Quantum Mechanics for Mathematicians

is called thek’th weight space of the representation. All vectors in it are eigen-
vectors ofπ′(S 3 )with eigenvaluek 2.
The dimensiondim Vkis called the multiplicity of the weightkin the rep-
resentation(π,V).

S 1 andS 2 don’t commute withS 3 , so they won’t preserve the subspacesVk
and we can’t diagonalize them simultaneously withS 3. We can however exploit
the fact that we are in the complexificationsl(2,C) to construct two complex
linear combinations ofS 1 andS 2 that do something interesting:

Definition(Raising and lowering operators).Let

S+=S 1 +iS 2 =

(

0 1

0 0

)

, S−=S 1 −iS 2 =

(

0 0

1 0

)

We haveS+,S−∈sl(2,C). These are neither self-adjoint nor skew-adjoint, but
satisfy
(S±)†=S∓

and similarly we have
π′(S±)†=π′(S∓)

We callπ′(S+)a “raising operator” for the representation(π,V), andπ′(S−)
a “lowering operator”.

The reason for this terminology is the following calculation:

[S 3 ,S+] = [S 3 ,S 1 +iS 2 ] =iS 2 +i(−iS 1 ) =S 1 +iS 2 =S+

which implies (sinceπ′is a Lie algebra homomorphism)

π′(S 3 )π′(S+)−π′(S+)π′(S 3 ) =π′([S 3 ,S+]) =π′(S+)

For anyv∈Vk, we have

π′(S 3 )π′(S+)v=π′(S+)π′(S 3 )v+π′(S+)v= (

k

2

+ 1)π′(S+)v

so
v∈Vk =⇒π′(S+)v∈Vk+2

The linear operatorπ′(S+) takes vectors with a well-defined weight to vectors
with the same weight, plus 2 (thus the terminology “raising operator”). A
similar calculation shows thatπ′(S−) takesVktoVk− 2 , lowering the weight by
2.
We’re now ready to classify all finite dimensional irreducible unitary repre-
sentations (π,V) ofSU(2). We define:

Definition(Highest weights and highest weight vectors). A non-zero vector
v∈Vn⊂Vsuch that
π′(S+)v= 0

is called a highest weight vector, with highest weightn.