From Classical Mechanics to Quantum Field Theory

134 From Classical Mechanics to Quantum Field Theory. A Tutorial

We are in a position to clarify why, in this context, observables are PVMs. Exactly
as in CM, an observableAis a collection of elementary observables{PE}E∈B(R)
labelled on the Borel setsEofR. Exactly as for classical quantities, (2.73) we can
say that the meaning ofPEis

PE= “the value of the observable belongs toE”. (2.74)

We expect that all those elementary observables are pairwise compatible and that
they satisfy the same properties (Fi)-(Fiii) as for classical quantities. We can com-
plete{PE}E∈B(R)to a maximal set of compatible elementary observables. Taking
Proposition 2.3.11 into account (Fi)-(Fiii) translate into

(i)PR=I;

(ii)PEPF=PE∩F;

(iii) IfN⊂Nand{Ek}k∈N⊂B(R) satisfiesEj∩Ek=∅fork=j,then

∑

j∈N

PEjx=P∪j∈NEjx for everyx∈H.

(The presence ofxis due to the fact that the convergence of the series ifN
is infinite is in the strong operator topology as declared in the last statement of
Proposition 2.3.9.)In other words we have just found Definition 2.2.47, specialized
to PVM onR: Observables in QM are PVM overR!
We know that all PVM over Rare one-to-one associated to all selfadjoint
operators in view of the results presented in the previous section (see (e) in remark
2.2.60). We conclude that, adopting von Neumann’s framework, in QM observables
are naturally described by selfadjoint operators, whose spectra coincide with the
set of values attained by the observables.

2.3.3 Recovering the Hilbert spacestructure

A reasonable question to ask is whether there are better reasons for choosing to
describe quantum systems via a lattice of orthogonal projectors, other than the
kill-off argument “it works”. To tackle the problem we start by listing special
properties of the lattice of orthogonal projectors, whose proof is elementary.

Theorem 2.3.13.The bounded, orthocomplemented,σ-complete latticeL(H)of
Propositions 2.3.9 and 2.3.11 satisfies these additional properties:

(i)separability(forHseparable): if{Pa}a∈A⊂L(H)satisfiesPiPj=0,

i=j,thenAis at most countable;

(ii)atomicity and atomisticity:thereexistelementsinA∈L(H)\{ 0 },

called atoms,forwhich 0 ≤P ≤AimpliesP =0orP =A; for any