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436 10 Group Theory
we identify the rotation matrix as
d
j
m
'
m
Β
j m
'
!
j m
'
!
j m! j m!
1/ 2
Ν
jm
'
Ν
j m
j m
'
Ν
j m
Ν
Cos
Β
2
mm
'
2Ν
Sin
Β
2
2 jmm
'
2Ν
(10.439)
or
d
j
m
'
m
Β
jm
'
j m
'
!
j m
'
!
j m! j m!
1/ 2
Sin
Β
2
2 j
Cot
Β
2
mm
'
Ν
Ν
j m
j m
'
Ν
j m
Ν
Cot
Β
2
2Ν
(10.440)
The summation over Ν is expressible in terms of a hypergeometric function or a Jacobi
polynomial in CotΒ/2
2
, but that does not really simplify the result. One should verify
that our previous result for d
1/ 2
is contained within this formula. Carrying this one step
further, one obtains the rotation matrix for spin-1
d
1
3
4
4
4
4
4
4
4
4
4
4
4
5
Cos
Β
2
2
SinΒ
2
Sin
Β
2
2
SinΒ
2
CosΒ
SinΒ
2
Sin
Β
2
2
SinΒ
2
Cos
Β
2
2
6
7
7
7
7
7
7
7
7
7
7
7
8
3
4
4
4
4
4
4
4
4
4
4
5
1
2
1 CosΒ
SinΒ
2
1
2
1 CosΒ
SinΒ
2
CosΒ
SinΒ
2
1
2
1 CosΒ
SinΒ
2
1
2
1 CosΒ
6
7
7
7
7
7
7
7
7
7
7
8
(10.441)
and observes that half-angles are not required for integral j. Wave functions for half-
integral spin change sign upon rotation through 2Π, whereas those for integral spin do
not.
Below we tabulate some of the symmetries and special values for rotation matrices
but do not provide proofs. Note that is restricted to integers while j maybeintegral
or half-integral. Be aware that some authors employ different phase conventions for both
rotation matrices and spherical harmonics.
10.7 Quantum Mechanical Representations of the Rotation Group 437
D
m,0
Α, Β, Γ
4Π
2 1
Y
,m
Β, Α (10.442)
d
m,0
Β
m
m!
m!
1/ 2
P
,m
cos Β (10.443)
d
0,0
Β P
cos Β (10.444)
d
j
m
'
,m
Β d
j
m,m
'
Β (10.445)
d
j
m
'
,m
Π
jm
∆
m
'
,m
(10.446)
d
j
m
'
,m
Π Β
jm
'
d
j
m,m
'
Β (10.447)
d
j
m
'
,m
Β
m
'
m
d
j
m
'
,m
Β
m
'
m
d
j
m,m
'
Β (10.448)
D
j
m
'
,m
Α, Β, Γ
D
j
m
'
,m
Α, Β, Γ
m
'
m
D
j
m
'
,m
Α, Β, Γ (10.449)
10.7.5 Orthogonality Relations for Rotation Matrices
Now that we have an expression for the reduced rotation matrix, the matrices for arbitrary
rotations can be evaluated using
D
j
Α, Β, Γ Exp
ΑJ
z
d
j
Β Exp
ΓJ
z
(10.450)
D
j
m
'
m
Α, Β, Γ Exp
mΓm
'
Α
d
j
m
'
m
Β (10.451)
Let ΩΑ, Β, Γ denote the Euler angles. According to the orthogonality theorem for
compact Lie groups, we expect the representation matrices to satisfy an orthogonality
relation of the form
D
j
m
'
m
Ω
D
j
'
n
'
n
ΩΡΩ Ω
∆
j,j
'
∆
m,n
∆
m
'
,n
'
2 j 1
ΡΩ Ω (10.452)
where ΡΩ is an appropriate density function and where the integration includes all group
elements. It is rather difficult to derive the density function directly because the compo-
sition law for Euler angles is complicated, but some physical reasoning will provide the
required function. Consider the motion of the z-axis under the rotations described by Γ and
Β. The first acts as an azimuthal angle while Β acts as a polar angle, describing any point
on the unit sphere. Thus, we expect a uniform density for Γ and a SinΒ density for the
polar angle. The final rotation by Α is also azimuthal with a uniform density. Therefore,
we expect that ΡΩ SinΒ and write
D
j
m
'
m
Ω
D
j
'
n
'
n
ΩΡΩ Ω
d
j
m
'
m
Βd
j
'
n
'
n
Β Expm nΓ Exp
m
'
n
'
Α
SinΒ Α Β Γ (10.453)
where d
j
is real. Orthogonality with respect to magnetic quantum numbers must be pro-
vided by the integrations over Α and Γ.If j, j
'
are either both integral or both half-integral
438 10 Group Theory
the differences between magnetic quantum numbers are integral and integration over the
range (0, 2Π) ensures orthogonality, but if one is integral and the other half-integral, the
difference in magnetic quantum numbers is half-integral and integration over the range
(0, 4Π) is needed for both Α and Γ. This should not be surprising, at least in retrospect,
because the identity element for subgroups consisting of rotations about a fixed axis is a
rotation of 4Π for a half-integral spin. Therefore, the covering group capable of accommo-
dating both integral and half-integral angular momentum features an enlarged parameter
space with 0 Α4Π and 0 Γ4Π.
Orthogonality with respect to j is then left to integration over Β. Treating d
j
Β as
representations of a one-parameter subgroup, we expect
Π
0
d
j
m
'
m
Ωd
j
'
m
'
m
Ω SinΒ Β
∆
j,j
'
2 j 1
Π
0
SinΒ Β (10.454)
or
Π
0
d
j
m
'
m
Ωd
j
'
m
'
m
Ω SinΒ Β
2
2 j 1
∆
j,j
'
(10.455)
It would be rather difficult to obtain this result directly from the power series for d
j
without using the underlying theorems from group theory. We can now put everything
together to obtain
4Π
0
Α
Π
0
ΒSinΒ
4Π
0
ΓD
j
m
'
m
Α, Β, Γ
D
j
'
n
'
n
Α, Β, Γ
32Π
2
2 j 1
∆
j,j
'
∆
m,n
∆
m
'
,n
'
(10.456)
as the orthogonality theorem for representations of the rotational covering group without
performing arduous integrations.
10.7.6 Coupling of Angular Momenta
Direct products of two irreducible representations of the rotation group produce a Clebsch–
Gordan expansion
D
j
1
P j
2
j
j
1
j
2
j
D
j
(10.457)
Recall that all rotations through the same angle belong to the same class regardless of axis.
Thus, the easiest way to determine the reduction coefficients is to evaluate the character
Χ
j
1
P j
2
Ω Χ
j
1
Ω Χ
j
2
Ω (10.458)
for Ω0, 0, Φ to obtain
j
j
m j
j
1
j
2
j
ExpmΦ
j
1
m
1
j
1
j
2
m
2
j
2
Exp
m
1
m
2
Φ
(10.459)
10.7 Quantum Mechanical Representations of the Rotation Group 439
Multiplying both sides by ExpnΦ and integrating over 0 Φ4Π to account for possible
half-integral quantum numbers, we obtain
j
j
m j
j
1
j
2
j
∆
m,n
j
1
m
1
j
1
j
2
m
2
j
2
∆
m
1
m
2
,n
(10.460)
Satisfying these constraints limits j to the range j
1
j
2
j j
1
j
2
. The total number of
terms on the right-hand side is 2 j
1
12 j
2
1. If we assume, without loss of generality,
that j
1
2 j
2
, the total number of terms on the left hand side is
j
1
j
2
j j
1
j
2
2 j 1
2 j
1
1
2 j
2
1
(10.461)
Therefore, every ( j
1
j
2
j) in the allowed range equals 1, such that
D
j
1
P j
2
D
j
1
j
2
d D
j
1
j
2
1
d d D
j
1
j
2
(10.462)
shows that the rotation group is simply reducible.
The direct product of two angular momentum eigenstates can now be expanded in
terms of irreducible representations according to
Ψ
j
1
,m
1
Ψ
j
2
,m
2
j,m
j
1
j
2
m
1
m
2
j
m
!
Ψ
j,m
(10.463)
where the Clebsch–Gordan coefficients can be obtained by generalizing the relationship
gG
D
Μ
i
'
,i
gD
Ν
j
'
,j
gD
Γ
k
'
,k
g
N
G
n
Γ
ΜΝ
i
'
j
'
Γ
k
'
!
ΜΝ
ij
Γ
k
!
(10.464)
for discrete groups to the continuum according to
4Π
0
Α
Π
0
ΒSinΒ
4Π
0
ΓD
j
1
m
'
1
,m
1
ΩD
j
2
m
'
2
,m
2
ΩD
j
m
'
,m
Ω
32Π
2
2 j 1
j
1
j
2
m
'
1
m
'
2
j
m
!
j
1
j
2
m
1
m
2
j
m
!
(10.465)
Notice that for the special cases m
'
j, the rotation matrices have only a single term:
D
j
j,m
Ω
2 j!
j m! j m!
1/ 2
Sin
Β
2
jm
Cos
Β
2
jm
ExpmΓ jΑ (10.466)
D
j
j,m
Ω
jm
2 j!
j m! j m!
1/ 2
Sin
Β
2
jm
Cos
Β
2
jm
ExpmΓ jΑ
(10.467)
440 10 Group Theory
Consider the special case m
'
1
j
1
,m
'
2
j
2
,forwhich
4Π
0
Α
Π
0
ΒSinΒ
4Π
0
ΓD
j
1
j
1
,m
1
ΩD
j
2
j
2
,m
2
ΩD
j
j
1
j
2
,m
Ω
32Π
2
2 j 1
j
1
j
2
j
1
j
2
j
j
1
j
2
!
j
1
j
2
m
1
m
2
j
m
!
(10.468)
and let
I
4Π
0
Α
Π
0
ΒSinΒ
4Π
0
ΓD
j
1
j
1
,m
1
ΩD
j
2
j
2
,m
2
ΩD
j
m
'
,m
Ω
j
2
m
2
2 j
1
!
j
1
m
1
!
j
1
m
1
!
2 j
2
!
j
2
m
2
!
j
2
m
2
!
j m
'
!
j m
'
!
j m! j m!
1/ 2
Ν
jm
'
Ν
j m
j m
'
Ν
j m
Ν
4Π
0
Α
Π
0
ΒSinΒ
4Π
0
ΓSin
Β
2
j
1
m
1
j
2
m
2
2 jmm
'
2Ν
Cos
Β
2
j
1
m
1
j
2
m
2
mm
'
2Ν
Exp
m
1
m
2
m
Γ
Exp
j
1
j
2
m
'
Α
(10.469)
where Ν is an integer whose range is limited by the binomial coefficients in the expansion.
Integration over Γ requires m m
1
m
2
while integration over Α requires m
'
j
1
j
2
,
such that
I 4Π
2
∆
m,m
1
m
2
∆
m
'
,j
1
j
2
j
2
m
2
2 j
1
!
j
1
m
1
!
j
1
m
1
!
2 j
2
!
j
2
m
2
!
j
2
m
2
!
j j
1
j
2
!
j j
1
j
2
!
j m! j m!
1/ 2
Ν
j j
1
j
2
Ν
j m
j j
1
j
2
Ν
j m
Ν
4
Π/ 2
0
ΗSinΗ
2 j2 j
2
2m
1
2Ν1
CosΗ
2 j
1
2m
1
2Ν1
(10.470)
Using our old friend the beta function (recall Sec. 2.6.2)
Π/ 2
0
CosΘ
2p1
SinΘ
2q1
Θ
ApAq
2Ap q
1
2
Bp, q (10.471)
10.7 Quantum Mechanical Representations of the Rotation Group 441
the integral reduces to
I 32Π
2
∆
m,m
1
m
2
∆
m
'
,j
1
j
2
j j
1
2 j
2
m
2
2 j
1
!
j
1
m
1
!
j
1
m
1
!
2 j
2
!
j
2
m
2
!
j
2
m
2
!
j j
1
j
2
!
j j
1
j
2
!
j m! j m!
1/ 2
Ν
Ν
j m
j j
1
j
2
Ν
j m
Ν
A
j
1
m
1
Ν1
A
j j
2
m
1
Ν1
A
j j
1
j
2
2
(10.472)
and we obtain
j
1
j
2
j
1
j
2
j
j
1
j
2
!
j
1
j
2
m
1
m
2
j
m
!
∆
m,m
1
m
2
∆
m
'
,j
1
j
2
2 j 1
j j
1
2 j
2
m
2
2 j
1
!
j
1
m
1
!
j
1
m
1
!
2 j
2
!
j
2
m
2
!
j
2
m
2
!
j j
1
j
2
!
j j
1
j
2
!
j m! j m!
1/ 2
Ν
Ν
j m
j j
1
j
2
Ν
j m
Ν
A
j
1
m
1
Ν1
A
j j
2
m
1
Ν1
A
j j
1
j
2
2
(10.473)
as the product of CG coefficients. Substitution of m
1
j
1
,m
2
j
2
then gives
j
1
j
2
j
1
j
2
j
j
1
j
2
!
2
2 j 1
j j
1
j
2
Ν
Ν
j j
1
j
2
j j
1
j
2
Ν
j j
1
j
2
Ν
A
2 j
1
Ν1
A
j j
1
j
2
Ν1
A
j j
1
j
2
2
(10.474)
The preceding two equations provide enough information to determine the full set of CG
coefficients for any m
1
,m
2
,m, but we would prefer to perform the summation above in
closed form. Here we simply quote the result
j
1
j
2
j
1
j
2
j
j
1
j
2
!
2
2 j 1
2 j
1
!
2 j
2
!
j
1
j
2
j
!
j
1
j
2
j 1
!
(10.475)
because evaluation of the sum is a bit tricky and we are not especially interested, at this
point, in techniques for manipulating such expressions; the details are postponed to the
exercises. We are free to choose the positive root for this particular CG coefficient. Also
note that the fact that j j
1
j
2
is an integer helps to simplify the phase. The remaining
442 10 Group Theory
coefficients then become
j
1
j
2
m
1
m
2
j
m
!
∆
m,m
1
m
2
$
j
1
,j
2
,j
j
1
jm
2
2 j 1
j
1
j
2
j
!
j j
1
j
2
!
j j
1
j
2
! j m! j m!
j j
1
j
2
1
!
j
1
m
1
!
j
1
m
1
!
j
2
m
2
!
j
2
m
2
!
1/ 2
Ν
Ν
j
1
m
1
Ν
!
j j
2
m
1
Ν
!
Ν! j m Ν!
j j
1
j
2
Ν
!
j
1
j
2
m Ν
!
(10.476)
where the triangle test
$
j
,
j
2
,j
L
M
M
N
M
M
O
1
j
1
j
2
j j
1
j
2
0 otherwise
(10.477)
denotes the selection rule for j. Notice that all CG coefficients are real.
Explicit formulas without summation can be found for small angular momenta in stan-
dard compilations. Those references also provide useful symmetry relations for CG coef-
ficients and related quantities, but we will not pursue those details here.
10.7.7 Spherical Tensors
Operators T
j,m
that transform among themselves under rotations as components of an irre-
ducible representation of the rotation group, such that
RT
j,m
R
1
j
m
'
j
T
j,m
'
D
j
m
'
,m
R (10.478)
are known as spherical tensors of rank j. We also require phases to ensure that complex
conjugation properties are the same as those for spherical harmonics, such that
T
j,m
m
T
j,m
(10.479)
and we define the scalar product of two spherical tensors according to
A
j
, B
j
j
m j
A
j,m
B
j,m
j
m j
A
j,m
B
j,m
(10.480)
Similarly, we define tensor products based upon coupling of angular momentum states
according to
A
j
1
P B
j
2
j,m
m
1
,m
2
j
1
j
2
m
1
m
2
j
m
!
A
j
1
m
1
B
j
2
m
2
(10.481)
10.7 Quantum Mechanical Representations of the Rotation Group 443
Matrix elements of spherical tensors are expressed most simply using the Wigner–Eckart
theorem in the form
j
f
m
f
T
j,m
j
i
m
i
j
i
j
m
i
m
j
f
m
f
!
j
f
)T
j
)j
i
(10.482)
where the reduced matrix element j
f
)T
j
)j
i
is independent of magnetic quantum numbers
and can be evaluated using whatever choices for those quantum numbers that prove sim-
plest. Obviously, these matrix elements vanish unless j
f
j
i
j j
f
j
i
and m m
f
m
i
satisfy the selection rules for angular momentum coupling.
For infinitesimal rotations we expand
R ! 1 ˆn ,
J O
2
RT
j,m
R
1
! T
j,m
ˆn ,
J,T
j,m
(10.483)
where ˆn is a unit vector and express the rotation matrix in terms of matrix elements of
J as
D
j
m
'
m
Rj, m
'
Expˆn ,
J j, m!∆
m,m
'
j, m
'
ˆn ,
J j, m (10.484)
to obtain commutation relations
J,T
j,m
m
'
T
j,m
'
j, m
'
J j, m (10.485)
between the components of the angular momentum operator and those of the spherical
tensor. Thus, we obtain commutation relations
J
0
,T
j,m
mT
j,m
(10.486)
J
,T
j,m
T
j,m1
j j 1mm 1 (10.487)
for spherical tensors that are often easier to verify than their transformation properties.
Naturally, we find
J,T
0,0
0 for scalar operators.
For example, the appropriate linear combinations of a cartesian vector should trans-
form as a spherical tensor of rank 1. To identify those combinations, we express cartesian
coordinates in terms of spherical harmonics as
x
r
SinΘ CosΦ
4Π
3
1
2
Y
1,1
Θ, ΦY
1,1
Θ, Φ (10.488)
y
r
SinΘ SinΦ
4Π
3
2
Y
1,1
Θ, ΦY
1,1
Θ, Φ (10.489)
z
r
CosΘ
4Π
3
Y
1,0
Θ, Φ (10.490)
444 10 Group Theory
Thus, it is useful to define spherical unit vectors as
ˆe
1
1
2
ˆx ˆy
(10.491)
ˆe
1
1
2
ˆx ˆy
(10.492)
ˆe
0
ˆz (10.493)
with the same conjugation property
ˆe
m
m
ˆe
m
(10.494)
as spherical harmonics and with the orthonormality condition
ˆe
m
'
ˆe
m
ˆe
m
'
, ˆe
m
∆
m,m
'
(10.495)
Spherical components of a cartesian vector are then obtained using
r
m
ˆe
m
r (10.496)
such that
r
1
ˆe
1
rx y
4Π
3
rY
1,1
Θ, Φ (10.497)
r
1
ˆe
1
r x y
4Π
3
rY
1,1
Θ, Φ (10.498)
r
0
ˆe
0
r z
4Π
3
rY
1,0
Θ, Φ (10.499)
Thus,
r r
4Π
3
1
m1
m
Y
1,m
ˆe
m
r
4Π
3
Y
1
, ˆe
1
(10.500)
expresses the basic cartesian vector in a spherical basis using polar coordinates.
The angular momentum operator
J is itself a spherical tensor. Thus, we can express its
components in the form
J
1
ˆe
1
J
1
2
J
x
J
y
1
2
J
(10.501)
J
1
ˆe
1
J
1
2
J
x
J
y
1
2
J
(10.502)
J
0
ˆe
0
J J
z
(10.503)
where one must not confuse the raising and lowering operators, J
and J
, with the spher-
ical components, J
1
and J
1
, despite their notational similarities. Matrix elements then
10.8 Unitary Symmetries in Nuclear and Particle Physics 445
take the form
j, m
'
J
1
j, m
j m j m 1/ 2 ∆
m
'
,m1
(10.504)
j, m
'
J
0
j, m m∆
m
'
,m
(10.505)
and we can use the special value
j 1
m 0
j
m
!
m
j j 1
(10.506)
to deduce the reduced matrix element
j
'
)J
Μ
)j
∆
j,j
'
j j 1 (10.507)
where Μ0, 1. Thus, the commutation relations for a spherical tensor can now be
expressed in the form
J
Μ
,T
j,m
j j 1
j 1
m Μ
j
m Μ
!
T
j,mΜ
(10.508)
Finally, if T
j,m
V
m
is a vector operator with j 1, we can express these commutation
relations in cartesian form as
J
i
,V
j
i, j,k
V
k
(10.509)
where the summation convention is employed and i, j, k x, y, z. Notice that this rela-
tionship contains the SU(2) commutation relations as a special case. Although we leave
the algebra to the exercises, this form is very helpful in manipulating operators composed
of products of
r,
p, and
L.
10.8 Unitary Symmetries in Nuclear and Particle Physics
No discussion of group theory in physics would be complete without at least mentioning
the assignment of elementary particles to multiplets and the dynamical consequences of
their associated group symmetries. The most obvious multiplets are those associated with
isospin symmetry in nuclear physics. The difference between the masses of the proton
and neutron is so small, only 0.14 %, that they can be considered to be two states of the
same particle, the nucleon, that are degenerate with respect to the strong interaction while
the small splitting is attributed to the much weaker electromagnetic interaction. Thus, the
nucleon (N) is assigned an internal variable, called isospin , that is analogous to spin.
We assume that isospin states transform according SU(2) with generators T
i
satisfying the
commutation relations
T
i
,T
j
i, j,k
T
k
(10.510)