Quantum Mechanics for Mathematicians

By theorem 2.2, sinceU(1) is a commutative group, all irreducible repre-
sentations will be one dimensional. Such an irreducible representation will be
given by a differentiable map

π:U(1)→GL(1,C)

GL(1,C) is the group of invertible complex numbers, also calledC∗. A differ-
entiable mapπthat is a representation ofU(1) must satisfy homomorphism and
periodicity properties which can be used to show:

Theorem 2.3. All irreducible representations of the groupU(1)are unitary,
and given by

πk:eiθ∈U(1)→πk(θ) =eikθ∈U(1)⊂GL(1,C)'C∗

fork∈Z.

Proof.We will write theπkas a function of an angleθ∈R, so satisfying the
periodicity property
πk(2π) =πk(0) = 1

Since it is a representation,πwill satisfy the homomorphism property

πk(θ 1 +θ 2 ) =πk(θ 1 )πk(θ 2 )

We need to show that any differentiable map

f:U(1)→C∗

satisfying the homomorphism and periodicity properties is of this form. Com-
puting the derivativef′(θ) =dfdθwe find

f′(θ) = lim

∆θ→ 0

f(θ+ ∆θ)−f(θ)

∆θ

=f(θ) lim

∆θ→ 0

(f(∆θ)−1)

∆θ

(using the homomorphism property)

=f(θ)f′(0)

Denoting the constantf′(0) byc, the only solutions to this differential equation
satisfyingf(0) = 1 are
f(θ) =ecθ

Requiring periodicity we find

f(2π) =ec^2 π=f(0) = 1

which impliesc=ikfork∈Z, andf=πkfor some integerk.