From Classical Mechanics to Quantum Field Theory

140 From Classical Mechanics to Quantum Field Theory. A Tutorial

of|T|(it exists for the previous theorem since|T|is trace class). We have

|tr T|=

∣∣

∣∣

∣

∑

u∈N

〈u, U|T|u〉

∣∣

∣∣

∣=

∣∣

∣∣

∣

∑

u∈N

〈u, Uu〉λu

∣∣

∣∣

∣≤

∑

u∈N

|λu||〈u, Uu〉|.

Next observe that|λu|=λubecause|T|≥0and|〈u, Uu〉|≤||u||||Uu||≤ 1 ||Uu||≤
||u||=1andthus,|tr T|≤

∑

u∈Nλu=

∑

u∈N〈u,|T|u〉=tr|T|=||T||^1.

2.3.4.2 The notion of quantum state and the crucial theorem by Gleason

As commented in (a) in remark 2.2.68, the probabilistic interpretation of quantum
states is not well defined because there is no true probability measure in view
of the fact that there are incompatible observables. The idea is to redefine the
notion of probability in the bounded, orthocomplemented,σ-complete lattice like
L(H) instead of on aσ-algebra. Exactly as in CM, where the generic states are
probability measures on Boolean latticeB(Γ) of the elementary properties of the
system (Sect. 2.3.1), we can thinkof states of a quantum system asσ-additive
probability measures over the non-Boolean lattice of the elementary observables
L(H).

Definition 2.3.21.LetHbe a complex Hilbert space. Aquantum stateinH
is a mapρ:L(H)→[0,1]such that the following requirements are satisfied.

(1)ρ(I)=1.

(2)If{Pn}n∈N⊂L(H),forN at most countable satisfiesPk(H)⊥Ph(H)

whenh=kforh, k∈N,then

ρ(∨k∈NPk)=

∑

k∈N

ρ(Pk). (2.77)

The set of the states inHwill be denoted byS(H).

Remark 2.3.22.
(a)The conditionPk(H)⊥Ph(H)is obviously equivalent toPkPh=0.Since
(taking the adjoint) we also obtainPhPk=0=PkPh, we conclude that we are
dealing with pairwise compatible elementary observables. Therefore Proposition
2.3.11 permits us to equivalently re-write theσ-additivity (2) as follows.
(2) If{Pn}n∈N ⊂L(H),forN at most countable satisfiesPkPh=0when
h=kforh, k∈N,then

ρ

(

∑

k∈N

Pk

)

=

∑

k∈N

ρ(Pk), (2.78)