p600_03n

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```					Rotation Group
Metric Preserving
   A metric is used to measure
x3               x3            the distance in a space.
• Euclidean space is delta
x2
x2      An orthogonal transformation
x1                                          preserves the metric.
x1                              • Inverse is transpose
• Determinant squared is 1
gij uiu j   ij uiu j
   The special orthogonal
gij S m S i        j
n    gmn                  transformation has
S   S 
i
m
T             j
n
1            determinant of +1.
Special Orthogonal Group
    Group definitions: A, B  G          Rotation matrices form a
•   Closure: AB  G                  group.
•   Associative: A(BC) = (AB)C        • Inverse is the transpose
•   Identity: 1A = A1 = A             • Identity is  or I
•   Inverse: A-1 = AA-1 = 1           • Associativity from matrix
multiplication
• Closure from orthogonality
S ij  Aik Bkj
   For three dimensional
S S  Akj B ji B A  Akj jm A
T
ki il
T
im
T
ml
T
ml       rotations the group is
S ki Sil  Akj AT   kl
T
jl
SO(3,R).
SO(3) Algebra
   The Lie algebra comes from
a parameterized curve.
d
de
RR 
T  dR T
de

R R
dR T
de
0
• R(e)  SO(3,R)
• R(0) = I
dR
a
de e  0
   The elements a must be
antisymmetric.
• Three free parameters in   a  aT  0
general form
 0         3     2 
                      
a    3       0      1 
           1    0 
 2                    
Algebra Basis
   The elements can be written
in general form.                                          0 0 0
• Use three parameters as                                        
a1   0 0 1 
coordinates
 0 1 0
• Basis of three matrices                                        

a  1a1   2 a2   3a3                            0 0  1
        
a2   0 0 0 
1 0 0 
        
0 0 0 
                    a1 3  a1             0 1 0
a1 2     0 1 0                                              
a3    1 0 0 
 0 0  1
                    a1 4  a1 2         0 0 0
         
Subgroups
   The one-parameter
subgroups can be found                              1     0              0 
                         
through exponentiation.                   g1 e    0 cose            sin e 
 0  sin e         cose 
                         
gi (e )  exp(eai )  I  eai  1 e 2ai  
2
2                         cose      0  sin e 
                     
gi (e )  I  ai   ai  cose  ai sin e
2       2                    g 2 e    0         1    0 
 sin e     0 cose 
                     
 cose       sin e     0
   These are rotations about                                                   
g 3 e     sin e    cose      0
the coordinate axes.
 0                    1
              0        
Commutator
   The structure of a Lie
[ai , a j ]  e ijk ak                                  algebra is found through the
commutator.
• Basis elements squared
[(ai ) 2 , a j ]  ai [ai , a j ]  [ai , a j ]ai          commute

 e ijk ai ak  e ijk ak ai  0

   This will be true in any other
representation of the Lie
group.
Special Unitary
   If a space is complex-valued
x3            x3            metric preservation requires
Hermitian matrices
x2        • Inverse is complex
conjugate
x2
• Determinant squared is 1
x1
x1
   The special unitary
transformation has
gij uiu j   ij uiu j
determinant of +1.
gij S i m S j n  gmn
   SU(2) has dimension 3
S   S 
i
m
*         j
n
1
SU(2) Algebra
   The Lie algebra follows as it
did in SO(3,R).
d
de

RR 
*  dR *
de

R R
dR*
de
0

   The elements b must be          b  b*  0                     dR
b
Hermitian.                                                     de e  0
• Three free parameters in
general form                    i1                    2  i3 
b
   i                           
 2      3                i1    
   The basis elements
commute as with SO(3).
[bi , b j ]  e ijk bk
Homomorphism
   The SU(2) and SO(3) groups have the same algebra.
• Isomorphic Lie algebras

   The groups themselves are not isomorphic.
• 2 to 1 homomorphism

   SU(2) is simply connected and is the universal
covering group for the Lie algebra.

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