172 From Classical Mechanics to Quantum Field Theory. A Tutorial
is the generator of the strongly continuous one-parameter unitary group
Rt→Uexp{tA}=e−isAin the sense of Theorem 2.3.62.
The expected result is that these generators (with a factor−i) define a representa-
tion of the Lie algebra of the group. The utmost reason is that they are associated
with the unitary one-parameter subgroups exactly as the elements of the Lie alge-
bra are associated to the Lie one-parameter subgroups. In particular, we expect
that the Lie parenthesis correspond to the commutator of operators. The technical
problem is that the generatorsAmay have different domains. Thus we look for
a common invariant (because the commutator must be defined thereon) domain,
where all them can be defined. This domain should embody all the amount of
information about the operatorsAthemselves, disregarding the fact that they are
defined in larger domains. In other words we would like that the domain be acore
((3) in Def. 2.2.20) for each generator. There are several candidates for this space,
one of the most appealing is the so calledG ̊arding space.
Definition 2.3.72.LetGbe a (finite dimensional real) Lie group and consider a
strongly continuous unitary representationUofGover the complex Hilbert space
H.Iff∈C 0 ∞(G;C)andx∈H, define
x[f]:=∫
Gf(g)Ugxdg (2.104)where dgdenotes the Haar measure overGand the integration is defined in a
weak sense exploiting Riesz’ lemma: Since the mapHx→
∫
Gf(g)〈y|Ugx〉dgis
continuous (the proof being elementary),x[f]is the unique vector inHsuch that
〈y|x[f]〉=∫
Gf(g)〈y,Ugx〉dg , ∀y∈H.The complex span of all vectorsx[f]∈Hwithf∈C 0 ∞(G;C)andx∈His called
G ̊arding spaceof the representation and is denoted byD(GU).
The subspaceDG(U)enjoys very remarkable properties we state in the next the-
orem. In the followingLg:C 0 ∞(G;C)→C 0 ∞(G;C) denotes the standard left-
action ofg∈Gon complex valued smooth compactly supported functions defined
onG:
(Lgf)(h):=f(g−^1 h) ∀h∈G, (2.105)