17. Group Theory
1.
Introduction to Group Theory
2.
Representation of Groups
3.
Symmetry & Physics
4.
Discrete Groups
5.
Direct Products
6.
Symmetric Groups
7.
Continuous Groups
8.
Lorentz Group
9.
Lorentz Covariance of Maxwell’s Equations
10. Space Groups
1.
Introduction to Group Theory
Symmetry :
1. Spatial symmetry of crystals
~
X-ray diffraction patterns.
2. Spatial symmetry of molecules
~
Selection rules in vibrational spectra.
3. Symmetry of periodic systems
~
e-properties: energy bands, conductivity, …
Invariance under transformations :
1. Linear displacement
~
Conservation of (linear) momentum.
2. Rotation
~
Conservation of angular momentum.
3. Between (inertial) frames
~
General (special) relativity.
Theories of elementary particles begin with symmetries & conservation laws.
Group theory was invented to handle symmetries & invariance.
Definition of a Group
W.K.Tung, “Group Theory in Physics” (85)
M.Tinkham, “Group Theory & QM” (64)
Refs:
A group { G,  } is a set G with a multiplication  such that  a, b, c  G ,
1.
a  b G
Closure
2.
 a  b  c  a   b  c   a  b  c
Associativity
3.
 unique I  G  I  a  a  I  a
Identity
4.
 a 1  G  a 1  a  a  a 1  I
Inverse
Group { G,  } is usually called simply group G and a  b, ab.
Two easily proved theorems :
1. Every a1 is unique.
2.
a G  G
Rearrangement theorem
More Definitions
Finite group : Group with a finite number n of elements.
n = order of the group.
Discrete group :  1-1 map between set G & a subset of the natural number.
( label of elements of G is discrete )
Continuous group with n-parameter :  1-1 map between set G & subset of Rn .
Abelian group :  is commutative, i.e., ab  ba
Cyclic group Cn of order n :
Cn   a , a 2 ,
 a, b  G
, an  I

Cn is abelian
Group {G,  } is homomorphic to group { H ,  } :
 a map f : G  H that preserves multiplications, i.e.,
a b  c
 a, b, c  G

f  a   f  b  f  c 
If f is 1-1 onto ( f1 exists ), then {G,  } and { H ,  } are isomorphic.
Subgroup of group {G,  } : Subset of G that is closed under .
Example 17.1.1.
D3 Symmetry of an Equilateral Triangle
Table of gi gj for D3
gi
gj
Subgroups :
C3   I , C3 , C32
C2   I , C2
Mathematica
Dihedral group

 I , C2 
 I , C2 

Example 17.1.2.
Rotation of a Circular Disk
Rotation in x-y plane by angle  :
r  R   r

I  R  0
 x   cos 
 y      sin 
  
sin    x 
cos    y 
R   R    R      R   R  
R1    R   


G  R       ,  

1-D continuous abelian group.
Example 17.1.3.
An Abstarct Group
An abstract group is defined by its multiplication table alone.
Vierergruppe (4-group) :
I
A B C
I
I
A B C
A
A
I
C
B
B
B C
I
A
A
I
C C
B
Example 17.1.4.
Isomorphism & Homomorphism: C4
C4 = Group of symmetry operations of a square that can’t be flipped.
C4   I , C4 , C  C2 , C
2
4
3
4

G   1, i ,  1,  i 
abelian
C4 & G are isomorphic.
Subgroup:
C2   I , C2

 1,  1 
I
C4
C2
C43
I C4 C2 C43
I C4 C2 C43
C4 C2 C43 I
C2 C43 I C4
C43 I C4 C2
1
i
1  i
1
1
i
1  i
i
i
1  i
1
1 1  i
1
i
i
i
1
i
1
2.
Representation of Groups
A representation of a group is a set of linear transformations on a
vector space that obey the same multiplication table as the group.
Matrix representation : Representation in which the linear
transformations tak the form of invertible matrices
( done by choosing a particular basis for the vector space ).
Unitary representation : Representation by unitary matrices.
Every matrix representation is isomorphic to a unitary reprsentation.
Example 17.2.1.
Unitary representations for D3 
1 0
U E   I   

0 1
U
E
 1 0 
C2   

0
1


U
U
A Unitary Representation
 I, C
2
 
,
C
3
3 , C 2 , C2 , C2  :
 1

U  E   C3    2
 3

 2
3


2 
1 
 
2 
 1
 
U  E   C32    2

3


 2
3

2 
1
 
2


E
U  C2   



3

2 
1
 
2
 1

E
U  C2   2

3

 2
3


2 
1 
 
2 
1
2
3
2
 A1 
g  1
 A2 
 1 g  I , C 3 , C 32
g  
 1 g  C2 , C 2 , C2
 g G
Active transformations
[ Vectors rotated by U(g) ]
Eq.17.2 in Arfken is passive but
doesn’t agree with fig.17.1.
More Definitions & Properties
A representation U(G) is faithful if U(G) is isomorphic to G.

U G   U  g   g  G
Every group has a trivial representation with

U  g   1  g G
Let U(G) be a representation of G, then W G   V U G  V 1 is also a representation.
W(G) & U(G) are equivalent representations :
W ~U
A representation U(G) is reducible if every U(g) is equivalent to the same block
diagonal form, i.e.,
W  g   W1  g   W2  g  
We then write :
for some W ~ U
U  U1  U 2 
A representation U(G) is irreducible if it is not reducible.
Commuting matrices can be simultaneously digonalized

All irreducible representations (IRs)
of an abelian group are 1-D.





Wg  





W1
W2












Example 17.2.2.
A Reducible Representation
A reducible representation for
 0 1 0
U  C3    0 0 1 
1 0 0


1 0 0
U  I    0 1 0 
0 0 1


Using




W




1
3
1
2
1
6
1
3
0

2
3
1 

3 
1 


2
1 

6 
D3   I , C3 , C 32 , C 2 , C2 , C2  :
0 0 1
U  C    1 0 0 
 0 1 0


2
3
0 0 1
U  C2    0 1 0 
1 0 0


1 0 0
U  C2    0 0 1 
 0 1 0


1
& U  W U W ,
we get the equivalent block diagonal form
1 0 0
U  I    0 1 0 
0 0 1



1 0

1
U   C3    0 

2

3
0

2

1 0 0
U  C2    0 1 0 
0 0 1


 0 1 0
U  C2   1 0 0 
0 0 1



0 

3

2 

1 
 
2 

1

U  C2    0


0




1
0
0 


1
3
2

U  C3   0 

2
2 


0  3  1 


2
2



1
0
0
0 


1
1
3
U  C2   0

2
2
2 


3
1
0  3
 

2

2
2
Mathematica
U   A1  E

0 

3

2 

1 
 
2 
Example 17.2.3.
Representations of a Continuous Group

Symmetry of a circular disk : G  R       ,  

 cos 
R    
  sin 
G is abelian  R is reducible.
1  1 i 


2 1 i 
Let
V

 cos   i sin 
U    
0

&
U    V R   V 1
0
  ei


cos   i sin    0
Independent IRs :
Un    ei n 
n  0,  1,  2,  3,
Only U1 & U1 are faithful.
0 

e i  
sin  
cos  
H  r   E r 
3.
Symmetry & Physics
H  T V
Let R be a tranformation operator such as rotation or translation.
R  H    R  E 

i.e.,
&
R H R 1
R

R H R 1 R  ER
is the tranformed hamiltonian
is the transformed wave function
If H is invariant under R :
i.e.,

R H R 1  H 
R , H   0

H R  ER
R  is also an eigenfunction with eigenvalue E.
possibility of degeneracy.
Actual degeneracy depends on the symmetry group of H & can be calculated,
without solving the Schrodinger eq., by means of the representation theory .
Starting with any function , we can generate a set

S  Ri  ; Ri  G  with R1  I  , i  1,2,
, nG

nG  order of G
G  symm. group of H
Next, we orthonormalize S using, say, the Gram-Schmidt scheme, to get
   i ; i  1,2,
, d  nG

 = basis that spans an d –D space.


d
R i   j U ji  R 
 R G
j 1
 S G
d
d
S R i   S j U ji  R   
j 1

d

j 1 k 1
d
k
U k j  S  U ji  R     k U k i  S R 
k 1
d
U k i  SR    U k j  S  U ji  R 
j 1
Or, in matrix form :
i.e., U   U  R  ; R  G
U  SR   U  S   U  R 

U  R i j  U i j  R 
is a representation of G on the space spanned by .
Starting with any function , we can generate a basis
   Ri  ; Ri  G 
orthnormalized
  i ; i  1,2,
for a d-D representation U   U  R  ; R  G
U is in general reducible, i.e.,

,d
for G.
U   m U     m U   



where m = number of blocks equivalent to the same IR U ( ).
H , R   0


 R G

HEI
w.r.t. a basis for an IR of G.
( Shur’s lemma )
If  is an eigenfunction of H, then U is an IR.
For arbitary  , we can take one state from each U (  ) block to get
a basis to set up a matrix eigen-equation of H to calculate E.

Example 17.3.1.
multiplication table
An Even H
H x  H  x
H is even in x 
Let  be the operator
x  x
then
G  I ,
Cs
I
I
I




I
  Cs
IR
Cs is abelian  All IRs are 1-D.
Cs
I
For an arbitrary  (x) :
A1
A2
1 1
1 1
S     x ,    x   x   
I   x    x
I  x   x
   x   x
  x    x
V  

 = basis for W


 1 0
UI   

0 1
0 1
U    

1 0
1 1 
V

1

1


W  V U V 1

1 0
WI   

0 1

Mathematica
1 0 
W    

 0 1
 1 1     x      x      x    A1  Even

 1 1    x     x    x   









 
  A2  Odd
Generation of IR Basis
Using Schur’s lemma, one can show that (Tung, §4.2)
 
Pi j 
n
nG
where
 Ui j   g 1  R  g 
P   
g G
n
nG
    g  R  g 
1
g G
P ( ) = projector onto the space of IR U ( ).
 ( ) (g) = Character (trace) of U ( ) (g) .
n = dimension of IR.
nG = order of G.
R(g) = operator corresponding to g.
For any f (x),
 P   f  x  ; j  1,
 P  f  x  ; j  1,

ij

, n
, n


, if not empty, is the ith basis vector for the IR U ( ).
, if not empty, is a basis vector for the IR U ( ).
Example 17.3.2.
QM: Triangular Symmetry
3 atoms at vertices Ri of an equilaterial triangle :
G  D3
ri  r  Ri
Starting with atomic s-wave function  (r1) at R1 :
1
A
    r     r1     r2     r3  

Pi j   
U
 A1 
g  1
1 0
U E   I   

0 1
n
nG
 U    g  R  g 
1
g G
 g G
U
E
ij
U
 A2 
 1

 C3    2
 3

 2



1
0
E






U
C



U  E   C2   
2


 0 1


1
2
3
2
 1 g  I , C 3 , C 32
 1 g  C2 , C 2 , C2
 g   
 1
 
C

   2
3

 2
3

2 
1 
 
2 
U
3

2 
1
 
2
 1

U  E   C2   2

3

 2

E
2
3
1
3
1
A
    r     r1     r2     r3     r1     r3     r2    0
2
6
 11 E   r  

E
 12
2
6
1
1
1
1



r


r


r


r


r


r












2
3
1
3
2 
 1 2
2
2
2

1
 2  r1     r2     r3  
3
2
r  
6
 3

1
3
3
3
  r3     r2  
  r2  
  r3  
  r3  
  r2   

2
2
2
2
3


3

2 
1
 
2
3

2 
1 
 
2 

4.
Discrete Groups
Classes :
For any a  G, the set
C   g a g 1 ;  g  G

is called a class of G.
C is usually identified by one of its elements.
Rearrangement theorem 
A class can be generated by any one of its members.
( a can be any member of C ).
Example 17.4.1.
g
a
Classes of D3
D3   I , C3 , C 32 , C 2 , C2 , C2 
Table of g a g1 for D3

Classes of D3 are :
 I , C
2
 
,
C
3
3  , C 2 , C2 , C2 

Usually denoted as
 I , 2C3 , 3C2 
Mathematica
Tr  g a g 1   Tr  a 
Orthogonality
relations :
(Unitary IR)
g
All members of a class have
the same character(trace).


 

C



C   nG   

*
C
  C  
 
 
nG

C    C C
nC
*
Dimensionality
theorem :
 n  n
2
G
 n  n
2
G


C     C   nG   
*
C
  C  
 
 
 C  
*
nG
 C C
nC
Normalized full representation table of D3 :
n
nG
Take each row (column) as a vector :
They’re all orthonormalized.
Products of columns (rows) correspond to the completeness condition.
Example 17.4.2.
n
row
orthogonality

C

Orthogonality Relations: D3
C   C   nG   
*
 
C
1  1  2   2  1   1   3  1  0   0
A1 , E :
1   2  2   2   1   1   3   0  0   6
E,E:

Completeness

 
 C    C 
 
*

nG
 C C
nC
1  1  1   1   1  0  0
C3 , C2 :
D3
1 1  1 1   1   1  3 
C3 , C3 :
 n  n
2
G
12  12  22  6
6
2
Character table of D3
Mathematica
Example 17.4.3.
C4   I , C4 , C 42 , C43
 ~  I, a , a
2
, a3
Counting IRs
multiplication table

C4
th IR for cyclic group of order n :
U
 
  1, , n
  1 k 


 a   exp  2 i n 


k
k  0,
,n 1
Table of g b g1 for C4
g
Orthogonality & completeness
b
 Character table is a square array.
 # of IRs  N = # of classes  NC .
( Easily guessed from the relations
N
n  n


2
G
1
E.g., for abelian C4 , we must have
n
C 1
C
)
Character table
C4
n   1, 1, 1, 1 
For non-abelian D3 : Only solution to
n   1, 1, 2 

NC
 n  6
2
is
Mathematica
Decomposing A Reducible Repesentation
 nC  

*
C     C   nG   
C
U   c U   

Let U be reducible
U  g    c U    g 



  g    c      g 
  C    c     C 


c 
1
nG

 nC     C   C 
*
C
Example 17.4.4. Decomposing
A Reducible Representation
c 
1
nG
 nC     C   C 
*
C
D3
Consider the following basis for D3 :

  x2 , y2 , 2 x y
I

C3
D3
U

C32
C2
C2
I 2C3 3C2
3 0
1
1
1  1  3  2  1  0   3  1  1   1
6
1
c A2  1  1  3  2  1  0   3   1  1   0
6
1
cE  1   2  3  2   1  0   3   0  1   1
6
C2
Mathematica
c A1 
U   A1  E

 A1  x 2  y 2
E   x2  y 2 , 2 x y 
Other Discrete Groups
Point group : Group of symmetry operations that keep 1 point P fixed.
Operations may include
1. Discrete ( n-fold ) rotations about axes through P.
2. Inversions about P.
3. Reflections about planes containing P.
4. Reflection after rotation with P fixed.
5.
Direct Products
The basis  of a system may be the direct product of other basis { j } if
1. The system consists of more than one particle.
2. More than one degrees of freedom of a single particle are considered.
Group representation U w.r.t.  is therefore a
direct product of representations {Uj } w.r.t. { j }.
i.e
.
U  U j
j
U is in general reducible.
Taking the trace of U(g) gives
 g    j g
 g G
j

Decomposition of U is easily done using the charcter table of G.
Example 17.5.1.
Even-Odd Symmetry
Consider system of n particles in potential
n
U   U  j


where
j 1
n
Ug   U
j 1
 j
g
 Ag

 Au
V  r   V r 
I i
1 1
1 1
U  j   Ag or Au
n

Let there be k particles in Au , then

Ci
Ag
Au
k  even
k  odd
  g      j  g 
j 1
 I  1
i.e.,
 i     
k
 is always irreducible.
D3
Example 17.5.2.
2 Particles with D3 Symmetry
E.g., 2 valence electrons in a molecule with D3 symmetry.
Let both particles be in states of E symmetry ( basis =
Let
D3 I 2C3 3C2
ee 4 1
0
Projectors 

2
 y2 , 2x y

).
b ri   b i   2 xi yi
a ri   a i   xi2  yi2
The direct product basis is
x

  1    2  ;  ,   a, b
c A1  61  4  2  0   1
 c  61  4  2  0   1
A2


U  e  e  A1  A2  E
cE  61 8  2  0  1

A1
 a 1 a  2   b 1 b  2    x12  y12  x22  y22   4 x1 y1 x2 y2

A2
 a 1 b  2   b 1 a  2   2  x12  y12  x2 y2  2 x1 y1  x22  y22 
1E  a 1a  2  b 1b  2   x12  y12  x22  y22   4 x1 y1 x2 y2
Mathematica
 2E  a 1b  2  b 1a  2  2  x12  y12  x2 y2  2 x1 y1  x22  y22 
6.
Symmetric Groups
nG  n !
Group of permutations of n objects = Symmetric group Sn .
Consider a system of n identical (indistinguishable) particles.
Let Pi j be operator that interchanges the positions of the i & j particles.

Pi j  1,
, i,
, j,
, n    1,
, j,
     1,

 P  Sn
P  1,
, n       1,
P
, i,
, i,
, n
, j,
, n
, n
For scalar , only 1-D representations of Sn are needed.

Group treatment is essential only for spinor .
Bosons
Fermions
Example 17.6.1.
2 & 3 Fermions
Ground state of the 2 electrons of He :
 1, 2    f 1 g  2   g 1 f  2    1   2    1   2  
Space part even
Streamlined notation:
Spin part odd
   f g  g f    
A2  A1  A2
S2
I
A1
A2
1 1
1 1
S2 ~ C2
Ground state of the 3 electrons of Li :
  1  1   2  2
E:
1 
1
 2       
6
S3 ~ D3
2 
1
     
2
1
 1   gh f  h f g  hg f  g f h 
2
2 
1 
1

f
gh

f
hg

gh
f

h
f
g

hg
f

g
f
h



2
3 
Mathematica
see Eg.17.6.2
p
Contruction of 
Let the spin part transforms like the th IR of Sn with basis

i
; i  1,
, n
.
( This fixes the multiplicity of  )
n
P i   j U ji   P 

 P  Sn
j 1
Let  
n

i
i 1
where { i } is a basis for representation U ( ) with n  n .
i
n
i.e.
P  i    j U ji   P 

 P  Sn
j 1
U

Furthermore, let
i.e.
U
 

 U

nA2  1 & n  n
 P   
 A2


P
U

 P
*
 P  Sn
*
U ( ) is the dual of U ()
U ( ) is an IR since U () is.
n
    i i
U     U 

i 1
P
n
n
i 1
i 1
P     P i   Pi   
n

k 1
n

P

k
P
n

j 1
P    
P
P i   j U ji   P 
j 1
n


 n


 
 
   k U k i  P      j U ji  P  
 k 1
  j 1

 j   k j   
P
 
j 1
 n  
 n
 
 k  j  U k i  P  U ji  P    

j 1
 i 1
 k 1
n
n
P  i    j U ji  P 
n
k 1 j 1


n
*
j

P






k
j 
j 1

n
j
n

i 1

*
U k i   P  U ji   P  

U (  ) is unitary
i.e.,  transforms like A2 .
( Fermionic )
i can be generated from any n-particle function  using the projector method.
i j 
n
nG
Uij   P 1  P  rk  
P
n
nG
 
P
P
U ji   P  P  rk 
*
Example 17.6.2.
Construction of ManyBody Spatial Functions
i 
j
n
nG
 
P
U ji   P  P  rk 
*
P
S3 ~ D3
C2
C2
D3
I
C3
C32
C2
S3
I
P(312)
P(231)
P(132) P(321) P(213)
= p(32) = p(31) = p(21)
Let

P(abc) = {1a, 2b, 3c }
p(ab) = { a  b }
 r1 , r2 , r3   f 1 g  2 h  3
χ
 U  I  f gh  U  C32  h f g  U  C3  gh f
nG 
n
 
 
U  C 2  f hg  U C2 hg f  U C2 g f h 

Results already listed in E.g.17.6.1.
Mathematica