Kelly J.J. Graduate Mathematical Physics, With MATHEMATICA Supplements
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396 10 Group Theory
Table 10.10. Completed character table for symmetries of a square.
Χ
1
2
2
3
2
4
2
5
D
1
11 11 1
D
2
11 11 1
D
3
1 11 1 1
D
4
1 11 11
D
5
20 20 0
Next we construct a reducible dimension-four representation using the permutation
matrices for four objects; examples from each class are listed below.
D1, 2, 3, 4
3
4
4
4
4
4
4
4
4
4
4
4
5
1000
0100
0010
0001
6
7
7
7
7
7
7
7
7
7
7
7
8
Χ
1
4 (10.129)
D2, 3, 4, 1
3
4
4
4
4
4
4
4
4
4
4
4
5
0100
0010
0001
1000
6
7
7
7
7
7
7
7
7
7
7
7
8
Χ
2
0 (10.130)
D3, 4, 1, 2
3
4
4
4
4
4
4
4
4
4
4
4
5
0010
0001
1000
0100
6
7
7
7
7
7
7
7
7
7
7
7
8
Χ
3
0 (10.131)
D4, 3, 2, 1
3
4
4
4
4
4
4
4
4
4
4
4
5
0001
0010
0100
1000
6
7
7
7
7
7
7
7
7
7
7
7
8
Χ
4
0 (10.132)
D3, 2, 1, 4
3
4
4
4
4
4
4
4
4
4
4
4
5
0010
0100
1000
0001
6
7
7
7
7
7
7
7
7
7
7
7
8
Χ
5
2 (10.133)
Then
1
N
G
k
c
k
Χ
k
2
1
8
1 4
2
2 0
2
1 0 2 0
2
2 2
2
3 (10.134)
shows that this representation contains three irreps, once each (1
2
1
2
1
2
3). Thus, D
contains the two-dimensional irrep plus two of the one-dimensional irreps. We can deter-
mine which one-dimensional representations are within D using
10.3 Representations 397
a
1
1
N
G
k
c
k
Χ
1
k
Χ
k
1
8
1 4 2 0 1 0 2 0 2 21 (10.135)
a
2
1
N
G
k
c
k
Χ
2
k
Χ
k
1
8
1 4 2 0 1 0 2 0 2 20 (10.136)
a
3
1
N
G
k
c
k
Χ
3
k
Χ
k
1
8
1 4 2 0 1 0 2 0 2 20 (10.137)
a
4
1
N
G
k
c
k
Χ
4
k
Χ
k
1
8
1 4 2 0 1 0 2 0 2 21 (10.138)
such that
D D
1
d D
4
d D
5
Χ
k
1 Χ
4
k
Χ
5
k
(10.139)
is also satisfied. If one has suitable reducible representations at hand, relationships of this
type can sometimes be used to compute the character table, perhaps instead of using factor
groups.
10.3.6 Example: Vibrational eigenvalues of square
Suppose that, at equilibrium, four equal masses m are found at the vertices of a square and
that these masses are connected by four equal springs k along its sides and two springs
Κk along the diagonals. We consider small-amplitude vibrations and expand the potential
energy to second order in the spatial coordinates. Let Ξx
1
,y
1
,x
2
,y
2
,x
3
,y
3
,x
4
,y
4
rep-
resent a state vector for the system where the masses are labeled sequentially. The kinetic
and potential energies are given by
T
1
2
m
i
˙
Ξ
2
i
(10.140)
V
1
2
k
i, j
Ν
i, j
Ξ
i
Ξ
j
k
2
3
4
4
4
4
5
x
1
x
2
2
y
2
y
3
2
x
3
x
4
2
y
1
y
4
2
Κ
x
2
y
2
2
x
4
y
4
2
2
Κ
x
1
y
1
2
x
3
y
3
2
2
6
7
7
7
7
8
(10.141)
where the symmetric matrix
398 10 Group Theory
Ν
1
2
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
2 Κ Κ 20Κ Κ 00
Κ 2 Κ 00ΚΚ0 2
202Κ Κ 00Κ Κ
00Κ 2 Κ 0 2 Κ Κ
Κ Κ 002Κ Κ 20
ΚΚ0 2 Κ 2 Κ 00
00Κ Κ 202Κ Κ
0 2 Κ Κ 00Κ 2 Κ
6
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
8
(10.142)
describes small-amplitude vibrations. (Please check this matrix!) The equations of motion
are then
m
¨
Ξ
i
(V
(Ξ
i
k
j
Ν
i, j
Ξ
j
(10.143)
and normal modes satisfy eigenvalue equations of the form
Ξ
Μ
ˆ
Ξ
Μ
Exp
Ω
Μ
t
Ω
2
0
Ν
ˆ
Ξ
Μ
Ω
2
Μ
ˆ
Ξ
Μ
(10.144)
where Ω
2
0
k/m. Note that we use Greek indices to label the eigenvectors and Latin
indices to label components. In this section we will demonstrate that one can deduce the
eigenvalues using properties of the symmetry group for this system without solving a sec-
ular equation.
We begin by constructing representation matrices for typical members of each class.
Consider first the parity operation
DP
3
4
4
4
4
4
4
4
4
4
4
4
5
000p
00p 0
0 p 00
p 000
6
7
7
7
7
7
7
7
7
7
7
7
8
(10.145)
where
p
10
0 1
(10.146)
inverts the y coordinates at each vertex and DP reflects vertices across the horizontal
midplane of the system. Similarly, the basic 90° rotation of vertices is represented by
DR
3
4
4
4
4
4
4
4
4
4
4
4
5
0 r 00
00r 0
000r
r 000
6
7
7
7
7
7
7
7
7
7
7
7
8
(10.147)
where
r
01
10
(10.148)
operates on the coordinates at each vertex. You should verify, by explicit calculation, that
the potential energy is invariant with respect to every member of the symmetry group for
the square, such that ΝDgΝDg
1
.
10.3 Representations 399
The transformation matrices for this representation are obviously reducible. To deter-
mine their irrep content, we evaluate the composite characters for each class
1
IΧ
1
8 (10.149)
2
%
R, R
3
&
Χ
2
0 (10.150)
3
%
R
2
&
Χ
3
0 (10.151)
4
%
P, PR
2
&
Χ
4
0 (10.152)
5
%
PR, PR
3
&
Χ
5
0 (10.153)
to obtain
Μ
a
Μ
2
1
N
G
i
N
i
Χ
i
2
8 (10.154)
Thus, this representation turns out to be the regular representation
D D
1
d D
2
d D
3
d D
4
d 2D
5
(10.155)
Upon application of the similarity transformation that diagonalizes Ν, we expect to find
ΝDiag
%
Λ
1
, Λ
2
, Λ
3
, Λ
4
, Λ
5,1
, Λ
5,1
, Λ
5,2
, Λ
5,2
&
(10.156)
where Λ
5,1
and Λ
5,2
are the eigenvalues for the two occurrences of D
5
.
Recognizing that traces are invariant under similarity transformations, we expand
Tr DgΝ Λ
1
Χ
1
gΛ
2
Χ
2
gΛ
3
Χ
3
gΛ
4
Χ
4
g
Λ
5,1
Λ
5,2
Χ
5
g (10.157)
and use the character table to deduce five linear equations for the eigenvalues
Tr DIΝ 8 4ΚΛ
1
Λ
2
Λ
3
Λ
4
2
Λ
5,1
Λ
5,2
(10.158)
Tr DRΝ 0 Λ
1
Λ
2
Λ
3
Λ
4
(10.159)
Tr D
R
2
Ν4ΚΛ
1
Λ
2
Λ
3
Λ
4
2
Λ
5,1
Λ
5,2
(10.160)
Tr DPΝ 4 Λ
1
Λ
2
Λ
3
Λ
4
(10.161)
Tr DPRΝ 4ΚΛ
1
Λ
2
Λ
3
Λ
4
(10.162)
for our particular representation. Combining the first and third equations gives
Λ
5,1
Λ
5,2
2 (10.163)
such that
Λ
1
Λ
2
Λ
3
Λ
4
4 4Κ (10.164)
Λ
1
Λ
2
Λ
3
Λ
4
0 (10.165)
Λ
1
Λ
2
Λ
3
Λ
4
4 (10.166)
Λ
1
Λ
2
Λ
3
Λ
4
4Κ (10.167)
400 10 Group Theory
From the first pair we deduce
Λ
1
Λ
2
Λ
3
Λ
4
2 2Κ (10.168)
and from the second pair
Λ
1
Λ
2
2 2Κ (10.169)
Λ
3
Λ
4
2 2Κ (10.170)
Thus, we find
Λ
1
2, Λ
2
2Κ, Λ
3
2 2Κ, Λ
4
0,
Λ
5,1
Λ
5,2
2 (10.171)
Finally, we require an argument that distinguishes between the two occurrences of D
5
.
On physical grounds we expect to find three modes with Λ0 that correspond to either
rigid rotation or to horizontal and vertical displacements of the center of mass. The null
eigenvalue for D
4
can be identified with the rotational mode because this representation
is one-dimensional, leaving one of the two-dimensional representations to describe trans-
lations of the center of mass. Therefore, we conclude that the set of eigenvalues for this
system is
Λ2, 2Κ, 2 2Κ, 0, 2, 2, 0, 0 (10.172)
where we account for degeneracy and choose, arbitrarily, to place the rigid translations
last.
10.3.7 Direct-Product Representations
Suppose that D
Μ
R and D
Ν
R are two representations of the same symmetry opera-
tor R that act upon different coordinates and that operators acting on different coordinates
commute. Let u
i
,i 1,m and v
j
,j 1,n represent the basis vectors for these rep-
resentations, such that the m n-dimensional set of basis vectors for the direct-product
representation D
ΜPΝ
is given by u
i
v
j
with two indices. Thus,
Ru
i
v
j
m
k1
n
l1
u
k
D
Μ
k,i
Rv
l
D
Μ
l,j
R
m
k1
n
l1
u
k
v
l
D
ΜPΝ
kl,i j
R (10.173)
where
D
ΜPΝ
kl,i j
RD
Μ
k,i
RD
Ν
l,j
R (10.174)
are the matrix elements of the direct-product representation. Similarly, the character for a
direct product
Χ
ΜPΝ
RTr D
ΜPΝ
R
i, j
D
ΜPΝ
ij,ij
R
i, j
D
Μ
i,i
RD
Ν
j,j
R (10.175)
10.3 Representations 401
is simply the product
Χ
ΜPΝ
RΧ
Μ
RΧ
Ν
R (10.176)
of the characters of the two representations. Direct-product representations are usually
reducible as a Clebsch–Gordan expansion
D
ΜPΝ
Γ
Μ, ΝΓD
Γ
(10.177)
where the summation over irreducible representations D
Γ
is interpreted as d and where
the reduction coefficients
Μ, ΝΓ
1
N
G
gG
Χ
Μ
gΧ
Ν
gΧ
Γ
g
(10.178)
are nonnegative integers.
The unitary matrices D
ΜPΝ
can be reduced to the block diagonal form
CD
ΜPΝ
C
1
3
4
4
4
4
4
4
4
4
4
4
4
5
D
Γ,1
000
0 00
00D
Γ,p
0
000
6
7
7
7
7
7
7
7
7
7
7
7
8
(10.179)
where p Μ, ΝΓrepresents, schematically, the number of occurrences of D
Γ
in the
Clebsch–Gordan expansion and where the pattern is replicated for all irreps Γ. The unitary
matrix C is the matrix of column vectors
w
Γ,Ζ,k
i, j
ΜΝ
ij
Γ, Ζ
k
!
u
i
v
j
(10.180)
whose Clebsch–Gordan coefficients are arranged in a symbol based upon bra-ket notation.
Note that specification of the irreps generally requires two labels because there may be
multiple occurrences. Using the orthonormality of the bases and the unitarity of C, one
obtains the orthogonality relations
i, j
Γ
'
, Ζ
'
k
'
ΜΝ
ij
!
ΜΝ
ij
Γ, Ζ
k
!
∆
Γ,Γ
'
∆
Ζ,Ζ
'
∆
k,k
'
(10.181)
Γ,Ζ,k
ΜΝ
ij
Γ, Ζ
k
!
Γ, Ζ
k
ΜΝ
i
'
j
'
!
∆
i,i
'
∆
j,j
'
(10.182)
such that
u
i
v
j
Γ,Ζ,k
ΜΝ
ij
Γ, Ζ
k
!
w
Γ,Ζ,k
(10.183)
where we define
Γ, Ζ
k
ΜΝ
ij
!
ΜΝ
ij
Γ, Ζ
k
!
(10.184)
for the inverse coupling according to the conventions for bra-ket notation.
402 10 Group Theory
A product representation is described as simply reducible when ( Μ, ΝΓ) is either 0 or
1 for all Γ and when each g
1
is in the same class as g. The index Ζ is then superfluous and
one can determine the CG coefficients, up to a phase, relatively simply. If we invert the
similarity transformation
D
ΜPΝ
i
'
j
'
,i j
gD
Μ
i
'
,i
gD
Ν
j
'
,j
g
Γ,k,k
'
ΜΝ
i
'
j
'
Γ
k
'
!
D
Γ
k
'
,k
g
ΜΝ
ij
Γ
k
!
(10.185)
and then multiply by D
Γ
g
, sum over group elements, and apply the orthogonality the-
orem for irreps, we obtain
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
!
(10.186)
where N
G
is the number of group elements and n
Γ
is the dimensionality of D
Γ
. Taking
i
'
i, j
'
j, k
'
k will provide at least one nonzero CG coefficient and we are free to
choose it to be positive. Varying i
'
,j
'
,k
'
will then provide enough information to determine
the remaining CG coefficients. You are probably familiar with the Clebsch–Gordan (CG)
coefficients for coupling of two angular momenta, but we are using similar notation in a
more generic sense applicable to direct-product representations of any group; naturally the
CG coefficients depend upon the particular group involved. For many groups it is possible
to construct bases that make the Clebsch–Gordan coefficients real.
Example
Let us return once more to the problem of small-amplitude vibrations of an equilateral
triangle. The six-dimensional representation we employed is actually a direct product of
a three-dimensional representation of S
3
describing the permutations of the mass indices
and a two-dimensional representation of a group containing rotations and reflections that is
isomorphic to S
3
. The two-dimensional representation is irreducible, but the three-dimen-
sional representation is not. Suppose that we arrange the group elements as I, R,R
2
,P,PR,
PR
2
. As worked out in a previous section, there are two one-dimensional irreps with char-
acters Χ
1
1, 1, 1, 1, 1, 1 and Χ
1
'
1, 1, 1, 1, 1, 1 and a two-dimenional irrep
with character Χ
2
2, 1, 1, 0, 0, 0. A faithful three-dimensional representation con-
sists of simple permutations of the three mass indices, and a little thought, or explicit
construction, will reveal that its character is Χ
3
3, 0, 0, 1, 1, 1. Hence, the character of
the product representation is Χ
6
6, 0, 0, 0, 0, 0. The reduction coefficients are
3, 2 1
1
6
3, 0, 0, 1, 1, 12, 1, 1, 0, 0, 01, 1, 1, 1, 1, 11 (10.187)
3, 2 1
'
1
6
3, 0, 0, 1, 1, 12, 1, 1, 0, 0, 01, 1, 1, 1, 1, 11
(10.188)
3, 2 2
1
6
3, 0, 0, 1, 1, 12, 1, 1, 0, 0, 02, 1, 1, 0, 0, 01 (10.189)
10.3 Representations 403
where this notation indicates multiplication of corresponding elements of each list and
summation over the list of products. Thus, the Clebsch–Gordan expansion takes the form
D
32
D
1
d D
1
'
d D
2
.
10.3.8 Eigenfunctions
The Hamiltonian H commutes with every operator in its symmetry group , such that
H,
0. Let j denote the labels for an irreducible representation D
j
of ; then
the matrix representation of H will commute with each irreducible D
j
R, such that
H,D
j
R
0 for all R . According to Schur’s lemma, the matrix representation
of H within the group space of D
j
must be a multiple of the unit matrix. In other words,
there must be a set of n linearly independent eigenfunctions Ψ
j
m
,m 1,n with the same
eigenvalue
j
, such that
HΨ
j
m
j
Ψ
j
m
(10.190)
for 1 m n
j
. This set of eigenfunctions is described as a multiplet that is degenerate
with respect to H with degeneracy n
j
. If we start with one eigenfunction in a multiplet, the
others may be generated by applying symmetry operations because, according to group
closure, RΨ
j
m
produces the linear combination
RΨ
j
m
n
j
m
'
1
Ψ
j
m
'
D
j
m
'
,m
R (10.191)
where the coefficients D
j
m
'
,m
R form a nonsingular n
j
-dimensional matrix representation
of operator R within the set of eigenfunctions belonging to eigenvalue
j
. Thus, if H and
R commute, sets of eigenfunctions that are degenerate with respect to H transform among
themselves under R. If we now enlarge the basis to include all possible values of j, H is
represented by a diagonal matrix with the set of
j
along the diagonal, each appearing n
j
times, while representations of R take the fully reduced block-diagonal form
DR
3
4
4
4
4
4
4
4
5
D
1
R 00
0 D
2
R 0
00
6
7
7
7
7
7
7
7
8
(10.192)
where the superscripts enumerate
j
. Each Ψ
j
m
is described as belonging to row m of
irrep j.
The energy eigenvalues
j
for each irrep j are usually distinct but occasionally one
finds two or more distinct irreps that are degenerate in energy. These situations generally
indicate the existence of an accidental or dynamical symmetry and not all eigenfunctions
with the same energy are connected by the symmetry operators within . Knowing the
degeneracy n
of an energy level is not always sufficient to determine the dimensionality
n
j
of the irreps at that energy; one must also verify that the corresponding DR are irre-
ducible. Testing for irreduciblity can be done using character tables for . On the other
404 10 Group Theory
hand, accidental degeneracies are often broken by either perturbations or by subtle effects
neglected by simpler models. A famous example is the Lamb shift in which the acciden-
tal degeneracy between the 2s and 2p
1/ 2
states of the hydrogen atom is broken by the
fluctuations of the electromagnetic field in vacuum that are predicted by quantum electro-
dynamics.
Assuming that our system does not possess accidental symmetries, the orthogonality
theorem for irreducible representations provides important selection rules. The unitarity of
R requires
Ψ
j
'
m
'
Ψ
j
m
RΨ
j
'
m
'
RΨ
j
m
Λ,Λ
'
Ψ
j
'
Λ
'
Ψ
j
Λ
D
j
'
Λ
'
,m
'
R
D
j
Λ,m
R (10.193)
and averaging over the group gives
'
Ψ
j
'
m
'
Ψ
j
m
(
Λ,Λ
'
'
Ψ
j
'
Λ
'
Ψ
j
Λ
(
1
N
G
R
D
j
'
Λ
'
,m
'
R
D
j
Λ,m
R
∆
j,j
'
∆
m,m
'
n
j
Λ,Λ
'
'
Ψ
j
'
Λ
'
Ψ
j
Λ
(
∆
Λ,Λ
'
(10.194)
where n
j
is the dimensionality of irrep j. Thus, eigenfunctions belonging to different irreps
or to different rows of the same irrep are orthogonal. Next, suppose that V is invariant with
respect to R, such that V,0, for any R , where is the symmetry group of H.
Then
'
Ψ
j
'
m
'
V
Ψ
j
m
(
'
Ψ
j
'
m
'
R
VR
Ψ
j
m
(
Λ,Λ
'
'
Ψ
j
'
Λ
'
V
Ψ
j
Λ
(
D
j
'
Λ
'
,m
'
R
D
j
Λ,m
R (10.195)
can again be averaged over R to obtain
'
Ψ
j
'
m
'
V
Ψ
j
m
(
∆
j,j
'
∆
m,m
'
n
j
Λ
'
Ψ
j
'
Λ
V
Ψ
j
Λ
(
(10.196)
Observe that these matrix elements vanish unless j j
'
, m m
'
and are independent of m
for specified j. Therefore, we can write
V,0
'
Ψ
j
'
m
'
V
Ψ
j
m
(
V
j
∆
j,j
'
∆
m,m
'
(10.197)
where V
j
depends only upon the irrep to which these states belong. This is also a conse-
quence of Schur’s lemma, but the foregoing derivation provides some practice using the
orthogonality theorem.
10.3.9 Wigner–Eckart Theorem
Although most operators of interest will not be invariant with respect to the symmetry
group of a Hamiltonian, it is often possible to express them in terms of operators that trans-
form among themselves according to an irreducible representation of that group. Consider
10.4 Continuous Groups 405
an operator V
j
m
that transforms with irrep j according to
RV
J
M
R
M
'
V
J
M
'
D
J
M
'
,M
R (10.198)
for any R . We assume that D
JP j
is simply reducible. Using the orthogonality theorem
to average matrix elements of the form
'
Ψ
j
'
m
'
V
J
M
Ψ
j
m
(
'
Ψ
j
'
m
'
R
RV
J
M
R
R
Ψ
j
m
(
Λ,Λ
'
,M
'
'
Ψ
j
'
Λ
'
V
J
M
'
Ψ
j
Λ
(
D
j
'
Λ
'
,m
'
R
D
J
M
'
,M
RD
j
Λ,m
R
(10.199)
over all R then gives
'
Ψ
j
'
m
'
V
J
M
Ψ
j
m
(
1
N
Λ,Λ
'
,M
'
'
Ψ
j
'
Λ
'
V
J
M
'
Ψ
j
Λ
(
D
j
'
Λ
'
,m
'
R
D
J
M
'
,M
RD
j
Λ,m
R
jJ
mM
j
'
m
'
!
1
n
j
'
Λ,Λ
'
,M
'
'
Ψ
j
'
Λ
'
V
J
M
'
Ψ
j
Λ
(
jJ
Λ M
'
j
'
Λ
'
!
(10.200)
where n
j
'
is the dimensionality of irrep j
'
. Notice that the summation on the right-hand
side depends upon j, J, j
'
but is independent of m, M, m
'
– the dependencies upon row
indices for each irrep are contained entirely within the Clebsch–Gordan coefficient, which
is hopefully known for . Therefore, we obtain the Wigner–Eckart theorem
'
Ψ
j
'
m
'
V
J
M
Ψ
j
m
(
j
'
m
'
jJ
mM
!
j
'
$
$
$
$
$
V
J
$
$
$
$
$
j
(10.201)
where the reduced matrix element j
'
)V
J
)jis independent of row and can be evaluated
for the most convenient selection of m, M, m
'
. The reduced matrix element can also be
obtained using
j
'
$
$
$
$
$
V
J
$
$
$
$
$
j
1
n
j
'
Λ,Λ
'
,M
'
Ψ
j
'
Λ
'
V
J
M
Ψ
j
Λ
(
jJ
Λ M
j
'
Λ
'
!
(10.202)
but that method offers no obvious economy of effort. Note that additional phase or normal-
ization factors are sometimes included in the definition of the reduced matrix elements.
The Wigner–Eckart theorem is usually presented in textbooks on quantum mechan-
ics in connection with rotational symmetry and angular momentum eigenstates, but the
group theoretical derivation is more general and applies to any group with simply reduc-
ible direct-product representations. Nor is it limited to quantum mechanics. Generaliza-
tions can also be made for direct-product representations that are not simply reducible.
10.4 Continuous Groups
10.4.1 Definitions
Suppose that R is a finite group containing n elements R
a
,a 1,n. The group multipli-
cation table R
a
R
b
R
c
can be described as a discrete function c Φa, b that produces a