From Classical Mechanics to Quantum Field Theory

126 From Classical Mechanics to Quantum Field Theory. A Tutorial

as an immediate consequence of Lebesgue’s dominate convergence theorem and
the first part of (h) in Proposition 2.2.66.

(2)If in the previous examplefis bounded onσ(A), andfn→funiformly on

σ(A), (orP-essentially uniformly||f−fn||(P

(A))

∞ →0forn→+∞)then

fn(A)→f(A) uniformly, asn→+∞,

again for the (second part of (h) in Proposition 2.2.66.

Exercise 2.2.73. Prove that a selfadjoint operatorAin the complex HilbertH
admits a dense set of analytic vectors in its domain.

Solution. Consider the class of functionsfn=χ[−n,n]wheren∈N.Asin
(1) of example 2.2.72, we haveψn:=fn(A)ψ=

∫

[−n,n]^1 dP

(A)ψ→∫

R^1 dP

(A)ψ=

PR(A)ψ=ψforn→+∞. Therefore the setD:={ψn|ψ∈H,n∈N}is dense inH.
The elements ofDare analytic vectors forAas we go to prove. Clearlyψn∈D(Ak)

sinceμ(P

(A))

ψn,ψn(E)=μ

(P(A))

ψ,ψ (E∩[−n, n]) as immediate consequence of the definition

of the measure μ(x,yP), therefore

∫

R|λ

k| (^2) dμ(P(A))
ψn,ψn(λ)=

∫

[−n,n]|λ|

2 kdμ(P(A))
ψ,ψ (λ)≤
∫
[−n,n]|n|

2 kdμ(P(A))

ψ,ψ (λ) ≤|n|

2 k∫

Rdμ

(P(A))

ψ,ψ (λ)=|n|

2 k||ψ|| (^2) < +∞. Similarly
||Akψn||^2 =〈Akψn,Akψn〉=〈ψn,A^2 kψn〉=

∫

Rλ

2 kdμ(P(A))

ψn,ψn(λ)≤|n|

2 k||ψ|| (^2) .We
conclude that

∑+∞

k=0

(it)k

k! ||A

kψn||conveges for everyt∈Cas it is dominated by

the series

∑+∞

k=0

|t|k

k!|n|

2 k||ψ|| (^2) =e|t||n|^2 ||ψ|| (^2).

2.3 More Fundamental Quantum Structures

The question we want to answer now is the following:

Is there anything more fundamental behind the phenomenological facts (1), (2),
and (3) discussed in the first section and their formalization presented in Sect.
2.2.8?

An appealing attempt to answer that question and justify the formalism based
on the spectral theory is due to von Neumann[ 7 ](and subsequently extended
by Birkhoff and von Neumann). This section is devoted to quickly review an
elementary part of those ideas, adding however several more modern results (see
also[ 11 ]for a similar approach).

2.3.1 TheBooleanlogicofCM

Consider a classical Hamiltonian system described in symplectic manifold (Γ,ω),
where ω =

∑n

k=1dqk∧dpk in any system of local symplectic coordinates