From Classical Mechanics to Quantum Field Theory

152 From Classical Mechanics to Quantum Field Theory. A Tutorial

(2)If adding the spin space (for instance dealing with an electron “without
charge”), we haveH =L^2 (R^3 ,d^3 x)⊗C^2. Referring to (2.11) a maximal set
of compatible observables is, for instance,A 1 ={X 1 ⊗I,X 2 ⊗I,X 3 ⊗I,I⊗Sz},
another isA 2 ={P 1 ⊗I,P 2 ⊗I,P 3 ⊗I,I⊗Sx}.Asbefore(A 1 ∪A 2 )′′is the von
Neumann algebra of observables of the system (changing the component of the
spin passing fromA 1 toA 2 is crucial for this result). Also in this case, it turns out
that the commutant of the von Neumann algebra of observables is trivial yielding
R=B(H).

2.3.5.4 Superselection charges and coherent sectors

We must have accumulated enough formalism to successfully investigate the struc-
ture of the Hilbert space (always supposed to be separable) and the algebra of the
observables when not all selfadjoint operators represent observables and not all
orthogonal projectors are intepreted as elementary observables. Re-adapting the
approach by Wightman[ 22 ]to our framework, we make two assumptions generally
describing the so calledsuperselection rulesfor QM formulated in a (separable)
Hilbert space whereRdenotes the von Neumann algebra of observables.

(SS1)There is a maximal set of compatible observables inR,sothatR′=
R′∩R.
(SS2)R′∩Rcontains a finite class of observablesQ={Q 1 ,...,Qn}, with
σ(Qk)=σp(Qk),k=1, 2 ,...,n, generating the centre:Q′′⊃R′∩R.
(If theQkare unbounded,Q⊂R′∩Rmeans that the PVM of theQjare included
inR′∩R.)

TheQkare calledsuperselection charges.

As the reader can easily prove, the joint spectral measureP(Q)inRnhas support
given exactly by×nk=1σp(Qk) and, ifE⊂Rn,

PE(Q)=

∑

(q 1 ,...,qn)∈×nk=1σp(Qk)∩E

P{(Qq 11 })···P{(qQnn}) (2.82)

We have the following remarkable result where we occasionally adopt the notation
q:= (q 1 ,...,qn)andσ(Q):=×nk=1σp(Qk).

Proposition 2.3.40.LetHbe a complex separable Hilbert and suppose that the
von Neumann algebra RinHsatisfies(SS1)and(SS2). The following facts
hold.