From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 167

One parameter unitary group generated by selfadjoint operators can be used
to check if the associated observables are compatible in view of the following nice
result[5; 6].

Proposition 2.3.64.IfAandBare selfadjoint operators in the complex Hilbert
spaceH, the identity holds

e−itAe−isB=e−isBe−itA ∀t, s∈R

if and only if the spectral measures ofAandBcommute.

2.3.6.4 Time evolution, Heisenberg picture and quantum Noether theorem

Consider a quantum system described in the Hilbert spaceHwhen an inertial
reference frame is fixed. Suppose that, physically speaking, the system is either
isolated or interacts with some external stationary environment. With these hy-
potheses, the time evolution of states is axiomatically described by a continuous
symmetry, more precisely, by a continuous one-parameter group of unitary pro-
jective operatorsRt→Vt. In view of Theorems 2.3.58 and 2.3.62, this group
is equivalent to a strongly continuous one-parameter group of unitary operators
Rt→Utand, up to additive constant, there is a unique selfadjoint operator
H, called theHamiltonian operatorsuch that (notice the sign in front of the
exponent)

Ut=e−itH,t∈R. (2.100)

Ut is calledevolution operatorThe observable represented by His usually
identified withthe energy of the systemin the considered reference frame.
Within this picture, ifρ∈S(H) is the state of the system att=0,asusual
described by a positive trace-class operator with unit trace, the state at timetis
ρt=UtρUt−^1. If the initial state is pure and represented by the unit vectorψ∈H,
the state at timetisψt:=Utψ.Inthiscase,ifψ∈D(H)wehavethatψt∈D(H)
for everyt∈Rin view of (c) in Theorem 2.3.62 and furthermore, for (b) of the
same theorem

−iHψt=dψt

dt

. (2.101)

where the derivative is computed wit respect to the topology ofH. One recognises
in Eq. (2.101) the general form ofSchr ̈odinger equation.

Remark 2.3.65. It is possible to study quantum systems interacting with some
external system which is not stationary. In this case, the Hamiltonian observable
depends parametrically on time as already introduced in remark 2.1.8. In these
cases, a Schr ̈odinger equation is assumed to describe the time evolution of the