From Classical Mechanics to Quantum Field Theory

120 From Classical Mechanics to Quantum Field Theory. A Tutorial

There is a uniquePVM,P(A^1 ×···×An),onRnsuch that

P(A^1 ×···×An)(E 1 ×···×En)=PE(A 11 )···PE(Ann), ∀E 1 ,...,En∈B(R).

For everyf:R→Cmeasurable, it holds
∫

Rn

f(xk)dP(A^1 ×···×An)(x)=f(Ak),k=1,...,n (2.63)

wherex=(x 1 ,...,xk,...,xn).

Definition 2.2.64. Referring to Theorem 2.2.63, the PVM P(A^1 ×···×An)
is called the joint spectral measure of A 1 ,A 2 ,...,An and its support
supp(P(A^1 ×···×An)), i.e. the complement inRnto the largest open setA with
PA=0, is called thejoint spectrumofA 1 ,A 2 ,...,An.

Example 2.2.65.The simplest example is provided by considering thenposi-
tion operatorsXminL^2 (Rn,dnx). It should be clear that thenspectral mea-
sures commute becausePE(Xk),forE∈B(R), is the multiplicative operator for
χR×···×R×E×R×···×Rthe factorEstaying in thek-th position among thenCarte-
sian factors. In this case, the joint spectrum of thenoperatorsXmcoincides with
Rnitself.
A completely analogous discussion holds for thenmomentum operatorsPk,
since they are related to the position ones by means of the unitary Fourier-
Plancherel operator as already seen several times. Again the joint spectrum of
thenoperatorsPmcoincides withRnitself.

2.2.7 Measurablefunctionalcalculus...............

The following proposition states some useful properties off(A), whereAis self-
adjoint andf:R→Cis Borel measurable. These properties define the so called
measurable functional calculus. We suppose here thatA=A†, but the statements
can be reformulated for normal operators[5; 6].

Proposition 2.2.66.LetAbe a selfadjoint operator in the complex Hilbert space
H,f,g:σ(A)→Cmeasurable functions,f·gandf+grespectively denote the
point-wise product and the point-wise sum of functions. The following facts hold.

(a)f(A)=

∑n

k=0akA

k where the right-hand side is defined in its standard

domainD(An)whenf(λ)=

∑n

k=0akλkwithan=0;

(b)f(A)=P(A)(E)iff=χEthe characteristic function ofE∈B(σ(A));

(c)f(A)†=f∗(A)where∗denotes the complex conjugation;

(d)f(A)+g(A)⊂(f+g)(A)andD(f(A)+g(A))⊂Δf∩Δg(the symbol

“⊂” can be replaced by “=”ifandonlyifΔf+g=Δf∩Δg);