From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 169

(a)Ais a constant of motion:At=A 0 for allt∈R.

(b)The one-parameter group of symmetries generated byA,Rs→e−isAis

agroup of dynamical symmetries: It commutes with time evolution

e−isAUt=Ute−isA for alls, t∈R. (2.102)

In particular transforms evolutions of pure states into evolutions of (other)

pure states, i.e.,e−isAUtψ=Ute−isAψ.

(c)The action on observables (2.90) of the one-parameter group of symmetries

generated byA,Rs→eisAleavesHinvariant. That is

e−isAHeisA=H, for alls∈R.

Proof. Suppose that (a) holds, by definitionUt−^1 AUt=A. By (i) in Proposition
2.2.66, we have thatUt−^1 e−isAUt=e−isA which is equivalent to (b). If (b) is
true, we have thate−isAe−itHeisA=e−itH. Here an almost direct application of
Stone theorem yieldse−isAHeisA=H. Finally suppose that (c) is valid. Again
(i) in Proposition 2.2.66 producese−isAUteisA=Utwhich can be rearranged into
Ut−^1 e−isAUt=e−isA. Finally Stone theorem leads toUt−^1 AUt=Awhich is (a),
concluding the proof.

Remark 2.3.68.
(a) In physics textbooks, the above statements are almost always stated us-
ing time derivatives and commutators. This is useless and involves many subtle
troubles with domains of the involved operators.
(b)The theorem can be extended to observablesA(t)parametrically depending
on time already in the Schr ̈odinger picture[ 5 ]. In this case (a) and (b) are equiv-
alent too. With this more general situation, (2.102) in (b) has to be re-written
as

e−isA(t)Ut=Ute−isA(0) for alls, t∈R

and Heisenberg evolution considered in (a) encompasses both time dependences

At=Ut−^1 A(t)Ut.

At this juncture, (c) can similarly be stated but, exactly as it happens in Hamilto-
nian classical mechanics, it has a more complicated interpretation[ 5 ].
An example is the generator of the boost one-parameter subgroup along the axis
nof transformations of the Galilean groupR^3 x→x+tvn∈R^3 , where the speed