From Classical Mechanics to Quantum Field Theory

174 From Classical Mechanics to Quantum Field Theory. A Tutorial

groupGadmittinggas Lie algebra. This is a much more difficult problem solved
by Nelson.
Given a strongly continuous representationUof a (real) Lie groupG,thereis

another spaceD(NU)with similar features toDG(U). Introduced by Nelson[ 24 ],it
turns out to be more useful than the G ̊arding space torecoverthe representation
Uby exponentiating the Lie algebra representation.

By definitionD(NU)consists of vectorsψ∈Hsuch thatGg→Ugψisana-
lyticing, i.e. expansible in power series in (real) analytic coordinates around every
point ofG. The elements ofD(NU)are calledanalytic vectors of the represen-

tationUandDN(U)is thespace of analytic vectors of the representation
U. It turns out thatD(NU)is invariant for everyUg,g∈G.

A remarkable relationship exists between analytic vectors inDN(U)and analytic
vectors according to Definition 2.2.36. Nelson proved the following important re-
sult[ 24 ], which implies thatD(NU)is dense inH, as we said, because analytic vectors
for a selfadjoint operator are dense (exercise 2.2.73). An operator is introduced,
calledNelson operator, that sometimes has to do with theCasimir operators[ 24 ]
of the represented group.

Proposition 2.3.74. LetGbe a (finite dimensional real) Lie group andG
g→Uga strongly continuous unitary representation on the Hilbert spaceH.Take
A 1 ,...,An∈ga basis and defineNelson’s operatoronD(GU)by

Δ:=

∑n

k=1

U(Ak)^2 ,

where the U(Ak)are, as before, the selfadjoint generatorsAk restricted to the

G ̊arding domainD(GU).Then:

(a)Δis essentially selfadjoint onD(GU).

(b)Every analytic vector of the selfadjoint operatorΔis analytic an element

ofDN(U),inparticularD(NU)is dense.

(c)Every vector inDN(U) is analytic for every selfadjoint operatorU(Ak),

which is thus essentially selfadjoint inDN(U)by Nelson’s criterion.

We finally state the well-known theorem of Nelson that enables to associate rep-
resentations of the only simply connected Lie group with a given Lie algebra to
representations of that Lie algebra.

Theorem 2.3.75(Nelson theorem). Consider a realn-dimensional Lie algebra
V of operators−iS–witheachSsymmetric on the Hilbert spaceH, defined on