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:

with m-1 rows. Dimension = m^2-1 = dimension of the group.

source

jl@crush.caltech.edu index

group_generation

covering_group

lorentz_group

Clebsh-Gordan

lie_algebra

lie_group

adjoint

tensor

weyl

SO