PHY 745 Group Theory
11-11:50 AM MWF Olin 102
Plan for Lecture 36:
1. Summary of what we learned about
linear Lie groups, especially SO(3)
and SU(2), their direct products
and Clebsch-Gordan coefficients
2. Review of topics in Group Theory
3. Course evaluation
Ref. J. F. Cornwell, Group Theory in Physics, Vol I
and II, Academic Press (1984)
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PHY 745 Spring 2017 -- Lecture 34
1
Jason and Ahmad
Taylor
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PHY 745 Spring 2017 -- Lecture 34
2
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PHY 745 Spring 2017 -- Lecture 34
3
SO(3) – all continuous linear transformations in 3dimensional space which preserve the length of
coordinate vectors -- R (w , nˆ )
SU(2) – all 2x2 unitary matrices
u(w , nˆ )
z
w
n̂
y
x
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PHY 745 Spring 2017 -- Lecture 34
4
z
w
n̂
V
V’
y
x
 cos w  n12 1  cos w 
n3 sin w  n1n2 1  cos w   n2 sin w  n1n3 1  cos w  


R (w , nˆ )    n3 sin w  n1n2 1  cos w 
cos w  n22 1  cos w 
n1 sin w  n2 n3 1  cos w  
2
 n2 sin w  n1n3 1  cos w   n1 sin w  n2 n3 1  cos w 

cos
w

n
3 1  cos w 


 cos w sin w 0 
For example: R (w , zˆ ) =   sin w cos w 0 
 0
0
1 

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  w  
PHY 745 Spring 2017 -- Lecture 34
5
z
w
V
V’
n̂
y
x
 cos w2  in3 sin w2
u(w , nˆ )  
w
in

n
sin


2
 1 2
 in1  n2  sin w2

w
w
cos 2  in3 sin 2 
 eiw / 2
For example: u(w , zˆ ) = 
 0
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0 
 iw / 2 
e

PHY 745 Spring 2017 -- Lecture 34
2  w  2
6
Class structure of these groups
 cos w  n12 1  cos w 
n3 sin w  n1n2 1  cos w   n2 sin w  n1n3 1  cos w  


R (w , nˆ )    n3 sin w  n1n2 1  cos w 
cos w  n22 1  cos w 
n1 sin w  n2 n3 1  cos w  
2
 n2 sin w  n1n3 1  cos w   n1 sin w  n2 n3 1  cos w 

cos
w

n
3 1  cos w 


Tr  R (w , nˆ )   3cos w   n12  n22  n32  1  cos w   1  2cos w  depends on w and not on nˆ
 cos w2  in3 sin w2  in1  n2  sin w2 
u(w , nˆ )  
w
w
w
i
n

n
sin
cos

in
sin
3
2
2
2 
  1 2
Tr  u(w , nˆ )   2cos w2 depends on w and not on nˆ
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PHY 745 Spring 2017 -- Lecture 34
7
“Generators” of the groups
 cos w  n12 1  cos w 
n3 sin w  n1n2 1  cos w 

R (w , nˆ )    n3 sin w  n1n2 1  cos w 
cos w  n22 1  cos w 
 n2 sin w  n1n3 1  cos w   n1 sin w  n2 n3 1  cos w 

 0
lim  R (w , nˆ )  1 
a
 n1a1  n2a 2  n3a3    n3


w  0
w

 n
 2
n3
0
 n1
 n2 
n1 
0 
 cos w2  in3 sin w2  in1  n2  sin w2 
u(w , nˆ )  
w
w
w
in

n
sin
cos

in
sin
3
2
2
2 
  1 2
lim  u(w , nˆ )  1 
a
 n1a1  n2a 2  n3a3 


w  0
w

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 n2 sin w  n1n3 1  cos w  

n1 sin w  n2 n3 1  cos w  
cos w  n32 1  cos w  
PHY 745 Spring 2017 -- Lecture 34
1  in3

2  in1  n2
in1  n2 
in3 
8
Irreducible representations of SO(3) and SU(2)
-- focusing on SU(2)
Consider the generator matrices
lim  u(w , nˆ )  1 
1  in3
a
 n1a1  n2a 2  n3a3  


w  0
w
2  in1  n2

1 0 i 
1  0 1
1i 0 
a1  
a2  
a3  



2  i 0
2  1 0 
2  0 i 
Commutation relations: [a1 , a 2 ]=  a3
in1  n2 
in3 
In order to determine irreducible representations of SU(2),
determine eigenstates associated with generators.
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PHY 745 Spring 2017 -- Lecture 34
9
Eigenfunctions associated with generator functions of SU(2)
1 0 i 
1  0 1
a1  
a2  
a3 


2  i 0
2  1 0 
Commutation relations: [a1 , a 2 ]=  a3
1i 0 


2  0 i 
It is convenient to map these generators into Hermitian matrices
A p  ia p = 12 σ p
1 0 1
1  0 i 
A1  
A2  


2 1 0
2 i 0 
Commutation relations: [A1 , A 2 ]=iA 3
1 1 0 
A3  

2  0 1
3 1 0
Define A =A +A  A = 

4 0 1
0 0
0 1
A   A1  iA 2  
A   A1  iA 2  


1
0
0
0




2
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2
1
2
2
2
3
PHY 745 Spring 2017 -- Lecture 34
10
Eigenfunctions associated with generator functions of SU(2)
Relationships between operators:
[A 2 , A p ]  0
A  A   A 2  A 32  A 3
A  A   A  A  A3
2
2
3
Note that this "algebra" is identical to that of the total
angular momentum operators
J x  A1
J y  A2
J z  A3
 Leap to eigenstates of J 2 and J z :
A 2 jm
jm
 j ( j  1) jm
A 3 jm  m jm
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PHY 745 Spring 2017 -- Lecture 34
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Eigenfunctions associated with generator functions of SU(2)
Additional relationships:
A  jm   j  m  j  m  1
j  m  1
A  jm   j  m  j  m  1
j  m  1
1/ 2
1/ 2
Mapping the representations back to the original generators
a p  iA p
of SU(2):
j
Dmm
 (a p )  jm a p jm '
j0
j
1
2
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j
Dmm
 (a p )  0
j
mm 
D
1 0 i 
(a1 )  

2  i 0
j
mm 
D
1  0 1
 a2   

2  1 0 
PHY 745 Spring 2017 -- Lecture 34
j
mm 
D
1i 0 
 a3   

2  0 i 
12
Comment of analogous equations for SO(3) -Eigenfunctions associated with generator functions of SO(3)
 cos w  n12 1  cos w 
n3 sin w  n1n2 1  cos w   n2 sin w  n1n3 1  cos w  


R (w , nˆ )    n3 sin w  n1n2 1  cos w 
cos w  n22 1  cos w 
n1 sin w  n2 n3 1  cos w  
2
 n2 sin w  n1n3 1  cos w   n1 sin w  n2 n3 1  cos w 
co
s
w

n
1  cos w  
3

 0
lim  R (w , nˆ )  1 
 n
a

n
a

n
a

n
a

3 3
 1 1 2 2
 3
w  0 
w

 n
 2
0 0 0
a1   0 0 1 
 0 1 0 


[a1 , a 2 ]  a3
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n3
0
 n1
 n2 
n1 
0 
 0 0 1
a 2   0 0 0 
1 0 0 


PHY 745 Spring 2017 -- Lecture 34
 0 1 0
a3   1 0 0 
 0 0 0


13
0 0 0
 0 0 1
a1   0 0 1 
a 2   0 0 0 
 0 1 0 
1 0 0 




Commutation relations: [a1 , a 2 ]=  a3
 0 1 0
a3   1 0 0 
 0 0 0


It is convenient to map these generators into Hermitian matrices
A p  i a p
0 0 0 
 0 0 i
A1   0 0 i 
A 2   0 0 0 
0 i 0 
 i 0 0 




Commutation relations: [A1 , A 2 ]=iA 3
Define
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 0 i 0 
A 3   i 0 0 
0 0 0


1 0 0
A 2 =A12 +A 22  A 32 =2  0 1 0 
0 0 1


PHY 745 Spring 2017 -- Lecture 34
14
However, in this case A3 is not diagonal; using a
similarity transformation:
 12

†
A 3  U A 3U   0
 1
 2
 12

†
A1  U A1U   0
 1
 2
 12

†
A 2  U A 2U   0
 1
 2
0

0 1
i

0
2

i
0
2

0 1
i

0
2

i
2
i
2
0
i
2
0

1
0

1

0

i
0

 2
 i 0 0  i

  2
0 0 0 

 0
1

0
0
0

 2
 0 0 i   i

  2
0 i 0  

 0
 0 0 i 
0 0 0 


i 0 0 


 12
 i
 2

0
 1 0 0 
 
i
0 2  =  0 0 0 

 
1 0   0 0 1
0 21   0 2i 0 

  i
i
i
0 2 = 2 0

2

 
i
1 0 0
0 
2
0
1
2
0
1
2
0
1
 0
  1
i
= 2
2 
 
0 0
1
2
0
1
2
0

1

2

0 
In accordance with the usual convention:
J z  A3
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J y  A1
J x  A2
PHY 745 Spring 2017 -- Lecture 34
15
In general, we can form a representation for SU(2) by
w D j ( a3 )
evaluating
e
:
The character of this representation is
j
 j (w ) 

eimw 
m  j
sin   j  12  w 
sin  12 w 
j1
j2
D
and
D
Direct product of irreducible representations
It can be shown that D j1  j2  D j1  j2  D j1  j2 1....D
j1  j2
This follows from the character relationships:

j1  j2
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(w ) 
sin   j1  12  w  sin   j2  12  w 
si n
2

1
2
w
PHY 745 Spring 2017 -- Lecture 34

j1  j2

 j (w )
j  j1  j2
16
For example: j1=1/2 and j2=1/2
It can be shown that D j1  j2  D j1  j2  D j1  j2 1....D
j1  j2
This follows from the character relationships:
 1/ 2 1/ 2 (w )   1 (w )   0 (w )
j1m1 j2 m2 j1 j2 jm 
1
2
 12 12 12
1
2
 12 12  12
1
2
 12 12 12
1 1
2 2
10 
1 1
2 2
1 1
2 2
1 1 1 1 1 1
2 2 2 2 2 2
1 1 1
2 2 2
 12
1 1
2 2
11  1
10 
1
2
11  1
00  
1 1 1
2 2 2
 12
1 1
2 2
00 
1
2
Review --
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PHY 745 Spring 2017 -- Lecture 34
17
Group theory
An abstract algebraic construction in mathematics
Definition of a group:
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PHY 745 Spring 2017 -- Lecture 34
18
Definition:
An element B  XAX 1 is defined as conjugate to element A,
where X is any element of the group.
Definition:
A class is composed of members of a group which
are generated by the conjugate construction:
C = X i1YX i where Y is a fixed group element and X i
are all of the elements of the group.
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PHY 745 Spring 2017 -- Lecture 34
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The great orthogonality theorem
Notation: h  order of the group
R  element of the group
 ( R )  ith representation of R
i

denote matrix indices
li  dimension of the representation
   ( R)    ( R)
i
R
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*
j
h
  ij  
li
PHY 745 Spring 2017 -- Lecture 34
20
Character of a representation:
 ( R)    ( R) 
j
j

In terms of classes C , each with N C elements :
 N   (C )  
i
C
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C
*
j
(C )  h ij
PHY 745 Spring 2017 -- Lecture 34
21
Some further conclusions:
j
N

(
C
)

 (C )  h ij
 C
*
i
C
The characters i behave as a vector space with the
dimension equal to the number of classes.
The number of characters=the number of classes
Second character identity:
h
i   (Ck )   (Cl )  N  kl
Ck
i
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*
i
PHY 745 Spring 2017 -- Lecture 34
22
Example of character table for D3
“Standard” notation for
representations of D3
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PHY 745 Spring 2017 -- Lecture 34
23
Extension of group notions to continuous groups
 Introduced notion of mapping group
members to finite number of continuous
parameters
 Introduced notion of “metric”
 Introduced notion of connected subgroups
 “Algebraic” properties of group generators
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PHY 745 Spring 2017 -- Lecture 34
24