From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 165

(a)γis equivalent to a strongly continuous unitary representationRr→Vr

of the same topological additive group onH.

(b)A strongly continuous unitary representationRr→Vr′is equivalent to

γif and only if

Vr′=eicrVr

for some constantc∈Rand allr∈R.

The above unitary representation can also be defined as astrongly continuous
one-parameter unitary group.

Definition 2.3.59.IfHis a Hilbert space,V:Rr→Vr∈B(H), such that:

(i)Vris unitary for everyr∈R;

(ii)VrVs=Vr+sfor allr, s∈R

is calledone-parameter unitary group.Itiscalledstrongly contin-

uous one-parameter unitary groupif in addition to (i) and (ii) we

also have

(iii)V is continuous referring to the strong operator topology. In other words

Vrψ→Vr 0 ψforr→r 0 and everyr 0 ∈Randψ∈H.

Remark 2.3.60.
(a)It is evident that, in view of the group structure, a one-parameter unitary
groupRr →Vr∈B(H)is strongly continuous if and only if it is strongly
continuous forr=0.
(b)It is a bit less evident but true that a one-parameter unitary groupR
r→Vr∈B(H)is strongly continuous if and only if it isweakly continuousat
r=0. Indeed, ifV is weakly continuous atr=0, for everyψ∈H, we have

||Urψ−ψ||^2 =||Urψ||^2 +||ψ||^2 −〈ψ,Urψ〉−〈Urψ,ψ〉

=2||ψ||^2 −〈ψ,Urψ〉−〈Urψ,ψ〉→ 0

forr→ 0.

2.3.6.3 One-parameter strongly continuous unitary groups:
von Neumann and Stone theorems

Theorem 2.3.58 establishes that, dealing with continuous unitary projective repre-
sentation of the additive topological groupR, one can always reduce to work with
proper strongly continuous one-parameter unitary groups. So, for instance, the
action on a quantum system of rotations around an axis can always described by
means of strongly continuous one-parameter unitary groups. There is a couple of