Chapter 8

Representations of SU(2)

and SO(3)

For the case ofG=U(1), in chapter 2 we were able to classify all complex
irreducible representations by an element ofZand explicitly construct each
irreducible representation. We would like to do the same thing here for repre-
sentations ofSU(2) andSO(3). The end result will be that irreducible repre-
sentations ofSU(2) are classified by a non-negative integern= 0, 1 , 2 , 3 ,···,
and have dimensionn+ 1, so we’ll (hoping for no confusion with the irreducible
representations (πn,C) ofU(1)) denote them (πn,Cn+1). For evennthese will
correspond to an irreducible representationρnofSO(3) in the sense that

πn=ρn◦Φ

but this will not be true for oddn. It is common in physics to label these
representations bys=n 2 = 0,^12 , 1 ,···and call the representation labeled bys
the “spinsrepresentation”. We already know the first three examples:

• Spin 0: π 0 orρ 0 is the trivial representation forSU(2) orSO(3). In
  physics this is sometimes called the “scalar representation”. Saying that
  states transform under rotations as the scalar representation just means
  that they are invariant under rotations.


• Spin^12 : Taking
  π 1 (g) =g∈SU(2)⊂U(2)
  gives the defining representation onC^2. This is the spinor representation
  discussed in chapter 7. It does not correspond to a representation of
  SO(3).


• Spin 1: SinceSO(3) is a group of 3 by 3 matrices, it acts on vectors inR^3.
  This is just the standard action on vectors by rotation. In other words,
  the representation is (ρ 2 ,R^3 ), withρ 2 the identity homomorphism

g∈SO(3)→ρ 2 (g) =g∈SO(3)