SO(n) = orthogonal matrix with determinant 1.
SO==special orthogonal
Dim(SO(n))=n(n-1)/2
For X a defining rep of SO(3):
(Xab)ij=delta(i,a)delta(j,b)-delta(j,a)delta(i,b)
=>Commutation relations:
[Xab,Xgd]=delta(b,g)Xad-delta(a,d)Xbg-delta(a,g)Xbg+delta(b,d)Xad
Xab=eabgLg for eabg=antisymmetric tensor.
SO(4) is special. It is the only SO(n) which decomposes.
SO(4) ~= SO(3) x SO(3)
Decomposition:
Xij=-Xji=SO(3) sub algebra
Define Lk as: Xij=i/2*fijkLk
Xi4=-X4i
Define Ki as:-iXij=Ki
Define J(1)=1/2(L+K) (SU(2) group)
Define J(2)=1/2(L-K) (Another SU(2) group)
Note that [J(1)i,J(2)i]=0
=> D(j1,j2) labeling ok.
For arbitrary D(j1,j2) j1+j2=integer => rep exists.
Example Of SO(4) symmetry in nature:
SO(4) is equivalent to the symmetries of a planet orbiting a star or
electron around a hydrogen atom (classically)
Define: M=1/2m(PxL - LxP)-kX/r where
* X= vector of current direction
* P=current momentum
* L=current angular momentum
* m = reduced mass of system
Note that L*M=0 and [M,H]=0
S0(2) = rotational symmetry of a circle. G={g(theta)|0<=theta<2Pi}
g(th1)g(th2)=g(th1+th2) if th1+th2<2Pi, g(th1+th2-2Pi) if th1+th2=>2Pi
Defining representation:
g(th)=
|cos(th) -sin(th)|
|sin(th) cos(th)|
=D(th)
Transpose(D)D=Id.=> D real and unitary.
This is an abelian group so its irreps must be 1-d.
M^DM=D~=
|e^ith 0 |
|0 e^-ith|
with M=1/sqrt(2)*
|1 -i|
|-i 1|
Irreps of SO(2): Dm(th)=e^imth for m any integer.
D=D1+D-1
D0 is the trivial rep.
for M!=0 g(th)-> Dm(th) is |m| to 1 mapping.
only D1 and D-1 are faithful.
Integral(G,dg)=1. Here dg=dth/2Pi.
Generator:
D(dth)=
|1 -dth|
|dth 1 |
+O(dth^2)
=Id-idthJ+O(dth^2)
where J=
| 0 -i |
| i 0 |
Note that adjoint(J)=J.
characters are orthogonal.
Integral(G,Xm*(g)Xn(g)dg)=delta(m,n)
<=> Integral(0,2Pi,e^imth * e^-inth dth/2Pi)=delta(m,n)
Completeness:
f(g)=Sum(-infinity,infinity,ame^-imth)
f(th)=Sum(m,cme^-imth) whre cm=Integral(0,2Pi,e^imth * f(th)dth/2Pi)
This is basically the Fourier series.
SO(1,1)=
|cosh t sinh t|
|sinh t cosh t|
-infinity Fourier transform
The conjugate rep. to a rep in SO(m) is found by filling in the missing spaces
in a young diagram.
For example the conjugate to:
###
#
is:
##
###