The uncertainty principle for operators determined by Lie groups

2 Pages·0.08 MB·English

by  Jens Gerlach Christensen

THE UNCERTAINTY PRINCIPLE FOR OPERATORS
DETERMINED BY LIE GROUPS
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0
0
2 JENSGERLACHCHRISTENSEN

n
a Abstract. ForunboundedoperatorsA,BandCingeneral,thecommutation
J relation [A,B] = C does not lead to the uncertainty relation kAukkBuk ≥
5 12|hCu,ui|.IfA,BandCarepartofthegeneratorsofaunitaryrepresentation
2 ofaLiegroupthentheuncertaintyprincipleaboveholds.

]
G
If A and B are self-adjoint or skew-adjoint operators then
D
1
. (1) kAukkBuk≥ |h[A,B]u,ui| for all u∈D([A,B])
h
2
t
a
Theoperator[A,B]neednotbe closed,andifC =[A,B],the extendedinequal-
m
ity
[
1
(2) kAxkkBxk≥ |hCx,xi| for all x∈D(A)∩D(B)∩D(C).
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2
v
2 need not be valid. K. Kraus [4] showed that (2) holds when A,B and C are
0 infinitesimal generators of a unitary representation of a Lie group of dimension
0 ≤3. The resultof K.Krauswas slightlyextended by G.B.Follandand A.Sitaram
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in[1,Theorem2.4],butthenecessityofthedimensionconstraintwasleftasanopen
0
problem. Inthepresentpaperthedimensionconstraintisshowntobeunnecessary,
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0 by means of some results by I.E. Segal in [2].
/ If (H,π) is a unitary representation of a Lie group G and R is a right-invariant
h
vector field, then by Stone’s theorem one can define the closed skew-adjoint (by
t
a Lemma3.1.13in[2])operatorπ(R)suchthatπ(exptR)=exp(π(tR))forallt∈R.
m
Then [π(X),π(Y)]⊆π([X,Y]).
:
v Theorem. Let G be a Lie group with Lie algebra g, and let (H,π) be a unitary
i
X representation of G. Suppose that X,Y ∈g. Then
r 1
a kAxkkBxk≥ |hCx,xi| for all x∈D(A)∩D(B)∩D(C)
2
holds with A=π(X),B =π(Y) and C =π([X,Y]).
Lemma. Let σ be the modular function on G. Given a right-invariant vector
field R on G, let L be the corresponding left-invariant vector field, and let γ =
ddtσ(exp(tR))|t=0.
(a) If f ∈C∞(G) and f∗(g)=f(g−1)σ(g), then π(Rf∗)∗ =π(Lf)+γπ(f).
c
(b) There is a sequence {fn} in Cc∞(G) such that π(fn)u→u and π(Rfn)u+
π(Lfn)u+γπ(fn)u→0 for all u∈H.
2000 Mathematics Subject Classification. Primary22E45,47B15,47B25,81Q10.
Theauthor wishestothankHenrikSchlichtkrullforhisproofreadingandkindsupervision.
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2 JENSGERLACHCHRISTENSEN
Proof. (a) is simply Lemma 3.1.7 in [2] using the fact that f∗∗ =f. (b) is Lemma
3.1.8 and Lemma 3.1.9 in [2], and π(fn)u→u has been proved in [3, p.56]. (cid:3)
ToprovethetheoremIwillfirstshowthatforR∈githoldsthatπ(R)π(fn)u→
π(R)uforu∈D(π(R)). FixR∈gandletLbetheleft-invariantvectorfieldcorre-
sponding to R. Note that for all f ∈C∞(G) it holds that π(f)π(R)⊆−π(Rf∗)∗,
c
since hπ(f)π(R)u,vi=hu,−π(Rf∗)vi for all u∈D(π(R)) and v ∈H. Hence
π(f)π(R)⊆−π(Lf)−γπ(f)
by (a) of the lemma. But then
π(R)π(fn)u−π(fn)π(R)u=π(Rfn)u+π(Lfn)u+γπ(fn)u
foranyu∈D(π(R)). By (b) inthe lemma this tends to0 andsinceπ(fn)π(R)u→
π(R)u it follows that π(R)π(fn)u→π(R)u.
Now let A,B and C be as in the theorem. For any u ∈ D(A)∩D(B)∩D(C)
it follows that π(fn)u → u, Aπ(fn)u → Au, Bπ(fn)u → Bu and Cπ(fn)u → Cu.
Sincethe G˚ardingvectorπ(fn)uisinD([A,B])by[3,p.56]the inequality(1)gives
(2) as desired.
References
[1] Gerald B. Folland and Alladi Sitaram, The Uncertainty Principle: A Mathematical Survey,
TheJournalofFourierAnalysisandApplications,Vol.3,Number3,207–238,1997
[2] I.E.Segal,Aclassofoperatoralgebraswhicharedeterminedbygroups,DukeMath.Journal,
Vol.18,221–265,1951
[3] Anthony W. Knapp, Representation Theory of Semisimple Groups, Princeton University
Press,Princeton,NewJersey,1986
[4] K.Kraus, A Further Remark on Uncertainty Relations, Zeitschriftfu¨rPhysik,201, 134–141,
1967
MatematiskInstitut, KøbenhavnsUniversitet
Current address: Universitetsparken5,2100KøbenhavnØ,Denmark
E-mail address: [email protected]
URL:http://www.math.ku.dk/~vepjan