In chapter 5, for finite dimensional unitary representations of a Lie group
Gwe found corresponding Lie algebra representations in terms of self-adjoint
matrices. For the case ofG=R, even for infinite dimensional representations
onH=L^2 (R^3 ) one gets an equivalence of unitary representations and self-
adjoint operators^2 , although now this is a non-trivial theorem in analysis, not
just a fact about matrices.
10.2 Translations in time and space
10.2.1 Energy and the groupRof time translations
We have seen that it is a basic axiom of quantum mechanics that the observ-
able operator responsible for infinitesimal time translations is the Hamiltonian
operatorH, a fact that is expressed as the Schr ̈odinger equation
i~d
dt|ψ〉=H|ψ〉WhenHis time-independent, this equation can be understood as reflecting the
existence of a unitary representation (U(t),H) of the groupRof time transla-
tions on the state spaceH.
WhenHis finite dimensional, the fact that a differentiable unitary repre-
sentationU(t) ofRonHis of the form
U(t) =e−
~itHforHa self-adjoint matrix follows from the same sort of argument as in theorem
2.3. Such aU(t) provides solutions of the Schr ̈odinger equation by
|ψ(t)〉=U(t)|ψ(0)〉The Lie algebra ofRis alsoRand we get a Lie algebra representation ofR
by taking the time derivative ofU(t), which gives us
~
d
dtU(t)|t=0=−iHBecause this Lie algebra representation comes from taking the derivative of a
unitary representation,−iHwill be skew-adjoint, soHwill be self-adjoint.
10.2.2 Momentum and the groupR^3 of space translations
Since we now want to describe quantum systems that depend not just on time,
but on space variablesq= (q 1 ,q 2 ,q 3 ), we will have an action by unitary trans-
formations of not just the groupRof time translations, but also the groupR^3
of spatial translations. We will define the corresponding Lie algebra representa-
tions using self-adjoint operatorsP 1 ,P 2 ,P 3 that play the same role for spatial
translations that the Hamiltonian plays for time translations:
(^2) For the case ofHinfinite dimensional, this is known as Stone’s theorem for one-parameter
unitary groups, see for instance chapter 10.2 of [41] for details.