Quantum Mechanics for Mathematicians

Digression.Dropping the requirement of finite dimensionality, the same con-
struction starting with a highest weight vector and repeatedly applying the lower-
ing operator can be used to produce infinite dimensional irreducible representa-
tions of the Lie algebrassu(2)orsl(2,C). These occur when the highest weight
is not a non-negative integer, and they will be non-integrable representations
(representations of the Lie algebra, but not of the Lie group).

Since we saw in section 8.1.1 that representations can be studied by looking
at the set of their weights under the action of our chosenU(1)⊂SU(2), we
can label irreducible representations ofSU(2) by a non-negative integern, the
highest weight. Such a representation will be of dimensionn+ 1, with weights

−n,−n+ 2,···,n− 2 ,n

Each weight occurs with multiplicity one, and we have

(π|U(1),V) =C−n⊕C−n+2⊕···Cn− 2 ⊕Cn

Starting with a highest weight or lowest weight vector, a basis for the repre-
sentation can be generated by repeatedly applying lowering or raising operators.
The picture to keep in mind is this

C−n C−n+2 Cn− 2 Cn

π′(S 3 ) π′(S 3 ) π′(S 3 ) π′(S 3 )

π′(S−) π′(S−) π′(S−) π′(S−)

lowest π′(S+) π′(S+) π′(S+) π′(S+)

weight

vectors

highest

weight

vectors

Figure 8.1: Basis for a representation ofSU(2) in terms of raising and lowering
operators.

where all the vector spaces are copies ofC, and all the maps are isomorphisms
(multiplications by various numbers).
In summary, we see that all irreducible finite dimensional unitarySU(2)
representations can be labeled by a non-negative integer, the highest weightn.
These representations have dimensionn+ 1 and we will denote them (πn,Vn=
Cn+1). Note thatVnis then’th weight space,Vnis the representation with
highest weightn. The physicist’s terminology for this uses notn, butn 2 and
calls this number the “spin”of the representation. We have so far seen the lowest
three examplesn= 0, 1 ,2, or spins=n 2 = 0,^12 ,1, but there is an infinite class
of larger irreducibles, with dimVn=n+ 1 = 2s+ 1.