Quantum Mechanics for Mathematicians

8.1.2 Lie algebra representations: raising and lowering op-

erators

To proceed further in characterizing a representation (π,V) ofSU(2) we need
to use not just the action of the chosenU(1) subgroup, but the action of
group elements in the other two directions away from the identity. The non-
commutativity of the group keeps us from simultaneously diagonalizing those
actions and assigning weights to them. We can however work instead with the
corresponding Lie algebra representation (π′,V) ofsu(2). As in theU(1) case,
the group representation is determined by the Lie algebra representation. We
will see that for the Lie algebra representation, we can exploit the complexifica-
tion (recall section 5.5)sl(2,C) ofsu(2) to further analyze the possible patterns
of weights.
Recall that the Lie algebrasu(2) can be thought of as the tangent spaceR^3
toSU(2) at the identity element, with a basis given by the three skew-adjoint
2 by 2 matrices

Xj=−i

1

2

σj

which satisfy the commutation relations

[X 1 ,X 2 ] =X 3 ,[X 2 ,X 3 ] =X 1 ,[X 3 ,X 1 ] =X 2

We will often use the self-adjoint versionsSj=iXjthat satisfy

[S 1 ,S 2 ] =iS 3 ,[S 2 ,S 3 ] =iS 1 ,[S 3 ,S 1 ] =iS 2

A unitary representation (π,V) ofSU(2) of dimensionmis given by a homo-
morphism
π:SU(2)→U(m)

We can take the derivative of this to get a map between the tangent spaces
ofSU(2) and ofU(m), at the identity of both groups, and thus a Lie algebra
representation
π′:su(2)→u(m)

which takes skew-adjoint 2 by 2 matrices to skew-adjointmbymmatrices,
preserving the commutation relations.
We have seen in section 8.1.1 that restricting the representation (π,V) to
the diagonalU(1) subgroup ofSU(2) and decomposing into irreducibles tells us
that we can choose a basis ofVso that

(π,V) = (πq 1 ,C)⊕(πq 2 ,C)⊕···⊕(πqm,C)

For our choice ofU(1) as matrices of the form

ei^2 θS^3 =

(

eiθ 0

0 e−iθ

)