Mathematical Foundations of Quantum Mechanics 161The meaning ofs∗(A) should be evident: the probability that the observable
s∗(A) produces the outcomeEwhen the state iss(ρ)(namelytr(P(s
∗(A))
E s(ρ)))
is the same as the probability that the observableAproduces the outcomeE
when the state isρ(that istr(PE(A)ρ)). Changing simultaneously and coherently
observables and states nothing changes. Indeed
tr(P(s∗(A))
E s(ρ)) =tr(VP(A)
E V− (^1) VρV− (^1) )=tr(VP(A)
E ρV
− (^1) )
=tr(PE(A)ρV−^1 V)=tr(PE(A)ρ).
Example 2.3.50.
(1)Fixing an inertial reference frame, the pure state of a quantum particle is
defined, up to phases, as a unit norm elementψofL^2 (R^3 ,d^3 x), whereR^3 stands
for the rest three space of the reference frame. The group of isometriesIO(3)
ofR^3 equipped with the standard Euclidean structure acts on states by means of
symmetries the sense of Wigner and Kadison. If (R, t):x→Rx+tis the action of
the generic element ofIO(3), whereR∈O(3) andt∈R^3 , the associated quantum
(Wigner) symmetrys(R,t)(〈ψ,〉ψ)=〈U(R,t)ψ,〉U(R,t)ψis completely fixed by the
unitary operatorsU(R,t). They are defined as
(U(R,t)ψ)(x):=ψ((R, t)−^1 x),x∈R^3 ,ψ∈L^2 (R^3 ,d^3 x), ||ψ||=1.
The fact that the Lebesgue measure is invariant underIO(3) immediately proves
thatU(R,t)is unitary. It is furthermore easy to prove that, with the given definition
U(I,0)=I, U(R,t)U(R′,t′)=U(R,t)◦(R′,t′), ∀(R, t),(R′,t′)∈IO(3). (2.91)
(2)The so-calledtime reversaltransformation classically corresponds to invert the
sign of all the velocities of the physical system. It is possible to prove[ 5 ](see also
(3) in exercise 2.3.69 below) that, in QM and for systems whose energy is bounded
below but not above, the time reversal symmetry cannot be represented by unitary
transformations, but only antiunitary. In the most elementary situation as in (1),
the time reversal is defined by means of the antiunitary operator
(Tψ)(x):=ψ(x),x∈R^3 ,ψ∈L^2 (R^3 ,d^3 x), ||ψ||=1.
(3)According to the example in (1), let us focus on the subgroup ofIO(3) of
displacements alongx 1 parametrized byu∈R,
R^3 x→x+ue 1 ,
wheree 1 denotes the unit vector inR^3 alongx 1. For every value of the parameter
u, we indicate bysuthe corresponding (Wigner) quantum symmetry,su(〈ψ,〉ψ)=
〈Uuψ, 〉Uuψwith
(Uuψ)(x)=ψ(x−ue 1 ),u∈R,