From Classical Mechanics to Quantum Field Theory

112 From Classical Mechanics to Quantum Field Theory. A Tutorial

wherewehaveusedPE†n,kPEn,k′=PEn,kPEn,k′=δkk′PEn,ksinceEn,k∩En,k′=∅
fork=k′. Next observe that, as|szn|^2 →|sh−^1 |^2 =|s|^2 , dominate convergence
theorem leads to
∫

X

|s|d|μxy|≤||x||

√∫

X

|s|^2 dμyy.

Finally, replacesabove for a sequence of simple functions|sn|→f∈L^2 (X,dμyy)
pointwise, withsn≤|sn+1|≤|f|. Monotone convergence theorem and dominate
convergence theorem, respectively applied to the left and right-hand side of the
found inequality, produce inequality (2.51).

(2)Prove that, with the hypotheses of Proposition 2.2.50, it holds
∫

X

χE(λ)dP(λ)=PE, ifE∈Σ(X) (2.52)

and in particular ∫

X

1 dP(λ)=I. (2.53)

Solution.It is sufficient to prove (2.52) since we know thatPX=I.Tothis
end, notice that, by direct inspection

〈x, PEy〉=

∫

X

χE(λ)μxy(P)(λ) ∀x∈H, ∀y∈ΔχE=H.

By the uniqueness property stated in Proposition 2.2.50, (2.52) holds.

(3)Prove that ifP,aPVMonHandTis an operator inHwithD(T)=Δf
such that

〈x, T x〉=

∫

X

f(λ)μ(xxP)(λ) ∀x∈Δf (2.54)

then

T=

∫

X

f(λ)dP(λ).

Solution.From the definition ofμxy, we easily have (everywhere omitting(P)
for simplicity)

4 μxy(E)=μx+y,x+y(E)−μx−y,x−y(E)−iμx+iy,x+iy(E)+iμx−iy,x−iy(E)

This identity implies that ifx, y∈Δf,

4

∫

X

fdμxy=

∫

X

fdμx+y,x+y−

∫

X

fdμx−y,x−y

−i

∫

X

fdμx+iy.x+iy+i

∫

X

fdμx−iy,x−iy