Mathematical Foundations of Quantum Mechanics 171Solution. We look for an operator, unitary or antiunitary such thatTUt=
U−tT for allt∈R. If the operator is unitary, the said identity easily implies
THT−^1 =−Hand therefore, with obvious notation,σ(THT−^1 )=−σ(H). (e) in
remark 2.2.42 immediately yieldsσ(H)=−σ(H) which is false ifσ(H) is bounded
below but not above.
2.3.6.5 Strongly continuous unitary representations of
Lie groups, Nelson theorem
Topological and Lie groups are intensively used in QM[ 24 ]. More precisely they
are studied in terms of their strongly continuous unitary representations. The
reason to consider strongly continuous representations is that they immediately
induce continuous representations of the group in terms of quantum symmetries
(Def. 2.3.54). In the rest of the section, we consider only the case of areal Lie
group,G, whose Lie algebra is indicated bygendowed with the Lie bracket or
commutator{, }.
Definition 2.3.70.IfGis a Lie group, astrongly continuous unitary rep-
resentationofGover the complex Hilbert spaceHis a group homomorphism
Gg→Ug∈B(H)such that everyUgis unitary andUg→Ug 0 ,inthestrong
operator topology, ifg→g 0.
We leave to the reader the elementary proof that strong continuity is equivalent to
strong continuity at the unit element of the group and in turn, this is equivalent
to weak continuity at the unit element of the group.
A fundamental technical fact is that the said unitary representations are
associated with representations of the Lie algebra of the group in terms of
(anti)selfadjoint operators. These operators are often physically interpreted as
constants of motion (generally parametrically depending on time) when the Hamil-
tonian of the system belongs to the representation of the Lie algebra. We want to
study this relation between the representation of the group on the one hand and
the representation of the Lie algebra on the other hand. First of all we define the
said operators representing the Lie algebra.
Definition 2.3.71.LetGbe a real Lie group and consider a strongly continuous
unitary representationUofGover the complex Hilbert spaceH.
IfA∈gletRt→exp(tA)∈Gbe the generated one-parameter Lie subgroup.
Theselfadjoint generator associated withA
A:D(A)→H