From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 173

and, ifA∈g,XA:C∞ 0 (G;C)→C 0 ∞(G;C) is the smooth vector field overG(a
smooth differential operator) defined as:

(XA(f)) (g) := lim

t→ 0

f(exp{−tA}g)−f(g)

t

∀g∈G. (2.106)

so that that map

gA→XA (2.107)

defines a representation ofgin terms of vector fields (differential operators) on
C∞ 0 (G;C). We conclude with the following theorem[ 24 ], establishing that the
G ̊arding space has all the expected properties.

Theorem 2.3.73.Referring to Definitions 2.3.71 and 2.3.72, the G ̊arding space
D(GU)satisfies the following properties.

(a)D(GU)is dense inH.

(b)Ifg∈G,thenUg(D(GU))⊂D(GU).Moreprecisely,iff∈C 0 ∞(G),x∈H,

g∈G, it holds

Ugx[f]=x[Lgf]. (2.108)

(c)IfA∈g,thenDG(U)⊂D(A)and furthermoreA(D(GU))⊂D(GU).More

precisely

−iAx[f]=x[XA(f)]. (2.109)

(d)The map

gA→−iA|D(U)

G

=:U(A) (2.110)

is a Lie algebra representation in terms of antisymmetric operators defined

on the common dense invariant domainD(GU).Inparticularif{,}is the

Lie commutator ofgwe have:

[U(A),U(A′)] =U({A,A′}) ifA,A′∈g.

(e)D(GU)is a core for every selfadjoint generatorAwithA∈g, that is

A=A|D(U)

G

, ∀A∈g. (2.111)

Now we tackle the inverse problem: We suppose a certain representation of a Lie
algebragin terms of symmetric operators defined in common invariant domain
of a complex Hilbert spaceH. We are interested in lifting this representation to
a whole strongly continuous representation of the unique simply connected Lie