156 From Classical Mechanics to Quantum Field Theory. A Tutorial
If there is a superselection structure, we have the decompositions so we re-write
down into a simpler version,
H=
⊕
k∈K
Hk, R=
⊕
k∈K
Rk, Rk=B(Hk),k∈K (2.86)
whereKis some finite or countable set. The latticeLR(H), as a consequence of
(2.85), decomposes as (the notation should be obvious)
LR(H)=
∨
k∈K
LRk(Hk)=
∨
k∈K
L(Hk) (2.87)
where
LRk(Hk)
∧
LRh(Hh)={ 0 } ifk=h.
In other wordsQ ∈LR(H) can uniquely be written asQ=+k∈KQkwhere
Qk ∈L(B(Hk)). In factQk =PkQk,wherePk is the orthogonal projector
ontoHk.
In this framework, it is possible to readapt Gleason’s result simply observing
that a stateρonLR(H) as above defines a stateρkonLRk(Hk)=L(Hk)by
ρk(P):=
1
ρ(Pk)
ρ(P),P∈L(Hk).
Ifdim(Hk)= 2 we can exploit Gleason’s theorem.
Theorem 2.3.44.LetHbe a complex separable Hilbert space and assume that the
von Neumann algebraRinHsatisfies(SS1)and(SS2), so that the decomposition
(2.86) in coherent sectors is valid where we supposedimHk=2for everyk∈K.
The following facts hold.
(a)IfT∈B 1 (H)satisfiesT≥ 0 andtr T=1then
ρT:LR(H)P→tr(TP)
is an elemeont ofSR(H)that is a state onLR(H).
(b)Forρ∈SR(H)there is aT∈B 1 (H)satisfiesT≥ 0 andtr T=1such
thatρ=ρT.
(c)IfT 1 ,T 2 ∈B 1 (H)satisfy same hypotheses asT in (a), thenρT 1 =ρT 2
is valid if and only if PkT 1 Pk =PkT 2 Pk for allk∈K,Pk being the
orthogonal projector ontoHk.
(d)Aunitvectorψ∈Hdefines a pure state only if it belongs to a coherent
sector. More precisely, a stateρ∈SR(H)is pure, that is extremal, if and
only if there isk 0 ∈K,ψ∈Hk 0 with||ψ||=1such that
ρ(P)=0 ifP∈L(Hk),k=k 0 and ρ(P)=〈ψ,Pψ〉ifP∈L(Hk 0 ).