From Classical Mechanics to Quantum Field Theory

110 From Classical Mechanics to Quantum Field Theory. A Tutorial

projectorPEsuch that, ifχEis thecharacteristic function ofE–χE(x)=0
ifx∈EandχE(x)=1ifx∈E–

(PEψ)(x):=χE(x)ψ(x) ∀ψ∈L^2 (R,dx).

MoreoverP∅:= 0. It is easy to prove that the collection of thePEis a PVM. In
particularμfg(E)=〈f,PEg〉=

∫

Ef(x)

∗g(x)dxandμff(E)=∫

E|f(x)|

(^2) dx.
We have the following fundamental result[8; 5; 6; 9].
Proposition 2.2.50.LetHbe a complex Hilbert space andP:Σ(X)→L(H)a
PVM. Iff:X→Cis measurable, define
Δf:=

{

x∈H

∣∣

∣∣

∫

X

|f(λ)|^2 μ(xxP)(λ)<+∞

}

.

Δfis a dense subspace ofHand there is a unique operator
∫

X

f(λ)dP(λ):Δf→H (2.47)

such that
〈
x,

∫

X

f(λ)dP(λ)y

〉

=

∫

X

f(λ)μ(xyP)(λ) ∀x∈H,∀y∈Δf (2.48)

The operator in (2.47) turns out to be closed and normal. It finally satisfies
(∫

X

f(λ)dP(λ)

)†

=

∫

X

f(λ)dP(λ) (2.49)

and
∣∣
∣∣

∣∣

∣∣

∫

X

f(λ)dP(λ)x

∣∣

∣∣

∣∣

∣∣

2

=

∫

X

|f(λ)|^2 dμ(xxP)(λ) ∀x∈Δf. (2.50)

Idea of the existence part of the proof. The idea of the proof of existence
of the operator in (2.47) relies upon the validity of the inequality ((1) in exercises
2.2.52 below)

∫

X

|f(λ)|d|μ(xyP)|(λ)≤||x||

√∫

X

|f(λ)|^2 dμ(yyP)(λ) ∀y∈Δf,∀x∈H. (2.51)

This inequality also proves thatf∈L^2 (X,dμ(yyP)) impliesf∈L^1 (X,d|μ(xyP)|)for
x∈H, so that (2.48) makes sense. Since from the general measure theory
∣∣
∣∣

∫

X

f(λ)dμ(xyP)(λ)

∣∣

∣∣≤

∫

X

|f(λ)|d|μ(xyP)|(λ),