From Classical Mechanics to Quantum Field Theory

150 From Classical Mechanics to Quantum Field Theory. A Tutorial

indicated byR, called thevon Neumann algebra of observables(though
only the selfadjoint elements are observables). Including non-selfadjoint elements
B∈Ris armless, as they can always be one-to-one decomposed into a pair of
selfadjoint elements

B=B 1 +iB 2 =

1

2

(B+B†)+i

1

2 i

(B−B†).

The fact that the elements ofRare bounded does not seem a physical problem.
IfA=A†is unbounded and represents an observable it does not belong toR.
Nevertheless the associatedclassof bounded selfadjoint operators{An}n∈Nwhere

An:=

∫

[−n,n]∩σ(A)

λdP(A)(λ),

embodies the same information asAitself.Anis bounded due to Proposition 2.2.61
because the support of its spectral measures is included in [−n, n]. Physically
speaking, we can say thatAnis nothing but the observableAwhen it is measured
with an instrument unable to produce outcomes larger than [−n, n]. All real
measurement instruments are similarly limited. We can safely assume that every
Anbelongs toR. Mathematically speaking, the whole (unbounded) observable
Ais recovered as the limit in thestrong operator topologyA= limn→+∞An((1)
in examples 2.2.72). Moreover the union of the spectral measures of all theAn
is that ofA. Finally the spectral measure ofAbelongs toRsince the spectral
measure of everyAn∈Rdoes, as has been established in the proof of Proposition
2.3.35 above.
Within this framework the orthogonal projectorsP∈Rrepresent all elemen-
tary observables of the system. The lattice of these projectors,LR(H), encompass
the amount of information about observables as established in Proposition 2.3.35.
As said aboveLR(H)⊂Ris bounded, orthocomplemented,σ-complete, ortho-
modular and separable like the wholeL(H) (assuming thatHis separable) but
there is no guarantee for the validity of the other properties listed in Theorem
2.3.13. The natural question is whetherRis∗-isomorphic toB(H 1 )forasuit-
able complex Hilbert spaceH 1 , which would automatically imply that also the
remaining properties were true. In particular there would exist atomic elements
inLR(H) and the covering property would besatisfied. A necessary condition is
that, exactly as it happens forB(H 1 ), there are no non-trivial elements inR∩R′,
sinceB(H 1 )∩B(H 1 )′=B(H 1 )′={cI}c∈C.

Definition 2.3.36. A von Neumann algebraRis afactorwhen itscenter,the
subsetR∩R′of elements commuting with the whole algebra, is trivial:R∩R′=
{cI}c∈C.