160 From Classical Mechanics to Quantum Field Theory. A Tutorial
(a)For every Wigner symmetrysW there is an operatorU:H→H,which
can be either unitary or antiunitary, and this choice is fixed by sw if
dimH=∞, such that
sw:〈ψ,〉ψ→〈Uψ,〉Uψ, ∀〈ψ, 〉ψ∈Sp(H). (2.88)
U andU′are associated to the samesW if and only ifU′=eiaUfor
a∈R.
(b)For every Kadison symmetrysKthere is an operatorU:H→H,which
can be either unitary or anti unitary, and this choice is fixed byswif
dimH=∞, such that
sw:ρ→UρU−^1 , ∀ρ∈S(H). (2.89)
U andU′ are associated to the samesKif and only ifU′=eiaUfor
a∈R.(c)U : H→H, either unitary or antiunitary, simultaneously defines a
Wigner and a Kadison symmetry by means of (2.88) and (2.89) respec-
tively.Remark 2.3.49.
(a)It is worth stressing that the Kadison notion of symmetry is an extension of
the Wigner one, after the result above. In fact, a Kadison symmetryρ→UρU−^1
restricted to one dimensional projector preserves the probability transitions, as
immediately follows from the identity|〈ψ,φ〉|^2 =tr(ρψρφ)and the cyclic property
of the trace, where we use the notationρχ=〈χ, 〉χ. In particular we can use the
same operatorUto represent also the found Wigner symmetry.
(b)If superselection rules are present, in general, quantum symmetries are
described in a similar way with unitary or antiunitary operators acting in a single
coherent sector or also swapping different sectors[ 5 ].
If a unitary or antiunitary operatorV represents a symmetrys, it has an action
on observables, too. IfAis an observable (a selfadjoint operator onH), we define
thetransformed observablealong the action ofsas
s∗(A):=VAV−^1. (2.90)ObviouslyD(s∗(A)) =V(D(A)). It is evident that this definition is not affected
by the ambiguity of the arbitrary phase in the choice ofVwhensis given.
According with (i) in Proposition 2.2.66 the spectral measure ofs∗(A)is
P(s
∗(A))
E =VP(A)
E V− (^1) =s∗(P(A)
E )
as expected.