From Classical Mechanics to Quantum Field Theory

166 From Classical Mechanics to Quantum Field Theory. A Tutorial

technical results of very different nature which are very useful in QM. The former
is due to von Neumann (see, e.g.,[ 5 ]) and proves that the one-parameter unitary
group which are not strongly continuous are not so many in separable Hilbert
spaces.

Theorem 2.3.61. IfHis a separable complex Hilbert space andV :Rr→
Vr∈B(H)is a one parameter unitary group, it is strongly continuous if and only
if the mapsRr→〈ψ,Urφ〉are Borel measurable for allψ,φ∈H.

The second proposition we quote[5; 6]is a celebrated result due to Stone (and
later extend to the famousHille-Yoshida theoremin Banach spaces). We start
by noticing that, ifAis a selfadjoint operator in a Hilbert space,Ut:=eitA,for
t∈R, defines a strongly continuous one-parameter unitary group as one easily
proves using the functional calculus. The result is remarkably reversible.

Theorem 2.3.62(Stone theorem).LetRt→Ut∈B(H)be a strongly contin-
uous one-parameter unitary group in the complex Hilbert spaceH. The following
facts hold.

(a)There exists a unique selfadjoint operator, called the(selfadjoint) gen-

eratorof the group,A:D(A)→HinH, such that

Ut=e−itA,t∈R. (2.98)

(b)The generator is determined as

Aψ=ilimt→ 0

1

t

(Ut−I)ψ (2.99)

andD(A)is made of the vectorsψ∈Hsuch that the right hand side of

(2.99) exists inH.

(c)Ut(D(A))⊂D(A)for allt∈Rand

AUtψ=UtAψ ifψ∈D(A)andt∈R.

Remark 2.3.63.For a selfadjoint operatorA,theexpansion

e−itAψ=

+∑∞

n=0

(−it)n

n! A

nψ

generally doesnotwork forψ∈D(A). It works in two cases however: (i) ifψ
is ananalyticvector ofA(Def. 2.2.36 and this result is due to Nelson), (ii) if
A∈B(H)which is equivalent to say thatD(A)=H. In the latter case, one more
strongly findse−itA=

∑+∞

n=0

(−it)n

n! A

n, referring to the uniform operator topology.

[ 5 ].