176 From Classical Mechanics to Quantum Field Theory. A Tutorial
dense invariant domainS(R^3 )
Lk=∑^3
i,j=1(^) kijXiPj|S(R^3 )
where (^) ijkis completely antisymmetric inijkand 123 = 1. By direct inspection,
one sees that
[−iLk,−iLh]=
∑^3
r=1(^) khr(−iLr)
so that the finite real span of the operatorsiLkis a representation of the Lie
algebra of the simply connected real Lie groupSU(2) (the universal covering of
SO(3)). Define the Nelson operatorL^2 :=−
∑ 3
k=1L^2 konS(R^3 ). Obviously this
is a symmetric operator. A well-known computation proves that
L^2 ψn(r)Yml=l(l+1)ψn(r)Yml.We conclude thatL^2 admits a Hilbertian basis of eigenvectors. Corollary 2.2.38
implies thatL^2 is essentially selfadjoint. Therefore we can apply Theorem 2.3.75
concluding that there exists a strongly continuous unitary representationSU(2)
M→UMofSU(2) (actually it can be proven to be also ofSO(3)). The three
selfadjoint operatorsLk:=Lkare the generators of the one-parameter of rota-
tions around the corresponding three orthogonal Cartesian axesxk,k=1, 2 ,3.
The one-parameter subgroup of rotations around the generic unit vectorn, with
componentsnk, admits the selfadjoint generatorLn=
∑ 3
k=1nkLk.Theobserv-
ableLnhas the physical meaning of then-component of the angular momentum
of the particle described inL^2 (R^3 ,d^3 x). It turns out that, forψ∈L^2 (R^3 ,d^3 x),
(UMψ)(x)=ψ(π(M)−^1 x),M∈SU(2),x∈R^3 (2.112)whereπ:SU(2)→SO(3) is the standard covering map. (2.112) is the action of the
rotation group on pure states in terms of quantum symmetries. This representation
is, in fact, a subrepresentation of the unitary representation ofIO(3) already found
in (1) of example 2.3.50.
(2)Given a quantum system, a quite general situation is the one where the quan-
tum symmetries of the systems are described by a strongly continuous representa-
tionV:Gg→Vgon the Hilbert spaceHof the system, and the time evolution
is the representation of a one-parameter Lie subgroup with generatorH∈g.So
that
Vexp(tH)=e−itH=:Ut.