Semiclassical Mechanics of the Wigner 6j-Symbol

Abstract

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 devoted to symplectic
reduction, the reduced phase space of the 6j-symbol (the 2-sphere of Kapovich
and Millson), and the reduction of Poisson bracket expressions for
semiclassical amplitudes. General principles for the semiclassical study of
arbitrary spin networks are laid down; some of these were used in our recent
derivation of the asymptotic formula for the Wigner 9j-symbol.

Full text

Page 1

arXiv:1009.2811v2 [math-ph] 25 Aug 2011
Semiclassical Mechanics of the Wigner 6j-Symbol
Vincenzo Aquilanti
Dipartimento di Chimica, Universit` a di Perugia, Perugia, Italy 06100
Hal M. Haggard, Austin Hedeman, Nadir Jeevanjee,
Robert G. Littlejohn, and Liang Yu
Department of Physics, University of California, Berkeley, California 94720 USA
E-mail: robert@wigner.berkeley.edu
Abstract.
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 devoted to symplectic
reduction, the reduced phase space of the 6j-symbol (the 2-sphere of Kapovich
and Millson), and the reduction of Poisson bracket expressions for semiclassical
amplitudes.General principles for the semiclassical study of arbitrary spin
networks are laid down; some of these were used in our recent derivation of the
asymptotic formula for the Wigner 9j-symbol.
The semiclassical mechanics of the Wigner 6j-symbol is examined
PACS numbers: 03.65.Sq, 02.20.Qs, 02.30.Ik, 02.40.Yy
1. 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.
Such applications and the definition of the 6j-symbol based on them are described by
Edmonds (1960). More recently the 6j- and other 3nj-symbols have found applications
in quantum computing (Marzuoli and Rasetti 2005) and in algorithms for molecular
scattering calculations (De Fazio et al
2003, Anderson and Aquilanti 2006), which
make use of their connection with discrete orthogonal polynomials (Aquilanti et al
1995, 2001a,b).
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 by Yutsis et al (1962), El Baz and Castel (1972), Lindgren
and Morrison (1986), Varshalovich et al (1981), Stedman (1990), Danos and Fano
(1998), Wormer and Paldus (2006), Balcar and Lovesey (2009) and others. The 6j-
symbol is the simplest, nontrivial, closed spin network (one that represents a rotational

Page 2

Semiclassical Mechanics of the Wigner 6j-Symbol
2
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 (1995), Baez (1996), Carlip (1998), Barrett
and Crane (1998), Regge and Williams (2000), Rovelli (2004) and Thiemann (2007),
among others.
Alongside the Yutsis school of graphical notation and the Clebsch-Gordan school
of algebraic manipulation there is a third approach to the evaluation of rotational
(SU(2)) invariants. The third method, sometimes called chromatic evaluation, grew
out of Penrose’s doctoral work on the graphical representation of tensors and is closely
related to knot theory. We will not have further occasion to mention this school, see
Penrose (1971) for its introduction, Rovelli (2004) for an overview and Kauffman and
Lins (1994) for its full development.
The asymptotics of spin networks and especially the 6j-symbol has played an
important role in many areas. By “asymptotics” we 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 ?. The asymptotic expression for the 6j-symbol
(the leading term in the asymptotic series) was first obtained by Ponzano and Regge
(1968), or, more precisely, they obtained several formulas, valid inside and outside
the classically allowed region and in the neighborhood of the caustics. In the same
paper those authors gave the first spin foam model (a discretized path integral)
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 (1961) 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.
For all these reasons, the asymptotic formula of Ponzano and Regge for the
6j-symbol has attracted a great deal of attention.
their formula by inspired guesswork, supporting their conclusion both with numerical
evidence and arguments of consistency and plausibility. The formula itself is of the
one-dimensional WKB-type, a reflection of the fact that the 6j-symbol fundamentally
represents a dynamical system of one degree of freedom.
The Ponzano-Regge formula was first derived by Neville (1971), using the
recursion relations satisfied by the 6j-symbol and a discrete version of WKB theory.
Similar techniques were later used by Schulten and Gordon (1975a,b), who also
presented stable algorithms for evaluating the 6j-symbol numerically.
a different sort was later given by Biedenharn and Louck (1981), based on showing
that the Ponzano-Regge formula satisfies a set of defining properties of the 6j-symbol.
More recently there have appeared more geometrical treatments of the
asymptotics of the 6j-symbol, that is, those based on geometric quantization (Kirillov
1976, Guillemin and Sternberg 1977, Woodhouse 1991), symplectic geometry and
symplectic and Poisson reduction (Abraham and Marsden 1978, Arnold 1989, Marsden
and Ratiu 1999) and other techniques. Among these are the works by Roberts (1999)
and by Charles (2008). In addition, the 6j-symbol has been taken as a test case
for asymptotic studies of amplitudes that occur in quantum gravity (Barrett and
Steele 2003, Freidel and Louapre 2003), in which the authors developed integral
representations for the 6j-symbol as integrals over products of the group manifold.
There have also been quite a few other studies of asymptotics of particular spin
Ponzano and Regge obtained
A proof of

Page 3

Semiclassical Mechanics of the Wigner 6j-Symbol
3
networks, including Barrett and Williams (1999), Baez et al (2002), Rovelli and
Speziale (2006), Hackett and Speziale (2007), Conrady and Freidel (2008), Alesci
et al (2008), Barrett et al (2009), among others. We also mention the works of Gurau
(2008), which applies standard asymptotic techniques (Stirling’s approximation, etc)
directly to Racah’s sum for the 6j-symbol; of Ragni et al (2010) on the computation
of 6j-symbols and on the asymptotics of the 6j-symbol when some quantum numbers
are large and others small; and of Littlejohn and Yu (2009) on uniform approximations
for the 6j-symbol.
In addition there has been some work on the q-deformed 6j-symbol, important
for the regularization of the Ponzano-Regge spin-foam model (Turaev and Viro 1992,
Ooguri 1992a,b) and for its possible connection to quantum gravity with cosmological
constant. In particular, Taylor and Woodward (2004) applied the recursion and
WKB method of Schulten and Gordon to the q-deformed 6j-symbol. The results
are geometrically interesting (the tetrahedron of Ponzano and Regge is moved from
R3to S3when the q-deformation is turned on), but it seems that at present there is
no geometrical treatment of the asymptotics of the q-deformed 6j-symbol, analogous
to what is available for the ordinary 6j-symbol. There is also the recent work of Van
der Veen (2010) on the asymptotics of general q-deformed spin networks, which treats
the problem from the standpoint of knot theory and representation theory. Among
other things, this work creates a broad generalization of the Schwinger-Bargmann
generating function of the 6j-symbol.
In Aquilanti et al (2007) we applied multidimensional WKB theory for integrable
systems to the asymptotics of the 3j-symbol, and in this paper we apply similar
techniques to the 6j-symbol.These methods bear the closest relationship to the
works of Roberts (1999) and of Charles (2008). The point of this paper is not another
derivation of the Ponzano-Regge formula, although one is provided, but rather to
clarify the relationship among some of the methods used in the past, to reveal useful
calculational techniques, and to lay the basis for the development of new results.
Among the latter we mention our own work on uniform approximations for the 6j-
symbol (Littlejohn and Yu, 2009), our recent derivation of the asymptotic form for
the 9j-symbol (Haggard and Littlejohn, 2010), both of which relied on techniques
explained in this paper, and our work on the Bohr-Sommerfeld quantization of the
volume operator in loop quantum gravity (Bianchi and Haggard, 2011). Previous and
current work on the volume operator includes Chakrabarti (1964), L´ evy-Leblond and
L´ evy-Nahas (1965), Lewandowski (1996), Major and Seifert (2001), Carbone et al
(2002), Neville (2006), Brunnemann and Rideout (2008, 2010) and Ding and Rovelli
(2010).
In addition, this paper is distinguished by its use of what we call the “4j-model”
for the 6j-symbol, in contrast to the “12j-model” used by Roberts (1999). The 4j-
model is less symmetrical than the 12-model, but it is closer to the manner in which
the 6j-symbol is commonly used in recoupling theory. In addition, in loop quantum
gravity (Rovelli 2004) angular momenta represent area vectors, which in the case of
four-valent nodes correspond by Minkowski’s (1897) theorem to a tetrahedron. In
this context the 4j-model is closer to the applications than the 12j-model, indeed, it
played an important role in the work of Bianchi and Haggard (2011).
In this paper we refer to Aquilanti et al (2007) as I, for example writing eqn. (I.13)
for an equation from that paper. We note two errata in I, namely, σ(x) in (I.89) should
read σ(u), and j3and j4should be swapped in the 6j-symbol in (I.112).

Page 4

Semiclassical Mechanics of the Wigner 6j-Symbol
4
2. Spin network notation
We begin by explaining our notation for spin networks, which is based on that of Yutsis
et al (1962) with modifications due to Stedman (1990). At the end of this section we
compare our conventions for spin networks with others in the Yutsis tradition.
2.1. The 3j-symbol and Wigner intertwiner
?
j1
m1
j2
m2
j3
m3
?
=
j1
j2
j3
m2
j2
m3
j3
j1
m1
=
j1
j2
j3
m2
m3
m1
Figure 1. The 3j-symbol contains the components of the standard three-valent
intertwiner.
The 3j-symbol is a number that can be regarded as the components of an
intertwiner W : Cj1⊗ Cj2⊗ Cj3→ C with respect to the standard basis |j1m1? ⊗
|j2m2? ⊗ |j3m3?, as indicated in Fig. 1. In this paper Cj denotes a carrier space for
unitary irrep j of SU(2), so that dimCj= 2j+1. We call W the “Wigner” intertwiner.
We will gradually explain the features of Fig. 1 as we proceed.
The standard notation for the 3j-symbol is on the left of Fig. 1, while the central
diagram is the standard Yutsis spin network for the 3j-symbol, with small arrows
presented as in the Yutsis notation. The indices (m1,m2,m3) in the central diagram
are covariant, that is, they transform under rotations as the components of a dual
vector (in contrast to an ordinary vector). In a Hilbert space we regard ordinary
wave functions or ket vectors as “vectors,” while bra vectors are regarded as “dual
vectors.” Thus, contravariant indices are those that transform as the components of
a vector. In the Yutsis notation the arrows indicate the transformation properties of
the corresponding m index, and there are rules for “raising and lowering” indices, that
is, reversing the direction of the arrow. The rules do not, however, make use of the
metric as in ordinary tensor analysis. Our definition of the arrow (explained below)
is different from that of Yutsis, but designed so that the two notations agree as much
as possible. In particular, our notation for the 3j-symbol is the same as the Yutsis
diagram in Fig. 1.
A trivalent node of a spin network such as those illustrated in Fig. 1 is assumed
to have a positive or counterclockwise orientation, unless otherwise indicated (thus we
dispense with the + sign used by Yutsis).
2.2. Bras, kets and scalar products
The diagram on the right of Fig. 1 makes use of the standard basis vectors |jimi? in
Cji, i = 1,2,3. The spin networks for these basis vectors and their duals are shown in

Page 5

Semiclassical Mechanics of the Wigner 6j-Symbol
5
|ψ? =
j
ψ
?ψ|
=
j
ψ∗
|jm? =
j
m
?jm| =
j
m∗
Figure 2. Spin networks for bras and kets.
Fig. 2. The large, broadly open arrow is a “chevron” (Stedman 1990). When pointing
outward (inward), the chevron indicates a ket (bra) vector. Also shown in Fig. 2 is the
spin network for an arbitrary vector |ψ? in Cj, and the dual bra vector ?ψ| obtained
by Hermitian conjugation. In the Dirac notation it is customary to label bras by
the same symbol as kets, it being understood that the two are related by Hermitian
conjugation. This convention is so deeply ingrained that we dare not change it. But in
spin networks there are two different ways of converting kets into bras and vice versa,
and this presents some notational challenges. One can see in Fig. 2 that Hermitian
conjugation applied to bras and kets is notationally the changing of bra chevrons to
ket chevrons and vice versa, and the starring of identifying symbols, with a double
star being removed. Full rules for Hermitian conjugation of any spin network are given
in Sec. 2.9.
The lines of a spin network will be referred to as “edges,” including cases like
those shown in Fig. 2.
An edge of a spin network ending in an unstarred m index represents a contraction
with the basis ket |jm?, so the m index transforms under rotations as a covariant
index. The explicit insertion of basis kets may be seen in Fig. 1. An edge ending in
a starred m index represents the insertion of a basis bra ?jm|, so starred indices are
contravariant. Thus, the star on an m index indicates its transformation property, and
invariant contractions can only take place between a pair of starred and unstarred m
indices.
?ψ|
=
j
ψ∗= ψ∗
j
Figure 3. Orientation of a spin network does not matter.
As illustrated in Fig. 3, the orientation of a spin network on the page does not
affect its value. The spin network in the figure has been rotated by 180◦.
?φ|ψ? = φ∗
jj
ψ = φ∗
j
ψ
?ψ|φ? =?
Figure 4. Spin networks for scalar products or contractions. Illustration of rules
for complex conjugation.
?∗
φ∗
j
ψ
= φ
j
ψ∗
As illustrated in the final diagram in Fig. 1 and in Fig. 4, when a bra chevron
and a ket chevron are juxtaposed, it represents the scalar product or contraction. In
effect, the bra chevron acts as a receptacle for the ket chevron, and vice versa. After

Page 6

Semiclassical Mechanics of the Wigner 6j-Symbol
6
the contraction the two edges may be joined, with a small arrow remaining to indicate
which was the bra and which the ket in the contraction. One might suppose that the
star would carry the same information, but, as shown below, it is possible to change
the direction of the arrow without changing the stars. Figure 4 also presents another
example of Hermitian conjugation (complex conjugation, in this case). Since the small
arrow represents the contraction of a bra and a ket chevron, when the chevrons are
reversed, the direction of the arrow changes.
?jm|jm′? = m∗
m′
jj
= m∗
m′
j
= m′∗
m
j
δmm′
=
Figure 5. Orthonormality relations for basis vectors.
As a special case of the scalar product, the orthonormality relations of the basis
vectors are illustrated in Fig. 5. The final spin network follows from the symmetry of
the Kronecker delta, or, alternatively, by the reality of δmm′ and the rules for complex
conjugation.
The j labels on the edges of the spin networks indicate which carrier space Cj
the ket or bra lies in, or in which carrier space a contraction has taken place. If
there are distinct carrier spaces with the same j label, then additional distinguishing
information must be supplied.
2.3. Intertwiners
?W| =
j1
j2
j3
Figure 6. Spin network for the Wigner or 3j-intertwiner.
An SU(2) intertwiner (the only kind we are interested in) is a linear map between
two vector spaces that commutes with the action of SU(2) on the two spaces. The
only case of interest here is where the target space is C, consisting of scalars, that is,
invariants under rotations. The Wigner intertwiner W : Cj1⊗Cj2⊗ Cj3→ C is of this
type. But a linear map from a Hilbert space to C can be thought of as a dual or bra
vector, for example, we can associate the map W with a bra vector ?W| belonging to
C∗
of the Wigner intertwiner, that is, the 3j-symbol, are obtained by inserting basis kets,
the ket chevron first, into the bra chevrons of the intertwiner, as in the final diagram
of Fig. 1.
More generally, a C-valued intertwiner on a Hilbert space H can be regarded as
an SU(2)-invariant bra vector on this space, that is, a member of H∗. By Hermitian
conjugation we obtain an SU(2)-invariant ket vector in H. Thus, there is a one-to-one
correspondence between the subspace of H of rotationally invariant vectors and the
j1⊗C∗
j2⊗C∗
j3. The spin network notation for ?W| is shown in Fig. 6. The components

Page 7

Semiclassical Mechanics of the Wigner 6j-Symbol
7
set of C-valued intertwiners on H. The subspace Z introduced in Sec. 3.4 below is a
subspace of this type, consisting of rotationally invariant vectors.
2.4. Tensor products and resolution of identity
m
?
m
|jm??jm| =
?
m
j
m∗
j
=
j
Figure 7. Resolution of identity, and an illustration of the outer product.
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
illustrated in Fig. 7. The orientations of the bra and ket spin networks is immaterial,
but in the figure they have been placed with their m indices adjacent in order
to emphasize the summation (a contraction over two indices, one covariant, one
contravariant). The final diagram is the spin network for the identity operator on
space Cj.
j
m∗
m
?
m
j
=
j
Figure 8. Summing on m to join two edges.
Figure 7 illustrates another technique, the replacement of a sum on m by a joining
of edges. The general usage is shown in Fig. 8. The directions of the arrows and the
stars on one of the m’s must be coordinated as shown for the identity to be used as
shown. The small arrow in the final diagram in Fig. 8 (a fragment of a spin network) is
a reminder of the directions of the arrows before the sum. This small arrow is omitted
in the final diagram of Fig. 7 (the identity diagram) because the chevrons already
indicate the direction of the edge. Recall that edges are also joined on contracting a
bra with a ket, as in Fig. 4.
jj
ψ
= φ∗
j
φ∗
jj
j
ψ =
φ∗
jjj
ψ = φ∗
j
ψ
Figure 9. Summing on m to join two edges.
The use of the identity diagram is illustrated in Fig. 9. The first row illustrates
its action on kets (as a map : Cj→ Cj), where bra and ket chevons are combined as in
Fig. 4. The small arrow is omitted after the joining of the bra and ket chevrons because
the remaining chevron indicates the direction of the edge. The second row of the figure
illustrates its action on bras (the map : C∗
action as a map : C∗
j→ C∗
j), and the third row illustrates its
j⊗Cj→ C, that is, the scalar product map : ?φ|⊗|ψ? ?→ ?φ|ψ?. All

Page 8

Semiclassical Mechanics of the Wigner 6j-Symbol
8
of these usages are encompassed by the same spin network. The identity spin network
can also be seen as an element of Cj⊗ C∗
it, viewed as a C-valued linear operator, acts (the third line of Fig. 9).
In general a spin network has some number of edges that terminate in incoming
or outgoing chevrons. Examples are the identity diagram in Fig. 7 and the Wigner
intertwiner in Fig. 6. In all cases there are multiple interpretations of the spin network
as a linear operator mapping one vector space to another, depending on how many
of the incoming and outgoing chevrons have ket and bra chevrons plugged into them
(specifying the domain), and how many are left free (specifying the range). The
domain is the tensor product of some number of Cjtimes some number of C∗
is the range. In the extreme case that all incoming and outgoing chevrons on the spin
network have kets and bras plugged into them the result is simply a number and the
range is C. In that case the spin network, as a C-valued linear operator, can be seen
as a vector in the space dual to the domain.
This facile identification of closely associated operators, and their reinterpretation
as elements of vector spaces, is an important advantage of spin networks. It is difficult
and awkward to do something similar with the Dirac notation.
In general, a tensor is a multilinear operator acting on a tensor product of some
set of vector spaces and their duals. Thus, a spin network is a notation for a tensor on
some product of the Cjand their duals. The edges of the spin network terminating in
ket or bra chevrons indicate the nature of the space the tensor acts on. In general, the
tensor product of two tensors in spin network notation is indicated by the placing of
the two spin networks together on the page, in any orientation. The outer product of
a bra and a ket illustrated in Fig. 7 is a special case. Partial or complete contractions
of tensors are indicated by joining some or all ket chevrons with bra chevrons.
j, that is, the space dual to C∗
j⊗Cjon which
j, and so
2.5. The 2j symbol and intertwiner
We now consider another intertwiner, which leads to an important mapping between
kets and bras, alternative to Hermitian conjugation. The intertwiner acts on the
Hilbert space Cj⊗ C′
general we wish to consider the second carrier space as distinct from the first, which is
the purpose of the prime on the second factor. To within a normalization and phase,
there is a unique vector in this space that is invariant under rotations; we call it |K?,
and express it in terms of the Clebsch-Gordan coefficients by
j, the tensor product of two carrier spaces of the same j. In
|K? =
?2j + 1
?
mm′
|jm? ⊗ |jm′?C00
jjmm′. (1)
This vector can also be expressed in terms of the “2j-symbol,” which we define in
terms of the usual 3j-symbol by
?
The terminology “2j-symbol” is not entirely standard, but it has been used by
Stedman (1990). The invariant vector |K? can also be written,
|K? =
mm′
j
m
j
m′
?
=
?
j
m
j0
0m′
?
= C00
jjmm′ =(−1)j−m
√2j + 1δm,−m′. (2)
?
|jm? ⊗ |jm′?(−1)j−mδm,−m′ =
?
m
|jm? ⊗ |j,−m?(−1)j−m. (3)
By Hermitian conjugation we convert |K? into the bra ?K|, which is otherwise
an intertwiner K : Cj⊗ C′
j→ C. Just as the components of the intertwiner W are

Page 9

Semiclassical Mechanics of the Wigner 6j-Symbol
9
m
jjjj
m′
m
jj
m′= (−1)j−mδm,−m′
=
Figure 10. Components of the map K, a standard bivalent intertwiner.
the 3j-symbol, the components of the intertwiner K are the 2j-symbol, multiplied
however by√2j + 1 because of a normalization convention. Figure 10 shows first the
spin network for the components of K, which is conceived of as a standard bivalent
node or intertwiner. The small arrows indicate that the components on the first line
can be considered the result of plugging basis kets into the intertwiner itself, as seen
on the second line. The short line extending above the node is a “stub” (Stedman
1990), whose purpose is to orient the node. The convention is that if we start at the
stub and move in a positive (counterclockwise) direction, the first and second edges
we encounter are respectively the first and second operands of K, conceived of as a
map : Cj⊗ C′
We note that if V and W are vector spaces, then V ×W is not the same as V ⊗W,
but if we have a bilinear map on V ×W it can be extended to a linear map on V ⊗W
by linear superposition. For example, in the previous paragraph we have regarded K
as a bilinear map : Cj× C′
The first and second operands of this expression correspond to the first and second
edges as specified by the stub.
j→ C.
j→ C, and computed its components as K(|jm?,|jm′?).
jj
=
?K|
Figure 11. Spin network for map K : Cj⊗ C′
j→ C.
Figure 10 also gives the numerical values of the components of K, and in the
final diagram, the network for K itself, with two kets inserted. The network for K in
isolation is illustrated in Fig. 11, regarded as a bra vector on Cj⊗ C′
element of C∗
j, that is, as an
j⊗ C′∗
j.
m
jj
m′=√2j + 1
0
j
m
m′
0
j
Figure 12. The stub can be regarded as a vestigial edge of a 3j-symbol with
value j = 0.
The stub in Fig. 10 can be regarded as a vestigial edge of a 3j-symbol or W-
intertwiner with the value j = 0, although one must beware of the normalization
convention. This is illustrated in Fig. 12, which is equivalent to (2). The value does
not depend on the direction of the arrow on the zero edge (see below for rules for

Page 10

Semiclassical Mechanics of the Wigner 6j-Symbol
10
reversing arrows).
φ
jjjj
ψ = φ
jj
ψ
= (−1)2jψ
jj
φ = (−1)2jφ
jj
ψ
Figure 13. The intertwiner K acquires a phase (−1)2jif the two operands are
swapped.
The components of K : Cj⊗C′
m and m′are swapped. This is equivalent to the statement
K(|φ?,|ψ?) = (−1)2jK(|ψ?,|φ?),
for all |ψ?, |φ?, which is illustrated in spin network language in Fig. 13. The final
diagram differs from the preceding simply by a 180◦rotation, so the value is the same.
But this leads to the rule, that a spin network acquires a phase of (−1)2jwhen the
stub at a bivalent node is inverted. In particular, the arrow on the null edge in Fig. 12
can be inverted without changing the value.
j→ C, seen in Fig. 10, acquire a phase of (−1)2jif
(4)
2.6. Kets to bras
jjj
ψ =
jj
ψ
≡
j
ψ ?=
j
ψ∗
Figure 14. The intertwiner K can be used to convert a ket to a bra.
The intertwiner K : Cj⊗ Cj → C has an alternative interpretation as a map
K1: Cj→ C∗
and where the 1-subscript distinguishes the new map from the old. This alternative
interpretation is natural in spin network language, as seen in Fig. 14. The spin network
for |ψ? ∈ Cj is plugged into the second operand of the spin network for K, resulting
in a spin network with one free bra chevron. The choice of the second operand of K
for this purpose is conventional. The result is an element of C∗
As indicated in the figure, we abbreviate this spin network by drawing the same spin
network for |ψ? that we started with, except the ket chevron is converted into a bra
chevron.
In other words, when we use K1to convert a ket into a bra, we just flip the ket
chevron, leaving everything else the same. In particular, we do not put a star on
the label of the ket. This distinguishes the map K1 : Cj → C∗
Hermitian conjugation, which is also a map : Cj → C∗
convert a ket to a bra, not only is the ket chevron flipped to a bra chevron, but a star
is appended to the label. These two maps are quite distinct; in particular, K1 is a
linear map, while the metric is an antilinear map. As indicated in Fig. 14, the results
are not the same.
When a ket is turned into a bra, the components with respect to some basis
change from contravariant to covariant. But since there is more than one way to do
j, where for simplicity we have dropped the prime on the second factor,
j, that is, it is a bra.
jfrom the metric or
j. When the metric is used to

Page 11

Semiclassical Mechanics of the Wigner 6j-Symbol
11
this, any notation based on the position (upper or lower) of the indices is inadequate
to represent the result.
j
X
Figure 15. An arbitrary spin network with one edge terminating in a ket chevron.
The map K1can be used to turn a ket chevron into a bra chevron on any spin
network, not only on kets themselves. A notation for an arbitrary spin network with
an edge terminating in a ket chevron is shown in Fig. 15. The circle around the X
indicates the rest of the spin network, which may include other edges terminating in
ket or bra chevrons.
jjj
X
=
jj
X
≡
j
X
Figure 16. Converting a ket chevron into a bra chevron with the map K1: Cj→
C∗
j.
To convert the ket chevron in Fig. 15 into a bra chevron we simply insert it into
the second operand of K : Cj⊗ Cj→ C, as shown in Fig. 16. The remainder of the
spin network, indicated by the X, does not change.
2.7. Bras to kets
jj
Figure 17. Spin network for the maps K−1: C∗
j⊗ C∗
J→ C and K−1
1
: C∗
j→ Cj.
φ∗
jjjj
ψ∗
jj
φ∗
=
ψ∗
Figure 18. Spin network for the quantity K−1(?φ|,?ψ|).
The map K1: Cj→ C∗
into kets. We associate K−1
is expressed by the spin network in Fig. 17. This spin network and the meaning of
the −1 on K−1are to be defined, but the spin network represents a linear operator
: C∗
action of K−1on two bras is illustrated in Fig. 18.
jhas an inverse, the map K−1
with a closely related map K−1: C∗
1
: C∗
j→ Cj, that takes bras
j⊗ C∗
1j→ C that
j⊗C∗
j→ C with the ordering of the two operands being specified by the stub. The

Page 12

Semiclassical Mechanics of the Wigner 6j-Symbol
12
φ∗
jjj
= φ∗
jj
≡ φ∗
j
?= φ
j
Figure 19. Action of K−1
1
on a bra.
We now define the network in Fig. 17 by requiring that K−1
inserting it into the first operand of that network, as illustrated in Fig. 19, and by
requiring that K−1
1
actually be the inverse of K1. Figure 19 shows the action of K−1
on an arbitrary bra ?φ|. The use of the first operand of K−1for this purpose is a
convention, but one that makes our overall notation for mapping kets to bras and
vice versa consistent (see Fig. 22 below). As indicated, we abbreviate the result by
taking the original network for the bra ?φ| and simply flipping the direction of the
chevron. We do not unstar the identifying symbol. As indicated, the result differs
from Hermitian conjugation applied to ?φ|, which is the ket |φ? (without the star).
jjj
1
act on a bra by
1
j
=
j
Figure 20. The requirement that K−1◦ K = Id.
The requirement that K−1
The stub on the network for K−1is inverted so that the output of the first step is fed
into the first operand of K−1.
1
actually be the inverse of K1is illustrated in Fig. 20.
jj
=
jj
Figure 21.
other.
Two arrow flips, with stubs oppositely oriented, annihilate each
The identity represented by Fig. 20 is usually encountered in practice in the form
shown in Fig. 21 (a fragment of a spin network). The 2j-node inverts the direction of
the arrow. Two such inversions, with stubs pointing in opposite directions, annihilate
one another.
jjjj
jjjj
==
jj
jj
Figure 22. Flipping the chevrons on K gives K−1.
We have made an independent definition of the spin network in Fig. 17, but it is
the same as the spin network for K, shown in Fig. 11, with both bra chevrons flipped.
Since we now have a convention for flipping bra chevrons (by applying K−1
consistency we must show that the two results are the same. This is done in Fig. 22,
which uses the identity of Fig. 21.
1), for

Page 13

Semiclassical Mechanics of the Wigner 6j-Symbol
13
m∗
jjjj
m′∗
= m∗
jj
m′∗= (−1)j−mδm,−m′
Figure 23. The components of K−1have the same numerical values as the
components of K.
By inserting resolutions of the identity into the diagram in Fig. 20 it is easy to
work out the components of K−1. These are displayed in Fig. 23. Notice that they
have the same numerical values as the components of K (see Fig. 10).
j
Y
jj
=
Y
jj
≡
j
Y
Figure 24. Converting a bra chevron to a ket chevron on an arbitrary spin
network.
By using K−1we can convert a bra chevron into a ket chevron on any spin
network, not only on bras and kets themselves. This is illustrated in Fig. 24, which
may be compared to Fig. 16.
=
j
Y
j
X
=
Y
jjjj
X
Y
jjj
X
=(−1)2j
Y
j
X
Y
j
X
=(−1)2j
Figure 25. Reversing the arrow on an edge of a spin network incurs a phase of
(−1)2j.
Finally, by using Figs. 16, 24 and 21, it may be shown that when we reverse the
arrow on an edge of a spin network, we incur a phase of (−1)2j. This is done in Fig. 25.
2.8. Raising and lowering indices
When we convert a ket to a bra by the action of K1, then the bra has components
with respect to the standard basis |jm? that are simple functions of the components
of the original ket with respect to the standard basis ?jm|. Mapping the one set of
components to the other is “lowering the index.” Using K−1
ket similarly amounts to “raising the index.” More generally the procedure can be
1
to convert a bra to a

Page 14

Semiclassical Mechanics of the Wigner 6j-Symbol
14
applied to an edge of any spin network terminating in a starred (contravariant) or
an unstarred (contravariant) index. The index can refer to any basis, not just the
standard one.
m∗
j
X
= m∗
j
= m∗
jjjj
X
=
?
m′
m∗
jj
m′∗m′
j
X
j
X
Figure 26.
components.
Expressing contravariant components in terms of covariant
m
j
X
= (−1)j−m(−m∗)
j
X
m∗
j
X
= (−1)j+m(−m)
j
X
Figure 27. Rules for raising and lowering indices.
Figure 26 shows how to the express contravariant components in terms of the
covariant components in the standard basis. By plugging in the numerical values of
the components of K, we obtain the first line of Fig. 27. Similarly we derive the
second line of Fig. 27 for expressing covariant components in terms of contravariant
components.
j1
j2
j3
m∗
3
m∗
2
m∗
1
j1
j2
j3
m3
m2
m1
==
?
j1
m1
j2
m2
j3
m3
?
Figure 28. The completely covariant and completely contravariant components
of the 3j-intertwiner are numerically equal.
m
jj
m′= m∗
jj
m′∗=(−1)j−mδm,−m′
Figure 29. The completely covariant and completely contravariant components
of the 2j-intertwiner are numerically equal.
If these rules are used to raise all three covariant components of the 3j-intertwiner
(Fig. 1), then we find that the completely contravariant components have the same

Page 15

Semiclassical Mechanics of the Wigner 6j-Symbol
15
values, namely, the 3j-symbol.
is necessary to use the symmetry of the 3j-symbol (see Varshalovich et al
Eq. (8.2.4.6)). Then by setting one of the j’s to zero and using Fig. 12, we find that
the same is true for the completely covariant and completely contravariant components
of the 2j-intertwiner, as shown in Fig. 29. The same result is obtained by comparing
Figs. 10 and 23.
This is illustrated in Fig. 28.To show this it
1981,
2.9. Hermitian conjugation of spin networks
Consider a spin network of arbitrary complexity involving only 2j- and 3j-nodes. The
network is allowed to have any number of edges terminating in bra or ket chevrons, or
in starred or unstarred labels such as m indices. By using the identities above, possibly
with the insertion or removal of 2j-nodes and the extraction of phases of the form
(−1)2j, it is possible to bring the spin network into a standard form, in which all edges
joining 3j-nodes have arrows pointing toward the 3j-node, all edges joining 2j-nodes
have arrows pointing away from the 2j-node, all edges terminating in a starred symbol
have arrows pointing toward that symbol, and all edges terminating in an unstarred
symbol have arrows pointing away from that symbol. Next, by inserting resolutions of
the identity, which involve m-sums, it is possible to express the spin network as sum
over the completely covariant components of 3j-symbols and completely contravariant
components of 2j-symbols, times a tensor product of bras and kets.
In this form it is easy to take the Hermitian conjugate.
conjugation, bras go to kets and vice versa, while the covariant components of 3j-
symbols and contravariant components of 2j-symbols do not change, since they are
real. By using Figs. 28 and 29, however, these components can be rewritten as the
completely contravariant components of 3j-symbols and the completely covariant
components of 2j-symbols.The m-sums can now be done, reversing the earlier
insertions of resolutions of the identity. Then the other steps leading to the standard
form can be reversed.
The result is a simple rule for the Hermitian conjugation of any spin network
of the given form: All ket chevrons are changed to bra chevrons and vice versa, the
directions of all arrows are reversed, and all edges terminating in a symbol have a star
added to the symbol, with a double star being removed.
Under Hermitian
2.10. Discussion of spin network rules
Our rules for spin networks differ from those of Yutsis et al and most of the literature
in the Yutsis tradition primarily by our ability to express abstract vectors (kets), dual
vectors (bras) and tensors in addition to the components of those objects. Also, we
indicate the nature of an m index (covariant or contravariant) by the presence or
absence of a star, rather than the direction of the arrow. One result is that our rules
for reversing the direction of the arrow are more uniform than in the Yutsis tradition,
where such a reversal picks up a phase (−1)2jonly on internal edges. In our approach,
the rule applies everywhere, including edges terminating in an m index. In addition,
our rules for Hermitian conjugation are simpler than those in the Yutsis tradition,
where phase factors must be introduced. The simplification is due to the explicit
introduction of 2j-symbols, and the use of stubs.
To translate a Yutsis spin network into one of ours, it is necessary only to put
stars on m indices terminating edges with outward pointing arrows.