled by  γ. The tetrahedron and its planar dual, together with an ordering of the vertices and a colouring of the edges of the dual is depicted below. When a more systematic notation for the edge labels is needed  they will be denoted them by dij it follows that

(a, b, c, d, e, f) = (d12, d23, d14, d34, d13, d24)

We can interpret the labels of the dual tetrahedron as edge lengths in a suitable flat geometry. A condition that allows one to realize ( Γ,γ )  in a flat metric space such that the edge lengths equal the edge labels. Labeling the vertices of ( Γ,γ ). We can formulate such a condition in terms of the Cayley-Menger determinant. This is
a homogeneous polynomial of degree 3 in the six variables a2, . . . , f2. A  definition of the determinant is:

Given numbers dij we can define the  Cayley-Menger matrix by

Cij = 1−d²ij/2 for i, j ≥1

and

Cij = sgn(i−j) for when i = 0 or j = 0.

In terms of the coloring = (a, b, c, d, e, f) of a tetrahedron, we have:

The sign of the Cayley-Menger determinant determines in what space the tetrahedron can be realized such that the edge labels equal the edge lengths

• If det(C) > 0 then the tetrahedron is realized in Euclidean space R³.
•  If det(C) = 0 then the tetrahedron is realized in the Euclidean plane R².
• If det(C) < 0 then the tetrahedron is realized in Minkowski space R²’¹.

In each case the volume of the tetrahedron is given by

The 6 dimensional space of non-degenerate tetrahedra consists of regions of Minkowskian and regions of Euclidean tetrahedra. It turns out to be a cone that is made up from one connected component of three dimensional Euclidean tetrahedra and two connected components of Minkowskian tetrahedra. The three dimensional Euclidean and Minkowskian tetrahedra are separated by Plane tetrahedra. The Plane tetrahedra also form two connected components, representatives of which are depicted below:

The tetrahedra in the Plane component that look like a triangle with an interior point are called triangular and the Plane tetrahedra from that other component that look like a quadrangle together with its diagonals are called quadrangular.  The same names are used for the corresponding Minkowskian components.

An integer representative of the triangular Plane tetrahedra is not easy to find as the smallest example is (37, 37, 13, 13, 24, 30).

Let’s look at the dihedral angles of a tetrahedron realized in either of the three above spaces. The cosine and sine of these angles can be expressed in terms of certain minors of the Cayley-Menger
matrix. Define the adjugate matrix ad(C) whose ij entry is (−1)i+j times the determinant of the matrix obtained from C by deleting the i-th row and the j-th column. Define Ciijj to be the matrix obtained from C by deleting both the i-th row and column and the j-th row and column.

The Law of Sines and the Law of Cosines are well-known formulae for a triangle in the Euclidean plane. let’s look at the Law of Sines and the Law of Cosines for a tetrahedron in all three flat geometries.

If we let θkl be the exterior dihedral angle at the opposite edge ij. The following formula is valid for all non-degenerate tetrahedra:

Related articles

# 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.

Related articles

# 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.

Related articles

# 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:

# 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
Generators of SU(2) use the τ-matrices given by τk := − iσk, with σk being the Pauli-matrices:

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

also

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
relations

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):

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

Standard Basis

3nj-Symbol

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.
Definition
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.

Introduction

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

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

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:  is 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 the conditions at the extrema fixed by  are

The formal solution of

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

# Semiclassical Mechanics of the Wigner 6j-Symbol by Aquilanti et al

This week I have been continuing my research on the the expectation values of the Ricci curvature on Carlo-Speziale semi-classical states, for different values of the spins in the case of a simple graph : a monochromatic 4 – valent node dual to an equilateral tetrahedron. I will be posting about this later.

As part of this work I am studying a paper on the semiclassical mechanics of the wigner 6j symbol.  In this paper the semiclassical mechanics of the Wigner 6j-symbol is examined from the standpoint of WKB theory for multidimensional, integrable systems, to explore the geometrical issues surrounding the Ponzano-Regge formula. The relations among the methods of Roberts and others for deriving the Ponzano- Regge formula are discussed, and a new approach, based on the recoupling of four angular momenta, is presented. A generalization of the Yutsis-type of spin network is developed for this purpose. Special attention is given to symplectic reduction, the reduced phase space of the 6j-symbol and the reduction of Poisson bracket expressions for semiclassical amplitudes.

Introduction
The Wigner 6j-symbol or Racah W-coefficient)is a central object in angular momentum theory, with many applications in atomic, molecular and nuclear physics. These usually involve the recoupling of three angular momenta, that is, the 6j symbol contains the unitary matrix elements of the transformation connecting the two bases that arise when three angular momenta are added in two different ways.

More recently the 6j- and other 3nj-symbols have found applications
in quantum computing and in algorithms for molecular
scattering calculations , which make use of their connection with discrete orthogonal polynomials.
The 6j-symbol is an example of a spin network, a graphical representation for contractions between tensors that occur in angular momentum theory. The graphical notation has been developed since the ’60s.

The 6j symbol is the simplest, nontrivial, closed spin network -one that represents a rotational invariant. Spin networks are important in lattice QCD and in loop quantum gravity where they provide a gauge-invariant basis for the field. Applications in quantum gravity are described by Rovelli and Smolin , Baez, Carlip , Barrett and Crane, Regge and Williams , Rovelli and Thiemann .

There are three approaches to the evaluation of rotational SU(2)invariants.

• The Yutsis school of graphical notation
• The Clebsch-Gordan school of algebraic manipulation
•  The chromatic evaluation – which grew out of Penrose’s doctoral work on the graphical representation of tensors and is closely related to knot theory.

The asymptotics of spin networks and especially the 6j-symbol has played an important role in many areas. Aymptotics refer to the asymptotic expansion for the spin network when all j’s are large, equivalent to a semiclassical approximation since large j is equivalent to small h/2π. The asymptotic expression for the 6j-symbol the leading term in the asymptotic series)was first obtained by Ponzano and Regge. In the  same paper those authors gave the first spin foam model for quantum gravity. The formula of Ponzano and Regge is notable  for its high symmetry and the manner in which it is related to the geometry of a tetrahedron in three-dimensional space. It is also remarkable because the phase of the asymptotic  expression is identical to the Einstein-Hilbert action for three-dimensional gravity integrated over a tetrahedron, in Regge’s simplicial approximation to general relativity. The semiclassical limit of the 6j-symbol thus plays a crucial role in simplicial approaches to the quantization of the gravitational field.

Spin network notation

The 3j-symbol and Wigner intertwiner

Bras, kets and scalar products

Intertwiners

An SU(2) intertwiner is a linear map between two vector spaces that commutes with the action of SU(2) on the two spaces

Tensor products and resolution of identity

The outer product of a ket with a bra is represented in spin network language simply by placing the spin networks for the ket and the bra on the same page as shown below

The 2j symbol and intertwiner

|Ki> is  expressed in terms of the Clebsch-Gordan coefficients by

This vector can also be expressed in terms of the 2j-symbol,which is defined in terms of the usual 3j-symbol by

The invariant vector |Ki> can also be written,

Kets to bras

Bras to kets

Raising and lowering indices

Models for the 6j-symbol

For given values of the six j’s, the 6j-symbol is just a number, but to study its semiclassical limit it is useful to write it as a scalar product <B|A>of wave functions in some Hilbert space. This can be done in many different ways, corresponding to different models of the 6j-symbol.

The 12j-model of the 6j-symbol

The 6j-symbol is represented as a product of six copies of a 2j-symbol and four of a 3j-symbol. Using the definition for the
2j-symbol, the result is

The Triangle and Polygon Inequalities

This is equivalent to the triangle inequalities when n = 3. In general, it represents the necessary and sufficient condition that line segments of the given, nonnegative lengths can be fitted together to form a polygon with n sides

The 4j-model of the 6j-symbol

The 6j-symbol is proportional to
a scalar product

The Ponzano-Regge Formula