for SU(2) M=
| a b |
|-b* a*|
det M = |a|^2 + |b|^2
The group manifold is S3.
Relationship to SU(2):
X=(X1,X2,X3)
S=(s1,s2,s3)=generator matrices
Note that:
* {si,sj}=anti-commutator=2delta(i,j)
* Trace(sisj)=2delta(i,j)
S*X=
|X3 X1-iX2|
|X1+iX2 -X3|
=general 2x2 hermitian traceless matrix.
properties:
* g(S*X)g^=S*X for g in SU(2).
* (S*X)(S*X)=X*X
if X'*X'=X*X => related by a 3-D rotation.
x'i=1/2 Tr(si(S*X'))=1/2 Tr(si(gS*Xg^))=Mij Xj for Mij = 1/2 Tr(si(gsjg^)) in SO3.
=> homomorphism from g in SU(2) -> M in SO(3). M(g) = 3-d rep of g.
g in SU(2) => -g in SU(2)
but M(g)=M(-g) So homomorphism 2-1.
K = kernel of homeomorphism = {I2,-I2} = normal subgroup of SU(2)
g1(t) = exp(.5is1t)
M(g1) =
|1 0 0 |
|0 cos t -sin t|
|0 sin t cos t|
= rotation by angle t about x1 axis.
X1 in standard rep =
i *
|0 0 0 |
|0 0 -1|
|0 1 0 |
X2=
i * |0 0 1|
|0 0 0|
|-1 0 0|
X3=
i * |0 -1 0|
|1 0 0|
|0 0 0|
[Xi,Xj]=eijk Xk
=> SO3 and SU2 have the same Lie algebra
irreps:
There exist irreps of every dimension.
Dim 1: g->1
Dim 2: g->defining matrix
Dim 3: SU(2)->SO(3) matrices
SU(2) has its terminology derived from use in quantum mechanics.
irreps are labeled by j=0,1/2,1,..
Dim D(j+1)mn=2j+1
with m,n=-j,1-j,...,j
For j=n/2, n odd irrep is faithful => not irreps of SO(3)
For j=n, n integer irrep is faithful rep of SO(3)
Use euler coordinates to derive irreps.
D(j)mn(a,b,g)=[D(j)(0,0,g)D(g)(0,b,0)Dj(a,0,0)]mn
=e^(-ima)d(j)mn(b)e^(-img)
dg = sin b dbdadg
Proof: show that d(g0g)=dg => jacobian = 1.
d(j)mn(b)= jacobi polynomials.
spherical harmonics:
for l integer,
Ylm(theta,phi)=((2l+1)/4Pi)^.5 * D(l)m0(phi,theta)
legendre polynomials:
Pl(cos theta)= D(l)00(theta)
The addition theorem for legendre polynomials is easily provable with its
group representation.
conjugacy classes:
l a 3-d vector with |l|<=2Pi
for g(l)= e^(.5 s*l)
|l| = C = a conjugacy class of SU(2).
D(j)mn(phi,0,0)=e^(-im*phi)delta(m,n)=> X(j)=Sum(m=-j,j,e^(-im*phi))
=(Sin(j+1/2)phi)/sin(phi/2)
=> X(j)(0)=2j+1 as expected.
Adjoint rep:
The general adjoint of rep has the young diagram:
##
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.
.
.
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with m-1 rows. Dimension = m^2-1 = dimension of the group.