From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 115

Notation 2.2.57.In view of the said theorem, and (b) in particular, iff:σ(A)→
Cis measurable (with respect to theσ-algebra obtained by restrictingB(R)to
σ(A)), we use the notation

f(A):=

∫

σ(A)

f(λ)dP(A)(λ):=

∫

R

g(λ)dP(A)(λ)=:g(A). (2.56)

whereg:R →Cis the extension off to the zero function outsideσ(A)or
any other measurable function which coincides withfonsupp(P(A))=σ(A).
Obviouslyg(A)=g′(A)ifg,g′:R→Ccoincide insupp(P(A))=σ(A).

Exercise 2.2.58.Prove that ifAis a selfadjoint operator in the complex Hilbert
spaceH, it holdsA≥ 0 – that is〈x|Ax〉≥ 0 for everyx∈D(A)–ifandonlyif
σ(A)⊂[0,+∞).

∫ Solution. Suppose thatσ(A)⊂[0,+∞). Ifx∈D(A)wehave〈x, Ax〉=
σ(A)λdμx,x≥0 in view of (2.48), the spectral decomposition theorem, sinceμx,x
is a positive measure andσ(A)∈[0,+∞). To conclude, we prove thatA≥ 0
is false ifσ(A) includes negative elements. To this end, assume that conversely,
σ(A)λ 0 <0. Using (c) and (d) of Theorem 2.2.55, one finds an interval
[a, b]⊂σ(A)with[a, b]⊂(−∞,0) andP[(a,bA)] = 0 (possiblya=b=λ 0 ). If

x∈P(a,bA) withx=0,itholdsμxx(E)=〈x|PEx〉=〈x, P[†a,b]PExP[a,b]〉=

〈∫x, P[a,b]PEP[a,b]x〉=〈x, P[a,b]∩Ex〉=0if[a, b]∩E=∅. Therefore,〈x, Ax〉=

σ(A)λdμx,x=

∫

[a,b]λdμx,x≤

∫

[a,b]bμx,x<b||x||

(^2) <0.
Example 2.2.59.
(1)Let us focus on them-axis position operatorXminL^2 (Rn,dnx) introduced in
(1) of example 2.2.39. We know thatσ(Xm)=σc(Xm)=Rfrom example 2.2.44.
We are interested in the PVMP(Xm)ofXmdefined onR=σ(Xm). Let us fix
m= 1, where the other cases are analogous. The PVM associated toX 1 is
(PE(X^1 )ψ)(x)=χE×Rn− 1 (x)ψ(x) ψ∈L^2 (Rn,dnx), (2.57)
whereE∈B(R) is identified with a subset of the first factor ofR×Rn−^1 =Rn.
Indeed, indicating byPon the right-hand side of (2.57), one easily verifies that
Δx 1 =D(X 1 )and^9
〈ψ|X 1 ψ〉=

∫

R

λμ(ψ,ψP)(λ) ∀ψ∈D(X 1 )=Δx 1

(^9) More generally∫R∫Rn− 1 g(x 1 )|ψ(x)| (^2) dxdn− (^1) x=∫Rg(x 1 )dμ(ψ,ψP)(x 1 ) is evidently valid for sim-
ple functions and then it extends to generic measurable functions when both sides make sense in
view of, for instance, Lebesgue’s dominate convergence theorem for positive measures.