Quantum Mechanics for Mathematicians

Note that in the Schr ̈odinger representation

−i

1

2

(QP+PQ) =−i(QP−

i

2

) 1 =−q

d

dq

−

1

2

1

The operator will have as eigenfunctions

ψ(q) =qc

with eigenvalues−c−^12. Such states are far from square-integrable, but do have
an interpretation as distributions on the Schwartz space.

20.4 Representations ofNoK,Ncommutative

The representation theory of semi-direct productsNoKwill in general be rather
complicated. However, whenNis commutative things simplify considerably,
and in this section we’ll survey some of the general features of this case. The
special cases of the Euclidean groups in 2 and 3 dimensions were covered in
chapter 19 and the Poincar ́e group case will be discussed in chapter 42.
For a general commutative groupN, one does not have the simplifying fea-
ture of the Heisenberg group, the uniqueness of its irreducible representation.
On the other hand, whileNwill have many irreducible representations, they
are all one dimensional. As a result, the set of representations ofNacquires its
own group structure, also commutative, and one can define:

Definition(Character group).ForNa commutative group, letN̂be the set of
characters ofN, i.e., functions

α:N→C

that satisfy the homomorphism property

α(n 1 n 2 ) =α(n 1 )α(n 2 )

The elements ofN̂form a group, with multiplication

(α 1 α 2 )(n) =α 1 (n)α 2 (n)

WhenN is a Lie group, we will restrict attention to characters that are
differentiable functions onN. We only will actually need the caseN=Rd,
where we have already seen that the differentiable irreducible representations
are one dimensional and given by

αp(a) =eip·a

wherea∈N. So the character group in this case isN̂=Rd, with elements
labeled by the vectorp.