From Classical Mechanics to Quantum Field Theory

124 From Classical Mechanics to Quantum Field Theory. A Tutorial

ifAandBare incompatible, just because, in general, nothing exists which can be

interpreted as the event “EAANDFB”ifPE(A)andPF(B)do not commute! The
correct formula is

μψ(EA|FB)=

〈ψ,PF(B)PE(A)PF(B)ψ〉

||PF(B)ψ||^2

which leads to well known different properties with respect to the classical theory,
the so called combination of “probability amplitudes” in particular. As a matter
of fact, up to now we do not have a clear notion of (quantum) probability. This
issue will be clarified in the next section.
(b)The reason to pass from operators to their spectral measures in defining
compatible observables is that, ifAandBare selfadjoint and defined on different
domains,AB=BAdoes not make sense in general. Moreover it is possible to
find counterexamples (due to Nelson) where commutativity ofAandBon common
dense invariant subspaces does not imply that their spectral measures commute.
However, from general results again due to Nelson, one has the following nice
result (see exercise 2.3.76).

Proposition 2.2.69.If selfadjoint operators,AandB, in a complex Hilbert space
Hcommute on a common dense invariant domainDwhereA^2 +B^2 is essentially
selfadjoint, then the spectral measures ofAandBcommute.

The following result, much easier to prove, is also true[5; 6].

Proposition 2.2.70.LetAandBbe selfadjoint operators in the complex Hilbert
spaceH.IfB∈B(H)the following facts are equivalent,

(i)the spectral measures ofAandBcommute (i.e. (2.69) holds);

(ii)BA⊂AB;

(iii)Bf(A)⊂f(A)B,iff:σ(A)→Ris Borel measurable;

(iv)PE(A)B=BPE(A)ifE∈B(σ(A)).

Another useful result toward the converse direction[5; 6]is the following.

Proposition 2.2.71.LetAandBbe selfadjoint operators in the complex Hilbert
spaceHsuch that their spectral measures commute. The following facts hold.

(a)ABx=BAxifx∈D(AB)∩D(BA);

(b)〈Ax, By〉=〈Bx,Ay〉ifx, y∈D(A)∩D(B).