The representations we have found are all unitary, withπktaking values in
U(1)⊂C∗. The complex numberseikθsatisfy the condition to be a unitary 1
by 1 matrix, since
(eikθ)−^1 =e−ikθ=eikθ
These representations are restrictions to the unit circleU(1) of irreducible rep-
resentations of the groupC∗, which are given by
πk:z∈C∗→πk(z) =zk∈C∗Such representations are not unitary, but they have an extremely simple form,
so it sometimes is convenient to work with them, later restricting to the unit
circle, where the representation is unitary.
2.3 The charge operator
Recall from chapter 1 the claim of a general principle that, when the state space
His a unitary representation of a Lie group, we get an associated self-adjoint
operator onH. We’ll now illustrate this for the simple case ofG=U(1), where
the self-adjoint operator we construct will be called the charge operator and
denotedQ.
If the representation ofU(1) onHis irreducible, by theorem 2.2 it must be
one dimensional withH=C. By theorem 2.3 it must be of the form (πq,C) for
someq∈Z. In this case the self-adjoint operatorQis multiplication of elements
ofHby the integerq. Note that the integrality condition onqis needed because
of the periodicity condition onθ, corresponding to the fact that we are working
with the groupU(1), not the groupR.
For a generalU(1) representation, by theorems 2.1 and 2.3 we have
H=Hq 1 ⊕Hq 2 ⊕···⊕Hqnfor some set of integersq 1 ,q 2 ,...,qn(nis the dimension ofH, theqjmay not be
distinct), whereHqjis a copy ofC, withU(1) acting by theπqjrepresentation.
One can then define
Definition.The charge operatorQfor theU(1)representation(π,H)is the
self-adjoint linear operator onHthat acts by multiplication byqj on the irre-
ducible sub-representationHqj. Taking basis elements inHqjit acts onHas
the matrix
Q=
q 1 0 ··· 0
0 q 2 ··· 0
··· ···
0 0 ··· qn
Thinking ofHas a quantum mechanical state space,Qis our first example
of a quantum mechanical observable, a self-adjoint operator onH. States in the
subspacesHqjwill be eigenvectors forQand will have a well-defined numerical