From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 121

(e)f(A)f(B)⊂(f·g)(A)andD(f(A)f(B)) = Δf·g∩Δg

(the symbol “⊂” can be replaced by “=”ifandonlyifΔf·g⊂Δg);

(f)f(A)†f(A)=|f|^2 (A)so thatD(f(A)†f(A)) = Δ|f| 2 ;

(g)〈x, f(A)x〉≥ 0 forx∈Δfiff≥ 0 ;

(h)||f(A)x||^2 =

∫

σ(A)|f(λ)|

(^2) dμxx(λ),ifx∈Δf,inparticular,iffis bounded
orP(A)-essentially bounded^11 onσ(A),f(A)∈B(H)and
||f(A)||≤||f||P
(A)
∞ ≤||f||∞.
(i)IfU :H→His unitary, Uf(A)U†=f(UAU†)and, in particular,
D(f(UAU†)) =UD(f(A)) =U(Δf).
(j)Ifφ:R→Ris measurable, thenB(R)E→P′(E):=P(A)(φ−^1 (E))is
aPVMonR. Introducing the selfadjoint operator
A′=

∫

R

λ′dP′(λ′)

such thatP(A

′)

=P′, we have

A′=φ(A).

Moreover, iff:R→Cis measurable,

f(A′)=(f◦φ)(A) and Δ′f=Δf◦φ.

2.2.8 Elementary formalism for the infinite dimensional case

To complete the discussion in the introduction, let us show how practically the
physical hypotheses on quantum systems (1)-(3) have to be mathematically in-
terpreted (again reversing the order of (2) and (3) for our convenience) in the
general case of infinite dimensional Hilbert spaces. Our general assumptions on
the mathematical description of quantum systems are the following ones.

(1) A quantum mechanical systemSis always associated to complex Hilbert

spaceH, finite or infinite dimensional;

(2) observables are pictured in terms of (generally unbounded)selfadjoint

operatorsAinH;

(3) states are of equivalence classes ofunitvectorsψ∈H,whereψ∼ψ′iff

ψ=eiaψ′for somea∈R.

(^11) Remark 2.2.51.