From Classical Mechanics to Quantum Field Theory

118 From Classical Mechanics to Quantum Field Theory. A Tutorial

to take arbitrarily large values (think ofXorP), it cannot be represented by a
bounded selfadjoint operator just because its spectrum is not bounded.
(e)IfPis a PVM onRandf:R→Cis measurable, we can always write
∫

R

f(λ)dP(λ)=f(A)

where we have introduced the selfadjoint operatorAobtained as

A=

∫

R

id(λ)dP(λ), (2.62)

due to (2.49) and whereid:Rλ→λ.EvidentlyP(A)=Pdue to the uniqueness
part of the spectral theorem. This fact leads to the conclusion that,in a complex
Hilbert spaceH,allthePVMoverRare one-to-one associated to all selfadjoint
operators inH.
(f)An elementλ∈σc(A)is not an eigenvalue ofA. However there is the
following known result arising from (d) in Theorem 2.2.55[ 5 ]which proves that
we can have approximated eigenvalues with arbitrary precision: With the said hy-
potheses, for every> 0 there isx∈D(A)such that

||Ax−λx||<, but||x||=1.

(g)IfAis selfadjoint andUunitary,UAU†,withD(UAU†)=U(D(A)),is
selfadjoint as well (Proposition 2.2.30). It is very simple to prove that the PVM
ofUAU†is nothing butUP(A)U†.

The next theorem we state here concerns a general explicit form of the integral
decompositionf(A)=

∫

σ(A)f(λ)dP

(A)(λ). As a matter of fact, up to multiplicity,

one can always reduce to a multiplicative operator in aL^2 space, as it happens for
the position operatorX. Again, this theorem can be restated for generally normal
operators.

Theorem 2.2.62(Spectral representation theorem for selfadjoint operators).Let
Abe a selfadjoint operator in the complex Hilbert spaceH. The following facts
hold.

(a)Hmay be decomposed as a Hilbert sum^10 H=⊕a∈SHa, whose summands

Haare closed and orthogonal. Moreover:

(i)for anya∈S,

A(Ha∩D(A))⊂Ha

(^10) Sis countable, at most, ifHis separable.