From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 159

2.3.6.1 Wigner and Kadison theorems, groups of symmetries

Consider a quantum system described in the complex Hilbert spaceH. We assume
that eitherHis the whole Hilbert space in the absence of superselection charges
or it denotes a single coherent sector. LetS(H)andSp(H) respectively indicate
the convex body of the quantum states and the set of pure states, referred to the
sectorHif it is the case.

Definition 2.3.46.IfHis a complex Hilbert space, we have the following defini-
tions.

(a)AWigner symmetryis a bijective map

sW:Sp(H)〈ψ,〉ψ→〈ψ′, 〉ψ′∈Sp(H)

which preserves the probabilties of transition. In other words

|〈ψ 1 ,ψ 2 〉|^2 =|〈ψ′ 1 ,ψ 2 ′〉|^2 if ψ 1 ,ψ 2 ∈Hwith||ψ 1 ||=||ψ 2 ||=1.

(b)AKadison symmetryis a bijective map

sK:S(H)ρ→ρ′∈S(H)

which preserves the convex structure of the space of the states. In other

words

(pρ 1 +qρ 2 )′=pρ′ 1 +qρ′ 2 if ρ 1 ,ρ 2 ∈S(H) and p, q≥ 0 withp+q=1.

We observe that the first definition is well-posed even if unit vectors define pure
states just up to a phase, as the reader can immediately prove, because transition
probabilities are not affected by that ambiguity.
Though the definitions are evidently of different nature, they lead to the same
mathematical object, as established in a pair of famous characterization theorems
we quote into a unique statement. We need a preliminary definition.

Definition 2.3.47.LetHbe a complex Hilbert space. A mapU:H→His said
to be anantiunitary operatorif it is surjective, isometric andU(ax+by)=
aUx+bUywhenx, y∈Handa, b∈C.

We come to the celebrated theorem. The last statement is obvious, the difficult
parts are (a) and (b) (see, e.g.,[ 5 ]).

Theorem 2.3.48(Wigner and Kadison theorems).LetH={ 0 }be a complex
Hilbert space. The following facts hold.