154 From Classical Mechanics to Quantum Field Theory. A Tutorial
withR, so it must belong to the centre for (SS2) and thus it belongs toP(Q)for
(SS1) and, more precisely it must be of the formPK=Ps(Q)becausePK≤Ps(Q)
by hypothesis, but there are no projectors smaller thanPs(Q)in the PVM ofQ.
SoK=Hs.
(b)Rq:=
{
A|Hq∣∣
A∈R
}
is a von Neumann algebra onHsconsidered as
a Hilbert space in its own right as it arises by direct inspection. (2.84) holds by
definition. SinceHqis irreducible forRq,wehaveRs=R′′s=B(Hs). Each map
RA→A|Hq∈Rqis a representation of∗-algebras as follows by direct check.
Ifq=q′–for instanceq 1 =q 1 ′– there is no isometric surjective mapU:Hq→Hq′
such that
UA|HqU−^1 =A|Hq′If such an operator existed one would have, contrarily to our hypothesisq 1 =q′ 1 ,
q 1 IHq′=UQ 1 |HqU−^1 =Q 1 |Hq′=q′ 1 IHq′so thatq 1 =q′ 1.
We have found that, in the presence of superselection charges, the Hilbert space
decomposes into pairwise orthogonal subspaces which are invariant and irreducible
with respect to the algebra of the observables, giving rise to inequivalent repre-
sentations of the algebra itself. Restricting ourselves to each such subspace, QM
takes its standard form as all orthogonal projectors are representatives of elemen-
tary observables, differently from what happens in the whole Hilbert space where
there are orthogonal projectors which cannot represent observables: These are the
projectors which do not commute withP(Q).
There are several superselection structures as the one pointed out in physics.
The three most known are of very different nature: The superselection structure of
theelectric charge, the superselection structure ofinteger/semi integers values of
the angular momentum, and the one related to the mass in non-relativistic physics,
i.e.,Bargmann’s superselection rule.
Example 2.3.41. The electric charge is the typical example of superselction
charge. For instance, referring toan electron, its Hilbert space isL^2 (R^3 ,d^3 x)⊗
Hs⊗He. The space of the electric charge isHe=C^2 and thereinQ=eσz(see
(2.12)). Many other observables could exist inHein principle, but the elecrtic
charge superselection rule imposes that the only possible observables are functions
ofσz. The centre of the algebra of observables isI⊗I⊗f(σ 3 ) for every function
f:σ(σz)={ 1 , 1 }→C. We have the decomposition in coherent sectors
H=(L^2 (R^3 ,d^3 x)⊗Hs⊗H+)⊕
(L^2 (R^3 ,d^3 x)⊗Hs⊗H−),whereH±are respectively the eigenspaces ofQwith eigenvalue±e.