From Classical Mechanics to Quantum Field Theory

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 ).