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Some group Theory
A group is a set of elements
+
a composition rule ☻, such that
1. combining two elements under the composition
rule☻ gives another of the elements
2. there is an identity element I, so that for any
element E of the group
E☻I = I☻E = E,
3. Every element has a unique inverse E-1,
with E-1E= EE-1= I
4. ☻is associative
A☻(B☻C)=(A☻B) ☻C
An example interisting for us
• The set of all complex factors of a wave
function U() ,  real, form a continuous group,
with
☻=  (multiplication)
U    e , 0    2
i
check
check 1
U  U  '  ei ei '  ei   '  U    '
check 2: I element
U  U     U   U    U (0)  I
U 1    U   
check 3: there is an inverse
check 4:
associative low
U 1 U 2 U 3   U 1  U 2 U 3 
U 1 U  2 U 3   e
i 1  2  i 3
e
U 1 U  2 U 3   e ei   
1
2
3
U 1 U  2 U 3   ei    
1
2
3
The group we have seen is U(1)
• infinite continuous group
• one dimensional unitary group
• each element is characterized by a
continuous parameter  (0  2)
• each element is differentiable
Lie Groups
• U(1) is a Lie group
• a Lie group is one where the elements E are
differentiable functions of their parameters
• It may be demonstrated that for a Lie group with
more parameters i, any element E(12...n)
may be written in the form
n
E 1 , 2 ,... n   exp 
 ii Fi
i 1
• Fi are the generators of the Lie group
• U(1), F1=1 (only one generator)
• U(1) is a “unitary group”
The SO(n) Groups
O(n)  rotations in n-dimensional
Euclidean space
elements  nn matrices R of real
numbers
each R matrix must have n(n-1)/2
indipendent elements
if R deteminant is 1  SO(n) (“Special” O(n)
Group)
The SO(2) , SO(3) , SU(n) ,Groups
• the elements of SO(2) are the rotation in a
plane
• the elements of SO(3) are rotation in space
• The elements in SU(n) are rapresented by
nn unitary matrices , U†U, detU=+1
SO(2)
plane rotation
0  i

 2  
i 0


 cos  sin   


  sin   cos 


ei 2
 x'   cos  sin    x 
  
 


 y '    sin   cos  y 

2 is the generator of the rotation in SU(2)
SU(2)
 cos ei sin  ei 


i
  sin  e cos e i 



P'  e
i 2
P
SU(2)