From Classical Mechanics to Quantum Field Theory

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