From Classical Mechanics to Quantum Field Theory

168 From Classical Mechanics to Quantum Field Theory. A Tutorial

system giving rise to a groupoid of unitary operators[ 5 ]. We shall not enter into
the details of this technical issue here.

Adopting the above discussed framework, observables do not evolve and states
do. This framework is calledSchr ̈odinger picture. There is however another
approach to describe time evolution calledHeisenberg picture.Inthisrepre-
sentation states do not evolve in time but observables do. IfAis an observable at
t= 0, its evolution at timetis the observable

At:=Ut−^1 AUt.

ObviouslyD(At)=Ut−^1 (D(A)) =U−t(D(A)) =Ut†(D(A)). According with (i) in
Proposition 2.2.66 the spectral measure ofAtis

PE(At)=Ut−^1 PE(A)Ut

as expected. The probability that, at timet, the observableAproduces the out-
comeEwhen the state isρatt= 0, can equivalently be computed both using the
standard picture, where states evolve astr(PE(A)ρt), or Heisenberg picture where
observables do obtainingtr(PE(At)ρ). Indeed

tr(PE(A)ρt)=tr(PE(A)Ut−^1 ρUt)=tr(UtPE(A)Ut−^1 ρ)=tr(PE(At)ρ).

The two pictures are completely equivalent to describe physics. Heisenberg picture
permits to give the following important definition

Definition 2.3.66.In the complex Hilbert spaceHequipped with a strongly con-
tinuous unitary one-parameter group representing the time evolutionRt→Ut,
an observable represented by the selfadjoint operatorAis said to be aconstant
of motionwith respect toU,ifAt=A 0.

The meaning of the definition should be clear: Even if the state evolve, the prob-
ability to obtain an outcomeE, measuring a constant of motionA, remains sta-
tionary. Also expectation values and standard deviations do not change in time.
We are now in a position to state the equivalent of theNoether theoremin QM.

Theorem 2.3.67(Noether quantum theorem). Consider a quantum system de-
scribed in the complex Hilbert spaceHequipped with a strongly continuous unitary
one-parameter group representing the time evolutionRt→Ut.IfAis an ob-
servable represented by a (generally unbounded) selfadjoint operatorAinH,the
following facts are equivalent.