Mathematical Foundations of Quantum Mechanics 153
(a)Hadmits the following direct decomposition into closed pairwise orthogo-
nal subspaces, calledsuperselection sectorsorcoherent sectors,
H=
⊕
q∈σ(Q)
Hq (2.83)
where
Hq:=Pq(Q)H.
and eachHqis invariant and irreducible underR.
(b)An analogous direct decomposition occurs forR.
R=
⊕
q∈σ(Q)
Rq (2.84)
where
Rq:=
{
A|Hq
∣∣
A∈R
}
is a von Neumann algebra onHqconsidered as Hilbert space in its own
right. Finally,
Rq=B(Hq)
(c)Each map
RA→A|Hq∈Rq
is an∗-algebra representation ofR(Def.2.2.18). Representations associ-
ated with different values ofqare (unfaithful and) unitarily inequivalent:
In other words there is no isometric surjective mapU:Hq→Hq′such
that
UA|HqU−^1 =A|Hq′
whenq=q′.
Proof. (a)SincePq(Q)Ps(Q)=0ifq=sand
∑
q∈σp(Q)P
q(Q)=I,Hdecomposes
as in (2.83). SincePq(Q)belongs to the centre ofR, the subspaces of the decom-
position are invariant under the action of each element ofR. Let us pass to the
irreducibility. IfP∈R′∩Ris an orthogonal projector it must be a function of
theQkby hypotheses:P =
∫
Rnf(x)dP
(Q)(x)sinceP=PP≥0andP =P†,
exploiting the measurable functional calculus, we easily find thatf(x)=χE(x)
for someE⊂supp(P(Q)). In other wordsPis an element of the joint PVM ofQ:
that PVM exhausts all orthogonal projectors inR′∩R.Now,if{ 0 }=K⊂Hs
is an invariant closed subspace forR, its orthogonal projectorPKmust commute