Quantum Mechanics for Mathematicians

For a semi-direct productNoK, we will have an automorphism ΦkofN
for eachk∈K. From this action onN, we get an induced action on functions
onN, in particular on elements ofN̂, by

Φ̂k:α∈N̂→̂Φk(α)∈N̂

wherêΦk(α) is the element ofN̂satisfying

Φ̂k(α)(n) =α(Φ−k^1 (n))

For the case ofN=Rd, we have

Φ̂k(αp)(a) =eip·Φ−k^1 (a)=ei(Φ−k^1 )T(p)·a

so
Φ̂k(αp) =α
(Φ−k^1 )T(p)

WhenKacts by orthogonal transformations onN=Rd, ΦTk= Φ−k^1 so

Φ̂k(αp) =αΦk(p)

To analyze representations (π,V) ofNoK, one can begin by restricting
attention to theN action, decomposingV into subspacesVαwhereN acts
according toα.v∈Vis in the subspaceVαwhen

π(n, 1 )v=α(n)v

Acting byKwill take this subspace to another one according to

Theorem.
v∈Vα =⇒π( 0 ,k)v∈VΦ̂k(α)

Proof.Using the definition of the semi-direct product in chapter 18 one can
show that the group multiplication satisfies

( 0 ,k−^1 )(n, 1 )( 0 ,k) = (Φk− 1 (n), 1 )

Using this, one has

π(n, 1 )π( 0 ,k)v=π( 0 ,k)π( 0 ,k−^1 )π(n, 1 )π( 0 ,k)v

=π( 0 ,k)π(Φk− 1 (n), 1 )v

=π( 0 ,k)α(Φk− 1 (n))v

=Φ̂k(α)(n)π( 0 ,k)v

For eachα∈N̂one can look at its orbit under the action ofKbyΦ̂k, which
will give a subsetOα⊂N̂. From the above theorem, we see that ifVα 6 = 0,
then we will also haveVβ 6 = 0 forβ∈ Oα, so one piece of information that
characterizes a representationV is the set of orbits one gets in this way.
αalso defines a subgroupKα⊂Kconsisting of group elements whose action
onN̂leavesαinvariant: