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University ofBRISTOL
Department of Physics
Final year project
Angular momentum coupling: spin networksand their evaluation by integration over
SU(2).
Author: Supervisor:N. G. Jones J. H. Hannay
April 2012
H H Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL
Abstract
In this work, the use of group integrals in the evaluation of spin networks is considered.The introduction of integrations over SU(2) is motivated by considering symmetry propertiesof spinors, and how spinors and their coupling properties are related to spin networks.Several spin networks are evaluated using integral methods, agreeing with prior calculationsusing different techniques. The applicability of these methods is then discussed within thecontext of quantum mechanics and spin network theory.
Acknowledgements
I wish to thank John Hannay for suggesting such an interesting and rich topic to explore, as wellas for the hours he has spent helping and advising me with physics both for the project belowand for anything else I troubled him with throughout the year.
I also wish to thank two other Johns. Firstly John Barrett for providing some clues whenwe were drowning in formalism. Secondly I would have struggled far more if John Baez and hisseminarists from 2000-2002 had not published their discussions online. Thanks go to them forproviding such a wonderful resource.
I thank my friends Pete, Mike and Katie who have gone through the course with me and sharedmany interesting discussions.
Finally I thank Madeleine Fforde for her help and support; she has had to talk far more aboutquantum mechanics than any classicist should.
N. G. JonesApril 2012
Notation
The following notation is used throughout:
A generic element of a group G will be denoted g, gi or R.
|G| =∑g∈G =
∫Gdg is the size or volume of the group.
The spin-j representation space of SU(2) is a 2j + 1 dimensional vector space acted on by(2j + 1) × (2j + 1) representation matrices. This space may be represented by the symbol V j ,or just the symbol j.
The (mn)th element of the spin-j representation matrix of g ∈ SU(2) is given by:[j gm n
]= D(j)(g)mn = ρj(g)mn.
The summation convention is enforced, taking the first index as ‘contravariant’ and the sec-ond index as ‘covariant’.
The character of the element g ∈ SU(2) in the spin-j representation is written:
[j; g] = χj(g).
ii
Contents
Acknowledgements ii
Notation ii
1 Introduction 1
2 Reasoning with diagrams 22.1 Diagrammatic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3 Spin and groups 43.1 Symmetries of a physical system and representations . . . . . . . . . . . . . . . . 43.2 SU(2) as a symmetry of a particle with spin . . . . . . . . . . . . . . . . . . . . . 43.3 Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 Spin wavefunctions as symmetric spinors . . . . . . . . . . . . . . . . . . . . . . . 53.5 Orthogonality relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.6 The Wigner 3j and 6j symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.7 The projector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.8 The 3j symbol as the projector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Spin networks 104.1 Penrose networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Diagrammatic spinor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3 Spin networks as operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.4 The 3j and 6j symbols as networks . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5 Evaluating spin networks as integrals 155.1 General method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.2 Computing integrals over SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.3 The cylinder network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.4 The prism network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6 Discussion 21
7 Conclusions 22
Bibliography 23
A Straightening out a diagram kink 24
B The cylinder calculation in detail 25
iii
1 Introduction
The aim of this project is to understand how and why integrals over SU(2) appear in theevaluation of spin networks. The answer to this question requires an understanding of how tointerpret spin networks as a diagrammatic spinor calculus, which gives a meaning and justificationto introducing integrals into the picture.
In this document the main elements of the theory required to understand how to go from aspin network to an integral are outlined. A blind algorithm is given in section 5.1, and this wouldsuffice to solve a certain class of problems, however it is useful to try to see how it originates. Thefirst three sections are thus an attempt to summarise some very broad fields which are relevantto the problem at hand - the evaluation of spin networks.
The first section deals with diagrammatic reasoning: turning algebra into diagrams in orderto make things simpler. This kind of reasoning is common in category theory, and spin networkscan be interpreted as a particular type of diagram. This helps to justify later assertions aboutspin networks.
The second section discusses the theory of angular momentum coupling and why the groupSU(2) is relevant to the discussion at all. Representations of SU(2) are discussed and objectssuch as the Wigner 3j symbol are defined, both of which are essential for understanding spinnetworks.
Spin networks themselves are then introduced in the third section. Some discussion of the selfcontained “Penrose theory” is given, before a useful interpretation is given in terms of spinorsand operators. From this understanding the algorithm introduced in section 5.1, can then bewell justified.
Thus, with the background in place, some results are explained in detail. First the methodused to turn the network into an integral is given. This method is then illustrated in severalexamples. Each of these examples also has some interest in the physics of spin networks, andhence some discussions follow.
1
2 Reasoning with diagrams
2.1 Diagrammatic notation
Manipulating multi index quantities is often difficult notationally, and makes some argumentsmore opaque than necessary. One way of improving the situation - popularised by Penrose - isto draw a diagram to represent the mathematical operations in the manner of a flow chart. Infact, the whole of linear algebra can be reformulated in the language of these diagrams.
A linear map, f , from V1 to V2, which is given by a matrix of numbers once bases for thevector spaces are chosen, is drawn as:
f
�� V1
�� V2
.
The ‘direction’ of the diagram is top to bottom, so an element of the domain, V1, is fed in atthe top and an element of the image, V2, is given out at the bottom (be aware, however, thatthe opposite convention is often used in the literature). The whole diagram corresponds to the
map f in the following way: the symbol f corresponds to the actual map and the edges in
and out represent the domain and codomain. The edges should thus be labelled by the vectorspaces which the circled functions map into and from. The numbers of input and output edgescorrespond to how many ‘input’ and ‘output’ vectors there are - or more concretely a tensor ofvalence (p, q) will be represented by a circle with p output lines and q input lines. We can now seehow to represent an element of a vector space in this picture - simply as a function with a singleoutput, or as v : C → V . Function composition is vertical, so given a second map g : V2 → V3,we can draw gf = g ◦ f : V1 → V3 as:
f
�� V1
g
�� V2
�� V3
= gf
�� V1
�� V3
.
In matrix language, if the matrix of f is Fmn and the matrix of g is Gsm, then gf has matrixMsn = GsmF
mn where the abstract index1 m is summed over. Thus internal edges on the diagram
are ‘to be summed over’.The tensor product is the horizontal direction on the diagram. For example the map h =
f ⊗ g : V1 ⊗ V2 → V ′2 ⊗ V ′3 is represented by the diagram:
f
��V1
��V ′2
g
�� V2
�� V ′3
= h
�� V1⊗V2
�� V ′2⊗V ′3
.
1 See, for example, Penrose and Rindler [12].
2
CHAPTER 2. REASONING WITH DIAGRAMS 3
There are also some ‘natural’ functions which need not be labelled - they are really part ofthe picture.2 The identity map, for one, but there are also the counit and unit, which are naturalonce we consider the dual space of covectors (or linear functionals) on V , denoted V ∗. The dualspace is represented on the diagrams by an arrow labelled V but directed up rather than downthe diagram. These natural maps have the following definitions: the counit takes in a covectorand a vector and gives out a number:
eV : V ∗ ⊗ V → C
f ⊗ v → f(v) ��V
YYV
.
This is just the covector acting on the vector. Its representation on the diagram is reminiscentof a cup. Then the unit takes in a number and gives out an element of V ⊗ V ∗:
iV : C→ V ⊗ V ∗
1→ IdVYY V
��V .
This is just using that any vector space has an identity map. Similarly one may say it lookslike a cap. The power of diagrammatic reasoning can be seen by considering the following usefulresult which involves these two maps and also the dual space:
OO V
OO'OO V
i.e. that this bend in the diagram can be ‘straightened out’. To prove this algebraically, the lefthand side must be shown to also represent the identity map on V ∗, which is done in AppendixA. Clearly the the diagrammatic rule that edges can be straightened out is much simpler to usein practice than repeating the involved algebraic manipulations. This is one special case of amore general result: that any two diagrams which can be continuously deformed into one anotherrepresent identical maps. Diagrams that are not equivalent in this way can still represent thesame map, but care must be taken in ‘unknotting’ them. In fact when we consider spin networksbelow, the topological invariance only holds up to a sign.3
This brief discussion only gives the key notions in the theory of linear algebra as diagrams,as diagrammatic reasoning is used throughout in the following. For a longer exposition see forexample Kauffman [7] or Baez and Muniain [3].
2For example any straight line can be interpreted as the identity map, just as at any stage in multiplication wecan say a multiplication by 1 has occurred. The word natural can be made precise, but here we use it informally.
3 In the spinor case.
3 Spin and groups
3.1 Symmetries of a physical system and representations
A quantum system is specified by some Hamiltonian operator, H. If a transformation of theconfiguration space leaves this Hamiltonian invariant it is a symmetry of the system. Thisstatement can be made precise by saying that if we transform a wavefunction ψ and then operatewith H we have the same result as if we had applied the transformation after operating with H.That is, the operator PR which carries out the transformation commutes with H. This discussionis continued in detail in Landau and Lifshitz [9].
The set of transformations R, which leave the Hamiltonian invariant, will form a group.The transformation will in general take the wavefunction into a linear combination of equivalentwavefunctions. The equivalent ψ will belong to the same eigenvalue of the Hamiltonian (Hψ =Eψ implies HPRψ = PRHψ = EPRψ). This means that in general, if the eigenvalue is n-folddegenerate with eigenfunctions {ψj}, then we can write PRψj =
∑ni=1D(R)ijψi. The D(R)
will in this way form a representation of the group. A representation should be considered as apair, a vector space and a set of endomorphisms of this space. The endomorphisms are these Dmatrices, and the vector space here is the portion of the Hilbert space of states of the systemwith a particular energy.
In fact we can think of the Hilbert space of the whole system as a direct sum of theserepresentations. The state vectors will in general be orthogonal to all but one of these spaces ifthey have a particular energy. Then the action of a group on the state vector will be given bythe appropriate representation matrix on each part of the space. To clarify, following Wigner[14], the state vector is written as:
φ =∑l
a(l)νψ(l)ν
where l indexes the different representation spaces. Then after applying the transformation:
PRφ =∑l
a(l)ν[l Rµ ν
]ψ(l)µ
and we see that the operator has not mixed the vector components from the different subrepre-sentation spaces.
3.2 SU(2) as a symmetry of a particle with spin
From general considerations, in particular the isotropy of space and the equivalence of differentobservers, it can be shown how a spin-half wavefunction of the form:
ψ(x, y, z, s) = u1δs,1ψ(x, y, z) + u2δs,−1ψ(x, y, z) =
(u1u2
)ψ(x, y, z)
will transform after a rotation R in order to leave all physical quantities invariant.1 The possi-bility of a spin-free experiment (e.g. the measurement of the position of an electron) means thatall wavefunctions in non-relativistic quantum mechanics can be written in this form. The spatialpart of the wavefunction will transform as:
ψ′(x, y, z) = PRψ(x, y, z) = ψ(R−1(x, y, z)).
1 Here s is the two valued spin variable.
4
CHAPTER 3. SPIN AND GROUPS 5
The spin part
(u1u2
)must transform unitarily to conserve probabilities. In fact the set of all
transformation matrices D(R), together with the two wavefunctions (1, 0) and (0, 1), will form arepresentation of the group SU(2) (this is discussed extensively in Wigner [13])2. Higher spinscorrespond to higher dimensional representations of SU(2).
Now, because any object will have a spin of some value (perhaps zero), any physics whichchanges the spin of the object(s) will be mapping between representations of SU(2) in a waycompatible with this symmetry. That is, it will be represented by an ‘intertwining’ operator orintertwiner. Use of this term is common in the literature, and simply means a map betweenrepresentations which commutes with the group action. This commutativity could be motivatedby considering that redefining coordinates before an event or after an event should not make adifference to the physics. As an example, Schur’s lemma states that there are no intertwiningoperators between the spin-j and spin-k representations for j 6= k. Physically we interpret thisas the conservation of angular momentum of an isolated body of spin-j.
3.3 Spinors
Spinors3 are mathematical objects built up in the same way as tensors are built from vectors. Arank one spinor is an element ψa of V = C2, with V having a built in symplectic structure. Thissymplectic structure is an antisymmetric inner product, and will be given as an antisymmetrictwo-form ωij . In two dimensions this must be proportional to εij , the Levi-Cevita symbol.
This inner product allows us to define a dual space V ∗. It contains objects ψa = εabψb.
Then φa · ψb = φaεabψb = φaψa. From the antisymmetry of the product, ψaψa = 0. The group
SL(2,C) acts on spinors by matrix multiplication in the usual way: g · ψb = gabψb. The inner
product is invariant under this action: gacucεabg
bdvd = det gucεcdv
d (after doing the multiplicationin components) and det g = 1 for this group. Note that as SU(2) is a subgroup of SL(2,C), thisinner product is also invariant under the smaller group.
Rank one spinors are rotated by an angle φ about an axis n in space by operating withUn = exp(iφ2n · σ) ∈ SU(2), where σ is a list of Pauli matrices. Higher rank spinors are builtby the outer product: ψab =
∑n χ
a(n)φ
b(n). Their transformation properties under a rotation are
governed by those of the rank one spinors.
3.4 Spin wavefunctions as symmetric spinors
Any wavefunction of a spin-j particle can be represented as a symmetric spinor: ψ(s)↔ ψα1...α2j .This has the advantage that the SU(2) symmetry under rotations is explicitly built in to thetransformation properties of the spinor. Also, as the inner product is invariant, probabilities offinding a particle at a point will be unchanged - as they must be. The components of ψ will begiven by:
ψ(s) =
√(2j)!
(j + s)!(j − s)!ψ
1...1j+s
2...2j−s
where the index 1 appears j + s times and the index 2 appears j − s times.4 This builds up thehigher spins from spin-half components, which is justified in Landau and Lifshitz [9].
2 In fact they will form the defining representation of SU(2).3 Precisely, 2-spinors. There are different ways of defining spinors, this follows Baez and Alvarez in [2].4 We consider the wavefunction at the origin, which is left invariant under rotations, so that only the spin
part is relevant.
CHAPTER 3. SPIN AND GROUPS 6
A system of two particles will be a product of two symmetric spinors, which is in general notsymmetric. It can, however, be decomposed into symmetric spinor ‘parts’ by contracting indicespairwise (one from each spinor - contraction with two indices from the same spinor will yield zeroas their contraction is antisymmetric whereas the indices themselves are symmetric) and at eachstage symmetrising5 the result. This procedure is described in Chapter 3 of Penrose and Rindler[12]. In this way, we can interpret the product as a superposition of systems with a total spinnumber between |j1− j2| and j1 + j2. The higher limit is evident as it comes from symmetrisingthe product wavefunction. The lower limit comes from the lowest valence symmetric spinorthat can be made by contracting across the spinors - by summing over all of the indices of thewavefunction of the smaller spin.
3.5 Orthogonality relations
The spin-j representations are irreducible - that is they contain no proper subrepresentations.This leads to a number of useful orthogonality results, derived in Wigner [13], for coefficients ofthe representation matrices. The most general of which is:
∑R
[j1 Rm n
] [j1 Rr s
] √lj√lj′
|G|= δjj′δmrδns (3.1)
where lj is the dimension of the spin-j representation - in the case of SU(2) this is 2j+ 1. Theserelations underlie many results in group theory. For a continuous group, such as SU(2), this sumis the invariant (Haar) integral.
3.6 The Wigner 3j and 6j symbols
3.6.1 In quantum mechanics
Following Landau and Lifshitz, and in a similar way to the procedure described in section 3.4, asystem composed of three particles of spins j1, j2 and j3 can have a spin of zero if the productof the three symmetric spinor wavefunctions can be reduced to a scalar by contraction. This isonly possible if the admissibility conditions are satisfied: that the ji can form three sides of atriangle and that their sum is an integer. This product wavefunction is of the form
ψ(1)(α1...α2j1 )ψ(2)(β1...β2j2 )ψ(3)(γ1...γ2j3 ).
To reach a scalar, all indices must be contracted across pairs of wavefunctions. Any contrac-tion within one of the three ‘factor’ wavefunctions will yield zero, as each is symmetric. If theji satisfy the triangle inequality this can be done in a canonical way - take j1 + j2 − j3 upperindices from ψ(1) and contract these with lowered ψ(2) indices, then j2 + j3 − j1 upper indicesfrom ψ(2) are contracted with lowered ψ(3) indices and finally the remaining j1 + j3 − j2 upperindices of ψ(3) are contracted with the lowered ψ(1) indices. (We should compare this procedureto diagram 4.4.) When translated back to the wavefunction picture, this gives the 3j symbols asthe coefficients in the following sum:
ψsystem0 =
∑m1,m2,m3
(j1 j2 j3m1 m2 m3
)ψ(1)j1,m1
ψ(2)j2,m2
ψ(3)j3,m3
. (3.2)
5 I.e. adding combinations of the spinor with permuted indices such that the resulting combination is symmet-ric. For example ψ(ij) = 1
2(ψij +ψji), where the brackets around the indices on the left indicate symmetrisation.
CHAPTER 3. SPIN AND GROUPS 7
By considering the pair ψ(1) and ψ(2) as a single coupled system, and then coupling thiscoupled system to ψ(3), we can find two expressions for the scalar ψ0. This gives an equation forthe combined wavefunction of a system of two particles, expressed as a superposition of singlesystems of a certain total angular momentum, in terms of 3j coefficients:
ψjm =∑m1,m2
(j1 j2 jm1 m2 m
)ψ(1)j1,m1
ψ(2)j2,m2
. (3.3)
Equation 3.2 also invites the contravariant and covariant notion for the spin indices m. Theleft hand side of this equation is a scalar under rotation of the coordinates, so if the spinor com-ponents transform contravariantly under rotation, then the 3j symbol will transform covariantlyin order for the sum to remain invariant. This is slightly different to tensors in spacetime, as inthat case each set of components transforms in the same way. Here, however, this transformationdepends on which representation each spinor is in, and hence which j values appear in the 3jsymbol.
The 6j symbol is a scalar built from the contraction of four 3j symbols. As each 3j symbolmust have j values which form the sides of a triangle, in order for this contraction to work the6j symbol must have six j values which can together form the sides of an irregular tetrahedron;each face of which corresponds to a 3j symbol. The defining equation is:{
j1 j2 j3j4 j5 j6
}=∑mk
(−1)∑
i (ji−mi)
(j1 j2 j3−m1 −m2 −m3
)×(
j1 j5 j6m1 −m5 m6
)(j4 j2 j6m4 m2 −m6
)(j4 j5 j3−m4 m5 m3
) (3.4)
where the sum is taken over all m values.6 This symbol has multiple symmetries due to thesymmetries of the 3j symbol. In fact the tetrahedron picture is again useful - if the ji whichform an irregular tetrahedron is deformed into a regular tetrahedron, then any symmetry of thisregular tetrahedron is a symmetry of the 6j symbol on interchange of the relevant ji.
The symbols arise in physics when we consider the addition of three angular momenta. Ifthey are all mutually weakly coupled, then one can proceed by coupling the first two with a 3jsymbol and then coupling this composite system to the third. Another equally valid methodwould be to couple the second two, and then couple this composite system back to the first. The6j symbol facilitates the change from one method to the other.
3.6.2 In group theory
The 3j symbol of a simply reducible (S.R.) group7, such as SU(2), is defined by the unitarymatrix which brings the tensor product of two representations into reduced form as follows:
Uj3,m3;m1,m2
(2j3 + 1)12
=
(j1 j2 j3m1 m2 m3
)(3.5)
6The 3j symbol is taken to be zero if its m values do not sum to zero.7One in which the tensor product of two irreducible representations contains no irreducible representation
more than once, see Wigner [14].
CHAPTER 3. SPIN AND GROUPS 8
where the 3j symbol is the quantity in parentheses on the right hand side. U here is the unitarymatrix which ‘diagonalises’ the product representation:
[j1 g
] [j2 g
]= D(j1) ⊗D(j2) = U†
j1+j2⊕i=|j1−j2|
D(i)
U = U†
D(j1+j2)
. . .
D(|j1−j2|)
U.
(3.6)In fact, for a general S. R. group, the right hand side should contain the conjugate representations
D(j), but for SU(2) these two representations are essentially the same.As the spinors form irreducible representations of SU(2) this is just another way of looking
at how to find a particular spin state from the product of two spinors. From the group point ofview the 3j selects the appropriate representation of that spin; from the spinor point of view the3j symbol gives the appropriate symmetric spinor.
3.7 The projector
If we consider a representation of G, P =∫GdgD(g) is a projector. This is because if v = P u =∫
GdgD(g)u, then:
P 2u = P v =
∫G
∫G
dgdg′D(g)D(g′)u =
∫G
∫G
dgdg′D(gg′)u =
∫G
dgv = v = P u.
The fourth equality uses the translation property of the normalised Haar measure dg, which isguaranteed to exist for a certain class of groups (including SU(2)) as discussed in Kosmann-Schwarzbach [8]. As u was arbitrary, we have the projector property P 2 = P . Now considerv = P u. Then:
D(g′)v =
∫G
D(g′)D(g)udg =
∫G
D(g′g)udg =
∫G
D(h)udh = v.
As this is true for all g′ ∈ G it must be that D(g′) is the identity representation and that v is inthe spin-zero representation space. Thus P projects onto this representation. Note that as thespin-j representation is irreducible it will not contain this representation and so this operationwill send a vector in the spin-j representation to zero (not spin zero).
3.8 The 3j symbol as the projector
Now consider a product of representations: a ⊗ b ⊗ c. A representation matrix in this space(which will be reducible) will be given by:[
a gm1 n1
] [b gm2 n2
] [c gm3 n3
].
The projector onto the spin-zero subspace will then be:∫G
dg
[a gm1 n1
] [b gm2 n2
] [c gm3 n3
].
CHAPTER 3. SPIN AND GROUPS 9
Using equation 3.6, the orthogonality relations between representations and also the orthogonal-ity relations expressing the unitarity of U (which appears in equation 3.5) we can show:∫
G
dg
[a gm1 n1
] [b gm2 n2
] [c gm3 n3
]=
(a b cm1 m2 m3
)∗(a b cn1 n2 n3
)=
(m1 m2 m3
a b c
)(a b cn1 n2 n3
). (3.7)
This result is found in Wigner [14], along with a discussion of why the complex conjugated 3jsymbol is ‘contravariant’. As the left-hand side acts as a projector, the two 3j symbols on theleft do too.
4 Spin networks
Spin networks were introduced by Penrose in [11] in an attempt to establish spacetime as an emer-gent property of some underlying discrete physics. The inspiration seems to be the probabilisticnature of quantum mechanics - perhaps these fundamental probabilities of an event occurring aresimply the ratio of the number of ways circumstances will lead to the event and the number of allpossible circumstances. This particular fundamental research programme seems to no longer bepursued, however some concept of a spin network has survived and is useful in modern theories.This concept is somewhat expanded and generalised in the modern literature as discussed byBaez [1]. In what follows, only SU(2) spin networks are considered. They have a clear physicalinterpretation in standard quantum mechanics, as discussed below.
4.1 Penrose networks
121
1
12
1
12
1
12
12
‘Closed’ network
j1 j2
j3
‘Open’ network
Figure 4.1: Spin networks
In Penrose’s original conception, a spin network is a trivalent graph with a label, ji, on eachedge. Physically each edge would correspond to an isolated system of total angular momen-tum ji
1. The only ‘dynamical’ element of this picture is the interaction between two systems,represented at a vertex. Thus each vertex can be viewed as (any) two of the systems coming to-gether and combining to give a single system of the third given total angular momentum. Thesevertices obey the triangle inequality for angular momentum addition (conservation of angularmomentum) and also j1 + j2 + j3 ∈ Z, which can be interpreted as conservation of fermionnumber.
Associated to each closed spin network is a number, its norm. This number is a kind ofquantum mechanical probability amplitude associated to the results of a certain experiment:measuring the angle in space between two large spinning bodies (represented by some large spinnetwork)2. This norm is derived from a related concept, the value. The value is a number assigned
1 Actually ji2
in the original paper in order to have graphs labelled by integers only - emphasising combina-torics.
2 This is an important experiment in this model, as the whole point of the research programme was to see iftaking some basic object outside of space and seeing if space emerges. In fact, Penrose recovers Euclidean 3-space,but it is unclear how this is related to taking the combination rules from standard quantum mechanics whichitself depends on the geometry E3.
10
CHAPTER 4. SPIN NETWORKS 11
Normj1 j2
j3
= j1 j2
j3
j1 j2
j3
Norm
Figure 4.2: The norm of an open network
to a closed network, which is found combinatorially using a formula described by Penrose in [11].The formula is evaluated directly from the diagram and roughly involves counting crossings atvertices. The norm of a closed network is the modulus of this value, and the norm of an opennetwork is the modulus of the value of the ‘square’ of the open network. We square a network bydrawing a second copy of the same network, and connecting all loose edges of the first copy tothe corresponding loose edge on the second copy (see figure 4.2). This is now a consistent closednetwork and its value can be found.
4.2 Diagrammatic spinor algebra
As discussed above, rank one spinors come with a vector space structure - which gives us the iden-tity map, unit and counit. There is also the symplectic structure, which takes in two spinors andgives a complex number, antisymmetric in its two arguments. This is drawn diagrammaticallyas:
��V1�� V2
= −
��V2�� V1
(4.1)
where V1,2 = C2, the state space of a spin-half particle. Combining this operator with the counitgives a natural isomorphism3:
�� VOOV
(4.2)
going from V → V ∗. This is just like index lowering defined in section 3.3. Similarly there isan index raising operator with the arrows inverted. Because we can insert these operators intothe diagram anywhere, the arrows can be dispensed with altogether. All diagrams with differentdistributions of arrows (indicating the spin-half space and its dual) will be isomorphic. Thesymplectic structure makes unknotting diagrams more difficult - signs can change when pullinga loop straight. What look like thin strings in the diagram could hence be better represented byribbons, with a sign associated to a twist.
Higher spins are constructed as symmetrised products of this spin-half space. The Hilbertspace of a spin-j particle is V j = j = S2j(V ), which is the symmetric part of V ⊗2j . We cannow reformulate section 3.6 as diagrams, as all calculations simply involved products of spinorsand contractions. The only other operation required is symmetrisation. We indicate this by a
3 There is a subtlety depending on which index of epsilon the counit is coupled to. Here the second index ischosen.
CHAPTER 4. SPIN NETWORKS 12
wavey line over the indices (or spaces) which we are symmetrising over. As spin-j is built fromsymmetrised spin-half spinors, the whole diagram only requires edges labelled by the spin-halfspace. The convention is thus that an unlabelled line should be read as having a label of 1
2 . AsS2j(V ) ⊂ V ⊗2j , the following diagram projects into the symmetric part of V ⊗2j :
j.
This diagram also works upside down, which is the map including the spin-j space into V ⊗2j . Nowthat the higher representations can be broken up into spin-half ‘strands’, maps between theserepresentations can be constructed from the spin-half maps which we have already defined4 -diagrammatically the cup shape, the cap shape and the identity.5 In fact it turns out that allintertwining maps can be constructed in this way.
4.3 Spin networks as operators
Penrose networks were trivalent graphs with edges labelled by spin numbers, which had quantummechanics in the background. The previous section on spinor diagrams provides graphs withedges labelled by spin numbers which represent maps from products of representation spacesto other products of representation spaces. These two pictures are actually almost the same,and so Penrose networks can also be interpreted as maps (or multilinear maps - tensors). Thedifference is explained in Kauffman [7], and is to do with a sign ambiguity in the spinor diagrams(that only comes in when you identify V with V ∗). The solution is to change spinors to ‘binors’,and this is what Penrose did when he invented spin networks. This introduces a sign to eachcrossing of spin-half strands (and so symmetrisers become antisymmetrisers, which appear inPenrose’s evaluation of the network). This sign cancels out the sign in equation 4.1, so all thesign information is contained in the number of crossings. Penrose networks are fully topologicallyinvariant. The spinor networks, when arrows are dispensed with, will have sign ambiguities.
4.4 The 3j and 6j symbols as networks
4.4.1 The 3j network
The 3j network is essential to both pictures of spin networks. The symbol is the vertex inthe Penrose network; this is clear if one considers Penrose’s description of the vertex as twobodies slowly coupling their angular momenta. It is also the diagrammatic representation ofthe intertwining map from a triple product of irreducible representations of SU(2) into C, as inequation 3.2. The 3j symbol is given, up to normalisation, by the diagram:
a b c=
a b c. (4.3)
This is actually the only map built from epsilons from the product space a ⊗ b ⊗ c intoC which is non-vanishing. To avoid having any ‘loose’ strands at the bottom of the diagram,
4 Or intertwiners - the technical term for symmetry compatible maps discussed above.5 Whether a cup, for example, is a counit or an epsilon doesn’t matter thanks to 4.2.
CHAPTER 4. SPIN NETWORKS 13
every strand at the top of the diagram must be connected to another. If two strands fromthe same representation are linked up, then an antisymmetric product of two symmetric indicesis taken - causing the whole diagram to vanish.6 In fact all intertwining maps involving threerepresentations are proportional to the 3j symbol - this is the Wigner-Eckart theorem of quantummechanics.7 Now, using the invariance of the diagram under a deformation, the space c can berotated round to the bottom of the diagram to give a map from a⊗ b to c.
a b
c
=a b
c
. (4.4)
These diagrams are simply a different, perhaps clearer, way of writing the symmetric spinorcoupling rules given in section 3.4. The numbers of indices matched between each wavefunctionthere are exactly the same as the number of strands in the appropriate bundle above. Thesebundle numbers are clearly the only way of linking up the diagram without joining a represen-tation back to itself. Similar deformation possibilities give a map from one representation intotwo (a spontaneous decay) and no representations into three (correlated particle production).This is the same as Wigner’s co/contravariant formalism described in Wigner [14]. Diagram 4.3is the fully covariant 3j symbol, whereas 4.4 is a 3j symbol contravariant in the index labellingthe space c. A more mathematical way of looking at it, as stated in Moussouris [10], is thatHom(V1 ⊗ V2,C) is naturally isomorphic to Hom(V1, V
∗2 ), and that V ∗2 ' V2 in the diagrams.8
The theta network arises often in spin network calculations. θabc is just the 3j networkconnected to itself, with edges labelled a, b, c. The right-hand side of figure 4.2 is actually θj1j2j3 .
4.4.2 The 6j network
The Wigner 6j symbol is represented by a tetrahedral network.9 It appears naturally consideringa map from a⊗ b⊗ c to d:
f
b
d
ca.
This map can be expanded in the following s indexed basis (where the vertices are 3j networks):
a b c
d
s
6 In Penrose’s networks, this would be a symmetric product of two strands emerging from an antisymmetriser.7 This theorem and Schur’s lemma emerge naturally in the spin network formalism. In all cases which these
theorems forbid, you have a contraction of a symmetric object with an antisymmetric object, yielding zero. Thisis discussed by Baez and Alvarez in [2].
8 Hom(X,Y ) being the set of homomorphisms between X and Y .9 In fact the vector picture of the 3j symbol is a triangle, and the network picture is the dual of this - three
edges meeting at a vertex. The vector picture of the 6j symbol is a tetrahedron, and the dual of this is also atetrahedron.
CHAPTER 4. SPIN NETWORKS 14
but could equally well be expanded in the t indexed basis:
a b c
d
t .
The 6j symbol is the coefficient relating the two bases. It appears in the sum:
a
b s
c
d=∑t
{a b tc d s
}a
btc
d(4.5)
found for example in Kauffman [7]. Again, this equation is really just the same as in the canonicalangular momentum coupling theory. In one basis a is coupled to b first, then a⊗ b couples to c.In the second basis the coupling is the other way round. Diagram manipulations on both sidesof equation 4.5 give the 6j symbol as a tetrahedral network, modulo some simple factors:
{a b tc d s
}=
(−1)2s(2s+ 1)
θbctθtda
a bt
cd
s
. (4.6)
As motivation for its consideration, the 6j network is very important in Ponzano-Regge modelof (2+1) dimensional quantum gravity, as the spacetime manifold can always be built up out oftetrahedra. This is explained in Barrett and Naish-Guzman [4].
5 Evaluating spin networks as integrals
5.1 General method
The key diagrammatic identity for evaluating a spin network by group integral is given by:
1
θabc
a b c
ca b
=
∫dg
|G|g
a
a
g
b
b
g
c
c
. (5.1)
This is really just writing equation 3.7 in diagrammatic form. Seeing the equation as a diagramgives an heuristic picture of what the operation really is. Each side is a map from a ⊗ b ⊗ c toitself. On the left, the lower 3j symbol projects out the spin-zero part of the wavefunction in theproduct space, and then the upper 3j symbol ‘includes’ this scalar wavefunction back into theproduct space. On the right the wavefunction in the product space is ‘averaged over the group’,which only leaves the scalar part nonzero.
The θ−1abc factor on the left-hand side is there in order to normalise the projection operator
written in terms of 3j symbols. The |G|−1 factor on the right-hand side is there in order toemphasise that the integration should also be normalised. If the normalised invariant measureis used this factor is simply equal to one.
Given a network to evaluate, we look for pairs of vertices carrying the same representations.The diagram should then be deformed until the pairs are arranged as in the left-hand side of theequation above - then this equality allows us to introduce group integrals. If all the vertices inthe network can be paired off in this way, then the network will be reduced to a set of loops, onein each representation space, with each loop having possibly multiple representation matrices1
threaded along it. These can then all be moved round next to each other and composed - whichwill here be matrix multiplication - giving the representation matrix of one group element, whichis the group-product of all the others by the very nature of a representation. This matrix will beconnected to itself by a line labelled by the vector space it acts on, and this is simply the traceof the matrix - in other words, the character of the group element.
5.2 Computing integrals over SU(2)
In order to actually evaluate the group integral, we first need a parameterisation of the group.The preferred choice here is the ‘axis-angle’ parameterisation, with a group element correspondingto a direction n on the unit sphere, and an angle φ in the range [0, 2π]. The vector n is thenparameterised using standard spherical coordinates. This is a good parameterisation over theentire group (unlike, for example, the Euler angles, which are singular at the poles). The invariantmeasure dg which facilitates the extension of results in finite group theory to SU(2) was foundusing an argument due to Hannay. The result is:∫
SU(2)
f(g)dg =
∫S2
∫ 2π
0
f sin2 θ
2dθdn =
∫ 2π
0
∫ π
0
∫ 2π
0
f(θ, ρ, φ) sin2 θ
2sin ρdθdρdφ.
The derivation of the sin2 θ2 weight factor is explained in Jones [6].
1These circled g symbols are really group actions on the vector space, but they will be expressible as matricesin some basis.
15
CHAPTER 5. EVALUATING SPIN NETWORKS AS INTEGRALS 16
5.3 The cylinder network
5.3.1 4-Vertex Cylinder
The cylinder network, which is defined diagrammatically by:
C = af
b
c
d
e
is the simplest non-trivial spin network. Using Penrose’s methods its value was found by Hannayto be:
C = δafθabcθdef1
2a+ 1, (5.2)
where δ is the Kronecker delta. To evaluate by group integral, consider first the case where a = f(when the network is non-zero):
aa
b
c
d
e
= aa
b
c
d
e
= θabcθade
∫dg1dg2|G|2
g1
g1
g1
g2
g2
g2
aab
c
d
e
.
All the loops containing one group representation matrix are simply the trace of that matrix.The other loop, in the space a, is:
tr
([a g1m n
] [a g2n m′
])=
[a g1g2m m
].
CHAPTER 5. EVALUATING SPIN NETWORKS AS INTEGRALS 17
In the diagram, this corresponds to composing the two representation matrices. Continuing thediagrammatic argument:
C = θabcθade
∫dg1dg2|G|2
g1
g1
g2
g2
g1g2
aa
b
c
d
e
= θabcθade
∫dg1dg2|G|2
[a; g1g2][b; g1][c; g1][d; g2][e; g2]. (5.3)
After parameterising SU(2), this integral can be computed numerically. Some low dimensionalcases were computed analytically but no general integration method was found. The integralcan, however, be recognised as a sum over 3j symbols by using the algebraic identity (3.7) twice- once for g1 and once for g2. After several summations and uses of the orthogonality propertiesof representations one arrives at:
C = θabcθade
a∑i=−a
1
(2a+ 1)2= θabcθade
1
2a+ 1.
A detailed calculation is given in Appendix B. This is clearly the same as the strand result fora = f .
To fully recover the result (5.2), the square of this network must be considered. As a closednetwork is just a complex number, the square of the cylinder is simply two copies placed side byside (C⊗ C ' C).
C2 = af
b
c
d
e
fa
b
c
d
e
= af
cb
d
e
a f
cb
d
e
.
Inserting group integrals as before and taking the trace of the representations around each loop:
C2 =∏
Vertices, i
θi
∫ ( 4∏i=1
dgi|G|
)[a; g2g3][b; g1g2][c; g1g2][d; g3g4][e; g3g4][f ; g1g4]. (5.4)
Again this integral can be broken up into multiple simple sums over 3j symbols, which aftersome manipulations will become proportional to a term
∑fi=−f δaf . As this term will be zero
for a 6= f , this automatically gives the result (5.2) back at this stage; however continuing thesummation does lead to the square of that quantity, as it should.
CHAPTER 5. EVALUATING SPIN NETWORKS AS INTEGRALS 18
5.3.2 2n-vertex cylinder (ladder)
This technique is readily extended to a simple large network, defined diagrammatically as:
a1 a2 an−1
a1 a2 an−1
b0 b1 b2 b3 bn−1 bn bn+1 .
This diagram is deformed into:
a1 a2 an−1
a1 a2 an−1
b2 b3b0 b1 bn−1 bn bn+1 ,
which can now have group integrals inserted between similar, vertically opposing, vertices. Wecan then evaluate this integral using the orthogonality relations to give:
θb0a1b1
(n−2∏i=1
1
2ai + 1θaiai+1bi+1
)1
2an−1 + 1θan−1bnbn+1
. (5.5)
This expression holds for n > 2, otherwise the limits on the product are an issue. The case n = 2is the 4-vertex cylinder above and the case n = 1 is simply the theta network.
5.4 The prism network
P = ca
b
ca
b
kl
m
=
ca
b
ca
b
kl
m
CHAPTER 5. EVALUATING SPIN NETWORKS AS INTEGRALS 19
Then, using the group integral introduction:
∝∫dg1dg2dg3|G|3
g1
g1
g1
g3
g3
g3
g2g2g2 ca
b
ca
b
kl
m
Then, as above, we compose the representation matrices and then evaluate the diagram bytaking the trace giving:
P ∝∫dg1dg2dg3|G|3
[a; g1g2][b; g3g1][c; g2g3][l; g1][m; g2][k; g3]. (5.6)
The integral form of the square of the 6j symbol is:{j1 j2 j3j4 j5 j6
}2
=1
|G|3
∫dg1dg2dg3[j6; g1g
−12 ][j5; g3g
−11 ][j4; g2g
−13 ][j1; g1][j2; g2][j3; g3]. (5.7)
The similarity between this integral and the prism network integral is clear. Using the cyclicproperty of the trace and that the character is a class function, equation (5.6) can be rewrittenas:
P ∝∫dg1dg2dg3|G|3
[a; g1g−12 ][b; g3g
−11 ][c; g2g
−13 ][l; g1][m; g2][k; g3]. (5.8)
These considerations lead to:
P =
{l m kc b a
}2
up to some θ factors. The prism network’s evaluation is given by Moussouris in [10] (it shouldbe noted that Moussouris normalises all θ networks to one). The result for a general prism isthe product of two different 6js:
ca
b
c′a′
b′
kl
m
=
{l m kc′ b′ a′
}{l m kc b a
}.
CHAPTER 5. EVALUATING SPIN NETWORKS AS INTEGRALS 20
This agrees with the result above for the symmetric case a = a′, b = b′ and c = c′. The generalresult actually follows quite simply from the Wigner-Eckart theorem (stated in network formas in Baez and Alvarez [2]). We just show that the centre of the prism is proportional to a 3jnetwork with a 6j proportionality factor, and then the outer edges of the prism close up this 3jnetwork into a tetrahedral 6j network. The first step is illustrated here:
c′a′
b′
kl
m
=1
θmlk
a′ mc′
kb′
l
kl
m
.
6 Discussion
In all the concrete cases we looked at, the group integral was evaluated using the orthogonalityrelations, and not by doing the integral. A network containing 3j symbol pairs, as in the identity5.1 can be directly interpreted algebraically in terms of sums over these symbols, and hence thegroup integral step can be skipped to get the same results. Equally, having converted the diagramto an algebraic sum, the group integral can be introduced using equation 3.7. The relationshipbetween the methods of evaluation of a network of this form could be summarised as follows:
Closed Spin Network
Algebra Group IntegralPenrose norm
↓ Evaluate
�� **oo //
tt
Of course, the absolute value of any network can be found by squaring the network - placingtwo copies of the network side by side. Then there will be a pairing between each vertex andits copy. Also, the 3j symbol contains a certain arbitrariness in its definition - this is fixed bymaking its complex phase zero. A group integral over SU(2) has no such phase information,or ‘knowledge’ of this phase fixing, so any combination of 3j symbols which does contain thisarbitrariness cannot be expressed as an integral. Hence not all spin networks could be expressedin integral form. We should also note the one way nature of the evaluation. A spin network isnot recoverable from an equation or integral - for example two networks which differ by somecrossings of the edges can have the same value, as discussed in Barrett and Naish-Gutzman [4].
The integral form of the networks above were not easy to work with in general; this wasdue to the need to take characters of group products and the relative complexity of finding theparameters of this product (in fact only the angle was required, but this turned out to dependon the axes of the factor elements, so all the integration variables were coupled together). Theintegral form was, however, susceptible to solution in low dimensional cases, and this gave thesame results as the sum over 3j symbols. Of course from the group theory this was to beexpected, but it showed that there had not been an error in the parameterisation or in any ofthe diagrammatic manipulations. A similar conclusion regarding the use of the group integralform of sums over 3j symbols was also reached by Wigner in [14]:
“In most cases, it will be easier to compare the six-j-symbols by the [relations withthree-j-symbols] than by [the group integral relations], and use [the group integralrelations] for the evaluation of sums or integrals over the group.”
The 6j symbols (or combinations thereof) referred to by Wigner are representable as suitablysymmetric spin networks.
The integral form is still useful to be aware of and have available. Firstly it can be evaluatednumerically; Monte Carlo integrations were implemented successfully for the cylinder network.There are explicit formulae for the 3j symbols, for example in Landau and Lifshitz [9], but theseare over technically infinite sums (although they reduce to finite sums by noting that for n ∈ N,|(−n)!| =∞) and so may cause difficulty in implementation. In any case, having two numericalpossibilities is certainly preferable to one if either is causing computational problems. Secondly,integrals are susceptible to certain asymptotic analyses. For example, the asymptotics of the 6jsymbol are found using a modified stationary phase method by Friedel and Louapre in [5]. Whilethere are asymptotic methods available for these sums, the integral form is the most natural.Barrett and Naish-Gutzman [4] convert diagrams to integrals because the integrals are easier towork with mathematically for some purposes than diagrams.1
1In their case this was needed to show that ‘regularisation’ of the diagrams is possible.
21
It is also interesting to consider spin networks in two contexts - one as a diagrammaticreformulation of the standard spinor coupling theory of quantum mechanics, and one as a startingpoint for a fundamental theory. In the calculations in this project the first attitude was taken,justifying the introduction of group integrals to the diagrams. As Penrose did not explicitlyinclude these group integrals in his theory, perhaps they should not be introduced into thediagrams as freely. The rules for his diagram evaluations are derived from quantum mechanics,so of course would agree with the integral results above, however if a combinatorial theory wassought which diverged from the background quantum mechanics, this group integral introductionwould need further justification. (Penrose networks are naturally SU(2) invariant, so perhaps ifa certain symmetry of the theory were postulated, then this could bring in a natural group sumor integral.)
The examples we chose to evaluate both have some importance. The ladder network wasconsidered as a natural generalisation of the cylinder network; however, Hannay noted that itis an important network in Penrose’s theory, as it is the form of the network which essentiallymeasures the angles between two bodies of large spin. The angle-measuring experiment involvesthe exchange of single spin units between two bodies emerging from one even larger body, andmust be repeated in order to find the angle. Hence from some large spin network a ladder networkemerges with bi = 1
2 for all i - representing multiple exchanges between the two bodies whicheach have a total spin of ai before each exchange. In general, of course, the two spins may havedifferent total spin, so a less symmetric ladder than the one evaluated here represents the generalexperiment.
The prism network decomposes into two 6j symbols and is the simplest network which de-composes into tetrahedra. In general any large network can be decomposed into tetrahedra, asdiscussed by Barrett and Naish-Gutzman [4], but this is deduced from topological arguments -here we saw it by evaluating the network as an integral.
7 Conclusions
Spin networks were seen as a good way to formulate quantum angular momentum coupling- visualising and clarifying the spinor formalism. The diagrams illuminate certain propertieswhich would be more opaque in index form. The conditions for a spin network to be reducible toan integral were found, and how to reach this integral was explained. The integral form did notseem to be easier to work with analytically than the spin network; however it is certainly usefulto have available. We discussed its usage in numerical calculations, asymptotics and bringingsome mathematical methods of integrals to spin networks.
22
Bibliography
[1] J. C. Baez, Spin networks in nonperturbative quantum gravity, The Interface of Knots andPhysics (1996), 167–203.
[2] J. C. Baez and M. C. Alvarez, Quantum gravity, http://math.ucr.edu/home/baez/
qg-fall2000/QGravity/QGravity.pdf, 2005.
[3] J. C. Baez and J. P. Muniain, Gauge fields, knots and gravity, World Scientific, 1994.
[4] J. W. Barrett and I. Naish-Guzman, The Ponzano-Regge model, Class. Quant. Grav. 26(2009), no. 15, 155014.
[5] L. Friedel and D. Louapre, Asymptotics of 6j and 10j symbols, Class. Quant. Grav. 20(2003), 1267–1294.
[6] N. G. Jones, Interim Project Report: The Wigner 6j symbol’s group integral formula, Un-published (2011).
[7] L. H. Kauffman, Knots and physics, World Scientific, 2001.
[8] Yvette Kosmann-Schwarzbach, Groups and symmetries, Springer, 2010.
[9] L. D. Landau and E. M. Lifshitz, Quantum Mechanics (non-relativistic theory), Heinemann,1958.
[10] J. P. Moussouris, Quantum models of space-time based on recoupling theory, Ph.D. thesis,Oxford, 1983.
[11] R. Penrose, Angular momentum: an approach to combinatorial space-time, Quantum Theoryand Beyond (T Bastin, ed.), Cambridge University Press, 1971.
[12] R. Penrose and W. Rindler, Spinors and space-time, vol. 1, Cambridge University Press,1984.
[13] E. Wigner, Group theory and its application to the quantum mechanics of atomic spectra,Academic Press, 1959.
[14] , On the matrices which reduce the Kronecker products of representations of S.R.groups, Quantum Theory of Angular Momentum (L. C. Biedenharn and H. Van Dam, eds.),Academic Press, 1965.
23
A Straightening out a diagram kink
OO V
OO
=OO V
OO
=
p
I ⊗ iV
eV ⊗ I
π
�� V ∗⊗C
�� C⊗V ∗
�� V ∗⊗V⊗V ∗
�� V ∗
�� V ∗
The diagram on the left is the composition (vertical) of the four smaller diagrams in the centre.Each of these smaller diagrams is interpreted using the natural maps and the diagram rules. (Forexample, an undisturbed line going from top to bottom is an identity map. The arrow on theline indicates if it is an identity on V or V ∗). The map p takes a vector and adjoins 1 ∈ C - thisis like multiplying by one so doesn’t change the vector at all. The map π is the inverse to p andprojects out the interesting part of a product of a vector space with C - i.e. the vector space.
Now, to show that we can straighten out the diagram on the left, is to show that it is simplythe identity map on V ∗ (as the arrows point upwards in the diagram). Take some elementα ∈ V ∗, along with any basis {e(i)} for V . Then applying the diagram to α is simply evaluating:
D(α) = (π ◦ (eV ⊗ I) ◦ (I ⊗ iV ) ◦ p)(α) =
(π ◦ (eV ⊗ I) ◦ (I ⊗ iV ))(α⊗ 1) =
(π ◦ (eV ⊗ I))(α⊗ e(i) ⊗ e(i)).
Now expand α in the same basis to give:
D(α) = (π ◦ (eV ⊗ I))(αje(j) ⊗ e(i) ⊗ e(i)) =
(π)(αje(j)(e(i))⊗ e(i)),
where eV is evaluating e(j)(e(i)) This gives δji by the definition of the dual basis. Then,
D(α) = (π)(αi ⊗ e(i)) = π(1⊗ αie(i)) = αie(i) = α.
Hence the diagram is indeed the identity map on V ∗.
24
B The cylinder calculation in detail
The following orthogonality relation comes from the 3j symbols being unitary:∑m1,m2
(j1 j2 jm1 m2 m
)(j1 j2 j′
m1 m2 m′
)=δjj′δmm′
2j + 1. (B.1)
This is equation (24.19) in Wigner’s book [13].Now, the integral of interest is:
C = θabcθade
∫dg1dg2|G|2
[a; g1g2][b; g1][c; g1][d; g2][e; g2].
The following method to evaluate C is slightly ‘counterproductive’ as really we’re reversing thesteps which led from diagram to integral. These steps were:
• Turn 3j symbol pairs into integrals over representation matrices.
• Compose these matrices.
• Take the trace around the loops.
To evaluate the integral using the orthogonality relation above, we need to undo all of these, butalgebraically now rather than diagrammatically.
So first we write the characters as the trace of matrices:
C = θabcθade
∫dg1dg2|G|2
∑mi
[a g1g2m1 m1
] [b g1m2 m2
] [c g1m3 m3
] [d g2m4 m4
] [e g2m5 m5
]Then using the ‘representation property’ of these matrices, we can split up the spin-a matrix asfollows:
C = θabcθade
∫dg1dg2|G|2
∑mi,n1
[a g1m1 n1
] [a g2n1 m1
] [b g1m2 m2
] [c g1m3 m3
] [d g2m4 m4
] [e g2m5 m5
].
Now, we have separated the integration variables in some sense, so can rewrite this as follows:
C = θabcθade∑mi,n1
∫dg1|G|
[a g1m1 n1
] [b g1m2 m2
] [c g1m3 m3
] ∫dg2|G|
[a g2n1 m1
] [d g2m4 m4
] [e g2m5 m5
].
We can use equation (3.7) to turn each of those single integrals into a pair of 3j symbols:
C = θabcθade∑mi,n1
(a b cm1 m2 m3
)(a b cn1 m2 m3
)(a d en1 m4 m5
)(a d em1 m4 m5
).
To be thorough we really need to use the symmetry properties of the 3j symbol - that is, thatthey are unchanged under cyclic permutations of the columns. Hence:
C = θabcθade∑
m1,m2,m3,m4,m5,n1
(b c am2 m3 m1
)(b c am2 m3 n1
)(a d en1 m4 m5
)(a d em1 m4 m5
).
Now we use equation (2) and evaluate the sum over m2 and m3 to give:
C = θabcθade∑
m1,m4,m5,n1
δaaδm1n1
2a+ 1
(a d en1 m4 m5
)(a d em1 m4 m5
).
25
APPENDIX B. THE CYLINDER CALCULATION IN DETAIL 26
Taking the sum over n1 now just sets m1 = n1, and clearly δaa = 1, so:
C = θabcθade∑
m1,m4,m5
1
2a+ 1
(a d em1 m4 m5
)(a d em1 m4 m5
).
Permuting as before and summing now over m4 and m5, with a further use of equation (2) gives:
C = θabcθade∑m1
1
2a+ 1
δaaδm1m1
2a+ 1= θabcθade
a∑m1=−a
1
(2a+ 1)2= θabcθade
1
(2a+ 1).
This was the required result.Of course, in the final step we need not have permuted the columns - the indices m1,m4 and
m5 were symmetric in their appearance. The outcome of that ended up being 2a+12a+1 and this
would have happened for any a (i.e. we could equally well have picked out d or e as special, we’dstill have eventually multiplied by one). The reason a appears in the final result is because in theoriginal problem a appeared differently as it had two indices associated to it (m1 and n1). Theladder works in the same way - just with some more “coupled” indices, which result in deltas,which are then summed over, just like m1 and n1.
