From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 177

This is the case, for instance, of relativistic quantum particles, whereGis thespe-
cial orthochronous Lorentz group,SO(1,3)+, (or its universal coveringSL(2,C)).
Describing non-relativistic quantum particles, the relevant groupGis anU(1) cen-
tral extension of the universal covering of the (connected orthochronous) Galilean
group.
In this situation, every element ofgdetermines a constant of motion. Actually
there are two cases.

(i) IfA∈gand{H,A}= 0, then the Lie subgroups exp(tH) and exp(sA)

commute as, for example, follows fromBaker-Campbell-Hausdorffformula

(see[24; 5], for instance). ConsequentlyAis a constant of motion because

Vexp(tH)=e−itH andVexp(sA) =e−isA commute as well and Theorem

2.3.67 is valid. In this casee−isAdefines a dynamical symmetry in ac-

cordance with the aforementioned theorem. This picture applies in par-

ticular, referring to a free particle, toA=Jn, the observable describing

total angular momentum along the unit vectorncomputed in an inertial

reference frame.

(ii) A bit more complicated is the case ofA∈gwith{H,A}= 0. However,

even in this caseAdefines a constant of motion in terms of selfadjont

operators (observables) belonging to the representation of the Lie algebra

ofG. The difference with respect to the previous case is that, now, the

constant of motionparametrically depend on time. We therefore have a

class of observables{A(t)}t∈Rin the Schr ̈odinger picture, in accordance

with (b) in remark 2.3.69, such thatAt:=Ut−^1 A(t)Utare the correspond-

ing observables in the Heisenber picture. The equation stating that we

have a constant of motion is thereforeAt=A 0.

Exploiting the natural action of the Lie one-parameters subgroups on

g, let us define the time parametrized class of elements of the Lie algebra

A(t):=exp(tH)Aexp(−tH)∈g,t∈R.

If{Ak}k=1,...,nis a basis ofg,itmustconsequentlyhold

A(t)=

∑n

k=1

ak(t)Ak (2.113)

for some real-valued smooth functionsak=ak(t). By construction, the

corresponding class of selfadjoint generatorsA(t),t∈R, define a para-

metrically time dependent constant of motion. Indeed, since (exercise)

exp(sexp(tH)Aexp(−tH)) = exp(tH)exp(sA)exp(−tH),