From Classical Mechanics to Quantum Field Theory

178 From Classical Mechanics to Quantum Field Theory. A Tutorial

we have

−iA(t)=

d

ds

∣∣

∣∣

s=0

Vexp(tH)Aexp(−tH)=

d

ds

∣∣

∣∣

s=0

Vexp(tH)exp(sA)exp(−tH)

=

d

ds

∣∣

∣∣

s=0

Vexp(tH)Vexp(sA)Vexp(−tH)=−iUtAUt−^1

Therefore

At=Ut−^1 A(t)Ut=Ut−^1 UtAUt−^1 Ut=A=A 0.

In view of Theorem 2.3.73, as the mapgA→A|D(V)

G

is a Lie algebra

isomorphism, we can recast (2.113) for selfadjoint generators

A(t)|D(GV)=

∑n

k=1

ak(t)Ak|D(GV) (2.114)

(whereD(GV)may be replaced byDN(V)as the reader can easily establish,

taking advantage of Proposition 2.3.74 and Theorem 2.3.75). SinceDG(V)

(resp.D(NV))isacoreforA(t), it also holds

A(t)=

∑n

k=1

ak(t)Ak|D(GV), (2.115)

the bar denoting the closure of an operator as usual. (The same is true

replacingD(GV)forD(NV).) An important case, both for the non-relativistic

and the relativistic case is the selfadjoint generatorKn(t) associated with

the boost transformation along the unit vectorn∈R^3 , the rest space

of the inertial reference frame wherethe boost transformation is viewed

as an active transformation. In fact, referring to the Lie generators of

(aU(1) central extension of the universal covering of the connected or-

thochronous) Galilean group, we have{h, kn}=−pn=0,wherepnis the

generator of spatial translations alongn, corresponding to the observable

momentum along the same axis when passing to selfadjoint generators.

The non-relativistic expression ofKn(t), for a single particle, appears in

(2.103). For a more extended discussion on the non-relativistic case see[ 5 ].

A pretty complete discussion including the relativistic case is contained

in[ 24 ].

2.3.6.6 Selfadjoint version of Stone-von Neumann-Mackey theorem

A remarkable consequence of Nelson’s theorem is a selfadjoint operator version
of Stone-von Neumann theorem usually formulated in terms of unitary operators