From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 175

a common invariant subspaceDdense inHandV-invariant – with the usual
commutator of operators as Lie bracket.
Let −iS 1 ,···,−iSn∈V be a basis of V and define Nelson’s operator with
domainD:

Δ:=

∑n

k=1

Sk^2.

IfΔis essentially selfadjoint, there exists a strongly continuous unitary represen-
tation

GVg→Ug

onH, of the unique simply connected Lie groupGV with Lie algebraV.
Uis completely determined by the fact that the closuresS, for every−iS∈V,
are the selfadjoint generators of the representation of the one-parameter subgroups
ofGV in the sense of Def. 2.3.71.
In particular, the symmetric operatorsSare essentially selfadjoint onD,their
closure being selfadjoint.

Exercise 2.3.76.LetA, Bbe selfadjoint operators in the complex Hilbert spaceH
with a common invariant dense domainDwhere they are symmetric and commute.
Prove that ifA^2 +B^2 is essentially selfadjoint onD, then the spectral measures
ofAandBcommute.

Solution.Apply Nelson’s theorem observing thatA, Bdefine the Lie algebra
of the additive Abelian Lie groupR^2 and thatDis a core forAandB, because
they are essentially selfadjoint therein again by Nelson theorem.

Example 2.3.77.
(1)Exploiting spherical polar coordinates, the Hilbert spaceL^2 (R^3 ,d^3 x)can

be factorised asL^2 ([0,+∞),r^2 dr)⊗L^2 (S

2
,dΩ), wheredΩ is the natural rota-
tionally invariant Borel measure on the sphere∫ S^2 with unit radius inR^3 , with

S^21 dΩ=4π. In particular, a Hilbertian basis of L

(^2) (R (^3) ,d (^3) x) is therefore
made of the productsψn(r)Yml(θ,φ)where{ψn}n∈Nis any Hilbertian basis in
L^2 ([0,+∞),r^2 dr)and{Yml|l=0, 1 , 2 ,... ,m=0,± 1 ,± 2 ,...±l}is the standard
Hilbertian basis ofspherical harmonicsofL^2 (S^2 ,dΩ)[ 24 ]. Since the functionYml
are smooth onS^2 , it is possible to arrange the basis ofψnmade of compactly
supported smooth functions whose derivatives in 0 vanish at every order, in or-
der thatR^3 x→(ψn·Yml)(x) are elements ofC∞(Rn;C) (and therefore also
ofS(R^3 )). Now consider the three symmetric operators defined on the common