PHY 745 Group Theory
11-11:50 AM MWF Olin 102
Plan for Lecture 12:
Continuous groups, their representations, and the
great orthogonality theorem
Reading: Eric Carlson’s lecture notes
Additional reference: Eugene Wigner, Group Theory
1. Importance of commutation relations
2. Integral relations
2/8/2017
PHY 745 Spring 2017 -- Lecture 12
1
2/8/2017
PHY 745 Spring 2017 -- Lecture 12
2
2/8/2017
PHY 745 Spring 2017 -- Lecture 12
3
Group structure of SO(3) and SU(2)
OR ( , nˆ )  e
 i J nˆ /
Multiplication rule:
OR (1 , nˆ 1 )OR ( 2 , nˆ 2 )  OR ( 3 , nˆ 3 )
Example for nˆ 1  nˆ 2  zˆ
OR (1 , zˆ )OR ( 2 , zˆ )  OR ( 3 , zˆ )
e
 i1 J z /
2/8/2017
e
 i 2 J z /
e
 i (1   2 ) J z /
PHY 745 Spring 2017 -- Lecture 12
  3  1   2
4
Group structure of SO(3) and SU(2)
OR ( , nˆ )  e
 i J nˆ /
Multiplication rule:
OR (1 , nˆ 1 )OR ( 2 , nˆ 2 )  OR ( 3 , nˆ 3 )
Example for nˆ 1  xˆ
nˆ 2  yˆ
OR (1 , xˆ )OR ( 2 , yˆ )  OR ( 3 , nˆ 3 )
e
 i1 J x /
e
 i 2 J y /
e

 i 1 J x   2 J y  i 1 J x , 2 J y  /  2
 /
In this case:  J x , J y  =i J z
2/8/2017
PHY 745 Spring 2017 -- Lecture 12
5
e
 i1 J x /
e
 i 2 J y /
e

 i 1 J x   2 J y  i 1 J x , 2 J y  /  2
 /
1
1
2
3


LHS= 1  i1 J x /   i1 J x /    i1 J x /  ... 
2!
3!


2
3
1
1


 1  i 2 J y /   i 2 J y /    i 2 J y /  ... 
2!
3!


2
1
2
=1  i 1 J x   2 J y  / 
1 J x    2 J y  / 2  1 2 J x J y / 2  ...
2!
RHS=1  i1 J x /  i 2 J y /  1 J x , 2 J y  /  2 2 




2
1
2

i1 J x /  i 2 J y /  1 J x , 2 J y  /  2   .....
2!
2
1 2
1
2
2
=1  i 1 J x   2 J y  / 
JxJy  JyJx / 
  1 J x    2 J y  /

2
2!



1 2
2
J J
x
y
 J yJx /
2
 ...


2
1
2
= 1  i 1 J x   2 J y  /  1 2 J x J y / 
 1 J x     2 J y  /
2!
2/8/2017
PHY 745 Spring 2017 -- Lecture 12
2
2
+....
6
2
Summary of result for nˆ 1  xˆ
nˆ 2  yˆ
OR (1 , xˆ )OR ( 2 , yˆ )  OR ( 3 , nˆ 3 )
e
 i1 J x /
e
 i 2 J y /
e

 i 1 J x   2 J y  i 1 J x , 2 J y  /  2
 /
Since  J x , J y  =i J z :
 i 1 J x   2 J y  1 2 J z / 2  /
 i 2 J y /
 i1 J x /
e
e
e
2/8/2017
PHY 745 Spring 2017 -- Lecture 12
7
Now consider the great orthogonality theorem:


h
R D ( R)  D ( R)  l  nn '  
n
For continuous groups, the summation
*
n
n '
becomes an integral:
D
2/8/2017
n
( R ) 

*
n '
D ( R) dR 
PHY 745 Spring 2017 -- Lecture 12
 nn '  
ln
 dR
8
In terms of the characters of the representations:

n

*
( R) 
n '
( R )dR  nn '  dR
Procedure for carrying out integration over group elements
In general, there will be continuous parameter(s) which
characterize each group element R  R ( ,  ,...)
 dR   g ( R( ,  ...))d d  ...
where g ( R ( ,  ...)) represents
the density of group elements in the neighborhood of R in the
parameter space of  ,  ...
2/8/2017
PHY 745 Spring 2017 -- Lecture 12
9
Analysis of continuum properties of continuous groups,
following notes of Professor Eric Carlson.
Notation:
Group elements R, S
Continuous parameters x1 , x2 ...  R ( x1 , x2 ...)  R( x)
Identity E  R(e)
Continuous group based on nearby mapping of continuous
parameters
2/8/2017
PHY 745 Spring 2017 -- Lecture 12
10
Some details from Professor Carlson’s Notes:
Another example:
R (1 )  OR (1 , zˆ )
 (1 , 2 )  R 1  R (1 ) R ( 2 )   1   2
2/8/2017
PHY 745 Spring 2017 -- Lecture 12
11
Use of measure function to perform needed integrals
 dR
f ( R) 
Normalization constant
2/8/2017
Jacobian for left or
right measures
PHY 745 Spring 2017 -- Lecture 12
12
We are particularly interested in showing that the great
orthogonality theorem holds for continuous groups which
requires :
Proof from Professor Carlson’s notes:
2/8/2017
PHY 745 Spring 2017 -- Lecture 12
13
Changing variables --
2/8/2017
PHY 745 Spring 2017 -- Lecture 12
14
Using z varable:
2/8/2017
PHY 745 Spring 2017 -- Lecture 12
15
When the dust clears:
These arguments form the basis of the extension of the
great orthogonality theorem to continuous groups.
See Professor Carlson’s notes for more details.
2/8/2017
PHY 745 Spring 2017 -- Lecture 12
16