The Hamiltonian H=xp and classification of osp(1|2) representations

0.11 MB·English

by  G. Regniers

The Hamiltonian H = xp and classification of
osp(1 2) representations
|
G.RegniersandJ.VanderJeugt
DepartmentofAppliedMathematicsandComputerScience,GhentUniversity,
Krijgslaan281-S9,B-9000Gent,Belgium.
0
[email protected],[email protected]
1
0
Abstract
2
n Thequantizationofthesimpleone-dimensionalHamiltonianH = xp
a is of interest for its mathematical properties rather than for its physical
J relevance. In fact, the Berry-Keating conjecture speculates that a proper

8 quantizationofH = xpcouldyieldarelationwiththeRiemannhypoth-
esis. Motivated by this, we study the so-called Wigner quantization of

] H =xp,whichrelatestheproblemtorepresentationsoftheLiesuperalge-
h
braosp(1|2).Inordertoknowhowtherelevantoperatorsactinrepresenta-
p
tionspacesofosp(1|2),westudyallunitary,irreducible∗-representations
-
h ofthisLiesuperalgebra. SuchaclassificationhasalreadybeenmadebyJ.
t W.B.Hughes,butwereexaminethisclassificationusingelementaryargu-
a
m ments.
[

1 Introduction
1
v
5 ThesuggestionthatthezerosoftheRiemannzetafunctionmightberelatedto
8 thespectrumofaself-adjointoperatorH goesbacktoHilbertandPo´lyainthe
2
early20thcentury.ItwasnotuntiltheworksofSelberg[1]andMontgomery[2]
1
thatthis conjecturegainedmuch credibility. Due to papersby Connes[3] and
.
1 BerryandKeating[4,5]inthelate1990s,itappearsthattheHilbert-Po´lyacon-
0
jecturemightberelatedtotheclassicalone-dimensionalHamiltonianH = xp.
0
Moreprecisely,BerryandKeatingsuggestthatsomesortofquantizationofthis
1
v: aHraemthiletohneiiagnhtmsiogfhtthreesnuolnt-itnriavisapleRctireummancnonzseirsotisng1o+ftihte.vAalupersoptne,rwquhaenretizthaetiotnn
i 2 n
X revealingsuchacorrespondenceis,however,notknown.
r Theseinterestingobservationsstimulatedustoperformadifferentquantization
a
oftheHamiltonianH =xp.InWignerquantizationoneabandonsthecanonical
commutation relations and instead imposes compatibility between Hamilton’s
equations and the Heisenberg equations as operator equations. The result is a
setofcompatibilityconditionsthatareweakerthanthecanonicalcommutation
relations.ThiswasappliedforthefirsttimeinafamouspaperbyWigner[6].
Wigner’sapproachhasbeenappliedtomanydifferentHamiltonians,leadingto
various connectionswith Lie superalgebras[7–9]. In the present text, Wigner
quantizationwillleadtotheLiesuperalgebraosp(12). Sinceitisourinterestto
determinethespectrumoftheoperatorsHˆ andxˆ,o|neneedstheactionofthese
operatorsinrepresentationspacesofosp(12). Wepresentaclassificationofall
|
irreducible -representations of this Lie superalgebra, thus reconstructing and
∗
improvingsomeresultsbyHughes[10].
1
2 Wigner quantization of H = xp
The simplest Hermitian operator that correspondsto our Hamiltonian is given
by
1
Hˆ = (xˆpˆ+pˆxˆ). (1)
2
Withouttheassumptionofanycommutationrelationsbetweenthepositionand
momentumoperatorsxˆandpˆ,onecanstillcomputeHamilton’sequations
∂Hˆ ∂Hˆ
xˆ˙ = =xˆ, pˆ˙ = = pˆ
∂p −∂x −
andtheequationsofHeisenberg
i i
xˆ˙ = [Hˆ,xˆ], pˆ˙ = [Hˆ,pˆ]
~ ~
andimposethattheyareequivalent. Theresultingcompatibilityconditions(we
choose~=1)
[ xˆ,pˆ ,xˆ]= 2ixˆ, [ xˆ,pˆ ,pˆ]=2ipˆ (2)
{ } − { }
areweakerthantheusualcanonicalcommutationrelations[xˆ,pˆ] = i. Wewish
tofindself-adjointoperatorsxˆ andpˆsuchthatthecompatibilityconditions(2)
are satisfied. For that purpose we define new operators b+ and b−, satisfying
(b±)† =b∓,as
xˆ ipˆ
b± = ∓ .
√2
OnecanrewritetheHamiltonianHˆ intermsoftheb± asfollows:
i
Hˆ = ((b+)2 (b−)2).
2 −
Evidentlytheoperatorsxˆandpˆcanbeexpressedaslinearcombinationsofthe
b±. Eventhecompatibilityconditionscanbereformulated.Theyareequivalent
to[Hˆ,b±]= ib∓,whichinturncanbewrittenas
−
b−,b+ ,b± = 2b±. (3)
(cid:2) (cid:3)
{ } ±
TheseequationsarerecognizedtobethedefiningrelationsoftheLiesuperalge-
braosp(12), generatedbytheelementsb+ andb−. Sowehavefoundexpres-
|
sionsofallrelevantoperatorsintermsofLiesuperalgebragenerators.
A question one might ask is to find the spectrum of Hˆ and xˆ in an osp(12)
|
representation space, which is only possible once these representation spaces
are known. Thespectralproblemwill betackledin a subsequentpaper. Right
now, we wish to present a straightforward way of classifying the irreducible
-representationsofosp(12).
∗ |
2
3 Classification of irreducible -representations of
∗
osp(1 2)
|
Although we are aware of the classification by Hughes in [10], we think it is
possibletoachievehisresultsinamoreaccessibleway,basedon[11]. Inaddi-
tionwe willbeableto identifysomeequivalentrepresentationclasses. Before
giving the details of our classification, we providethe readers with the neces-
sary definitions and a general outline of how we will construct all irreducible
-representationsofosp(12).
∗ |
3.1 Basicintroduction andoutline
WewillbedealingwiththeLiesuperalgebraosp(12),generatedbytwoopera-
torsb+ andb− thataresubjecttotherelations(3). T| hegeneratingoperatorsb+
andb−aretheoddelementsofthealgebra,whiletheevenelementsare
1 1 1
h= b−,b+ , e= b+,b+ , f = b−,b− .
2{ } 4{ } −4{ }
Amongothers,thefollowingcommutationrelationscannowbecomputedfrom
thedefiningrelations(3):
[h,e]=2e, [h,f]= 2f, [e,f]=h.
−
One can define a -structure on osp(12), which is an anti-linear anti-
multiplicative involut∗ion X X∗. For X| ,Y osp(12) and a,b C we
have that (aX +bY)∗ = a¯X7→∗ +¯bY∗ and (XY∈)∗ = Y∗|X∗. Our -s∈tructure
is provided by the dagger operation X X†, so we have b± ∗ ∗= b∓ and
(cid:0) (cid:1)
7→
thereforeh∗ = h, e∗ = f and f∗ = e. Once we have constructedsuch a
− −
-algebra,weneedtodefinerepresentations.
∗
Definition1 Let be a -algebra, let be a Hilbert space and let be a
A ∗ H D
densesubspaceof . A -representationof on isamapπfrom intothe
H ∗ A D A
linearoperatorson suchthatπisarepresentationof regardedasanormal
D A
algebra,togetherwiththecondition
π(X)v,w = v,π(X∗)w (4)
h i h i
for all X and v,w . The representation space , together with the
∈ A ∈ D D
representationπ, iscalledan -module. Asubmoduleof isasubspacethat
A D
isclosedundertheactionof . Therepresentationπissaidtobeirreducibleif
A
the -module hasnonon-trivialsubmodules.
A D
Theevenoperatorsh,eandf,togetherwiththepreviouslydefined -structure,
∗
form the Lie algebra su(1,1). Both su(1,1) and osp(12) possess a Casimir
|
operator,denotedbyΩandC respectively:
1 1
Ω= (4fe+h2+2h), C = 4Ω+ (b−b+ b+b−).
−4 − 2 −
3
The Casimir elements generate the center of the respective (enveloping)alge-
bras. SoΩcommuteswitheveryelementofsu(1,1)andsimilaryforC. More-
over,wehaveΩ∗ =ΩandC∗ =C.
Wewillconstructallpossibleirreducible -representationsofosp(12)start-
∗ |
ing from one assumption: h has at least one eigenvectorin the representation
spacewitheigenvalue2µ,or
π(h)v0 =2µv0. (5)
Startingfromthisonevector,wewillbuildotherbasisvectorsoftherepresenta-
tionspaceV bylettingoperatorsofosp(12)actonit. Afterhavingdetermined
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theactionsofallosp(12)operatorsonallbasisvectorsofV,wewillextendthe
|
representationπ to a -representation. This is done by defining a sesquilinear
form .,. :V C,w∗hichistobeaninnerproductthatsatisfies(4).
h i →
The stipulationthat .,. shouldbe an innerproductwill becrucialin limiting
h i
thepossiblerepresentationspaces.However,wewillpostponethedetailsofthis
discussiontothepointwherewehaveenoughargumentsforthisend. Soletus
startwiththeactualconstructionoftherepresentationspaceV.
3.2 Construction ofthe representation space
In this section, the -structureis of no importance. We will constructan ordi-
∗
naryosp(12)representationspace thatwe willextendto a -representationin
| ∗
thenextsection.
Theembeddingofsu(1,1)inosp(12)impliesthatanyirreduciblerepresenta-
|
tion of osp(12)is a representationofsu(1,1), the latter beingnotnecessarily
|
irreducible.V canthereforebewrittenasadirectsumofirreduciblerepresenta-
tionspacesofsu(1,1),or
V =MWi.
i
Withoutlossofgenerality,wecanregardv asanelementofW . SinceW isa
0 0 0
representationspaceofsu(1,1),weknowthat
v2k =π(e)kv0 and v−2k =π(f)kv0
mustbeelementsofW .AllthesevectorsspanthespaceW ,whichisgenerated
0 0
byasinglevectorv .
0
The action of b+ on any vector of W must be a vector outside W , provided
0 0
thatthisactiondiffersfromzero. Letusdefine
v1 =π(b+)v0.
Wecansaythatv isanelementofW . Similarly,wecanlookattheactionof
1 1
b−onv :
0
v−1 =π(b−)v0.
Sinceb−b+ isa diagonaloperator(apparentfromthedefinitionoftheCasimir
operatorC),π(b−)v1 isacertainmultipleofv0. Atthispointhowever,wecan-
not be sure that π(b−)v1 is different from zero. Likewise, it is impossible to
4
tellwhetherπ(b+)v−1 = 0. Sincewecanneithersaythatπ(f)v1 isanonzero
6
multiple of v−1, nor that π(e)v−1 is a multiple of v1, we must regard v−1 as
anelementof adifferentsubspaceW . NotethatW andW are thesame
−1 1 −1
spaceswhen either π(b−)v1 or π(b+)v−1 differsfromzero. These actionsare
zerosimultaneouslyonlywhenµ=0.
WedenotethegeneratingvectorsofW−1 asv−2k−1 = π(f)kv−1 andthegen-
eratingvectorsofW1 asv2k+1 =π(e)kv1.
Lemma2 ThevectorsofW ,W andW areconnectedbytheactionsofb+
0 −1 1
andb− inthefollowingmanner
v2k+1 =π(b+)v2k and v−2k−1 =π(b−)v−2k, (6)
foreverypositiveintegervalueofk.
Proof: Applying π(b+) to the vector v1 results in a vector of W0 because
π(b+)v1 = 2π(e)v0. Thuswefindπ(b+)v2 =π(e)v1 = v3. Itisclearthatthis
can be generalizedto the stated formulafor v . The resultfor v can
2k+1 −2k−1
befoundanalogously. (cid:3)
Figure1helpstovisualizehowtherepresentationspaceisconstructed. We
emphasizethattherelationshipbetweenv andv isnotyetdetermined.
1 −1
(cid:27) f e -
v−q3 v−q1 vq1 vq3
W−1 Sob− (cid:19)Sob− b+(cid:19)7S b+(cid:19)7S W1
S S (cid:19) (cid:19)
(cid:19) S S
S S (cid:19) (cid:19)
S(cid:19)/q(cid:19)b− S(cid:19)q b+SSw(cid:19)q b+SSwq
v−2 v0 v2 v4 W0
(cid:27) -
f e
Figure1: TherepresentationV =W W W
−1 0 1
⊕ ⊕
The action of h on the entire representationspace V can already be deter-
mined.
Lemma3 TheactionofhonV isgivenby
π(h)v =(2µ+k)v , (7)
k k
forallk Z.
∈
Proof:Forevenvaluesofk,thisfollowsjustfromtherelations
[h,ek]=2kek, [h,fk]= 2kfk.
−
5
Fork = 1, thesearethe commutationrelations[h,e] = 2eand[h,f] = 2f,
−
andtherequiredidentitiesfollowbyinduction.Wethenobtain
π(h)v2k = π(h)π(e)kv0 = (2µ+2k)v2k.
Fortheoddvaluesofk,weneed[h,b±]= b±,whichisaninstantconsequence
±
ofequation(3). Fromthis,weobtain
π(h)v2k+1 = π(h)π(b+)v2k = (2µ+2k+1)v2k+1,
andsimilarlyforv . (cid:3)
−2k−1
Wewouldliketodeterminetheactionsofb+andb−oneveryvectorofW ,
0
W andW . OurmethodinvolvesdefiningtheactionoftheCasimiroperators
−1 1
ontherepresentationspace. Wewritetherespectivediagonalactionsas
π(C)v =λv ( v V),
∀ ∈
π(Ω)v2k = δ(δ+1)v2k ( k Z).
− ∀ ∈
Wewillarguethatthechoiceofλisnotindependentofδ. Itisaniceexercice
toshowwiththehelpofequation(3)that
(b−b+ b+b−)2 =4(b−b+ b+b−) 16Ω.
− − −
ThiscanbeusedtoshowthatC2 = (1 4Ω)(2C +4Ω). Ifweletbothsides
−
ofthisequationactonavectorv , wegetaquadraticequationinλ. Thetwo
2k
possiblesolutionsare
λ1 =2δ(2δ+1) and λ2 =2(δ+1)(2δ+1).
We choose λ = λ1 and remark that the results for the choice λ = λ2 can be
reproducedwiththetransformationδ δ 1.
Inordertobeabletodeterminetheac→tio−nso−fb+ andb− oneveryvectorofV,
westillneedtheactionofthesu(1,1)CasimiroperatorΩonW−1andW1.
Lemma4 TheCasimiroperatorΩactsonW−1andW1 asgivenby
1 1
π(Ω)v2k+1 = (δ )(δ+ )v2k+1, (k Z). (8)
− − 2 2 ∈
Asdesired,thesu(1,1)-CasimiroperatorisconstantonthesubspacesW−1and
W aswell. Moreover,theactionsonbothsubspacesarethesame.
1
Proof:Toproveequation(8),wewillcalculateπ(Ω)v2k+1asπ(Ωb+)v2k.From
(3)wecanimmediatelyderivethat
[b−,b+]b+ =2b+ b+[b−,b+].
−
UsingthisandtwicethedefinitionoftheCasimirelementC,weobtain
4Ωb+ =b+(1 2C 4Ω).
− −
6
The same formula holds if we change b+ into b− in both sides of the equa-
tion. All of the operators on the right hand side can be applied to vectors of
W0.Sonowπ(Ωb+)v2kcanbeeasilycalculated,withequation(8)asaresult.(cid:3)
Ithasnowbecomestraightforwardtofindtheactionsofb+andb−onallthe
vectorsofV.
Proposition5 Theactionsoftheoperatorsb+ andb− onthevectorsofV are
givenby
π(b−)v2k = (µ+k+δ)v2k−1,
π(b−)v2k+1 = 2(µ+k δ)v2k,
− (9)
π(b+)v−2k = (µ k δ)v−2k+1,
− − −
π(b+)v−2k−1 = 2(µ k+δ)v−2k.
−
Afterthechoiceλ = λ2 onewouldfindtheseactionsbymeansofthetransfor-
mationδ δ 1.
→− −
Sincetheactionsofh,eandf followdirectlyfromtheserelations,wehave
nowconstructedallrepresentationsofosp(12)generatedbyaweightvectorv0.
|
Itremainstoinvestigateirreducibilityandthe -condition.
∗
3.3 Extension to -representations
∗
RecallthatV isthespacespannedbyallthevectorsv ,k Z. Weintroducea
k
sesquilinearform .,. :V Csuchthat ∈
h i →
π(X)v,w = v,π(X∗)w
h i h i
for all X osp(12) and for all v,w V. We see that h∗ = h implies that
v ,v =∈0fork =| l. Thismeansthat∈theset = v k Z,v = 0 forms
k l k k
h i 6 S { | ∈ 6 }
anorthogonalbasisforV. Wedenoteby theindexsetsuchthatv forall
k
I ∈S
k .
∈I
Theform .,. isdefinedbyputting
h i
v ,v =a δ , k,l ,
k l k kl
h i ∈I
with ak to be determined and a0 = 1. The definition of a -representation
∗
requiresthattherepresentationspaceisaHilbertspace,sooursesquilinearform
needstobeaninnerproduct.Hence,wewanta >0fork . Fromtheaction
k
∈I
ofhandfromh∗ =hweobtain
2µ= π(h)v0,v0 = v0,π(h)v0 =2µ¯,
h i h i
so µ must be a real number. Similar calculations for the actions of Ω and C
revealthatbothδ(δ+1)andδ(2δ+1)arereal. Thesetwoconditionstogether
implythatδmustbereal.
Fromtheactionsofb+andb−andfrom(b±)∗ =b∓,wederive
a2k+1 =(cid:10)v2k+1,π(b+)v2k(cid:11)=(cid:10)π(b−)v2k+1,v2k(cid:11)=2(µ+k δ)a2k.
−
7
Inthesamewaywefind
1
a2k = (µ+k+δ)a2k−1.
2
Somereadersmightcareforaclosedexpressionforthea . Thisisgivenby
k
1
a = (3 ( 1)k)(µ δ) (µ+δ+1) ,
k 2 − − − ⌈k/2⌉ ⌊k/2⌋
where(x) =x(x+1) (x+k 1)istheclassicalPochhammersymbol.
k
··· −
We wish to determineunderwhich conditions .,. is an innerproduct. Alter-
h i
nativelyput,forwhichparametervaluesisa > 0forallk ? Startingfrom
k
∈ I
a0 = 1 this can be derived inductively using the two previousequations. We
findthatalla canbepositiveonlyifµ δ >0andµ+δ+1>0.
k
−
A similar reasoning should yield a positivity condition for the a for negative
k
k. However,theresultingconditionsµ δ+k > 0 canneverbesatisfied for
±
allnegativevaluesofk. Hence,therepresentationπmusthavealowestweight
vector,becauseotherwiseitwouldnotbepossibletodefineaninnerproducton
the entire representationspace. In this case, the restriction of π to an su(1,1)
subspaceisknownasapositivediscreteseriesrepresentation.
There are two choices for δ to obtain a lowest weight representation. One
choiceistohavev0asalowestweightvector,whichwillarisewhenδ = µas
oneseesfromtheactions(9). Forδ =µ 1weobtainπ(b+)v−2 =0,in−which
−
case v is the lowestweightvector. Afterone ofthese choicesProposition5
−1
must obviously be rewritten. Before we do this, let us make use of the inner
product .,. toconstructanorthonormalbasis e :
k
h i { }
v v
e2k = v22kk (k ≥0), e2k =(−1)k v22kk (k <0),
k k k k
and
v v
e2k+1 = v22kk++11 (k ≥0), e2k+1 =(−1)k−1 v22kk++11 (k <0),
k k k k
fork . Wecannowinvestigateallirreducible -representationsofosp(12).
∈I ∗ |
Proposition6 The only class of irreducible -representations of osp(12) is
∗ |
a direct sum of two positive discrete series representations of su(1,1), deter-
mined by a parameter µ. For 0 < µ 1, there is only one irreducible -
≤ 2 ∗
representation of osp(12). The actionsof the generatorson the basis vectors
|
e k =0,1,2,... oftherepresentationspacearedeterminedby
k
{ | }
π(b+)e2k = p2(2µ+k)e2k+1,
π(b−)e2k = √2ke2k−1, (10)
π(b+)e2k+1 = p2(k+1)e2k+2,
π(b−)e2k+1 = p2(2µ+k)e2k.
8
Forµ > 1,thisrepresentationcanoccuralongsideanotherone,forwhichthe
2
actionsofthegeneratorsonthebasisvectors e k = 1,0,1,2,... aregiven
k
{ | − }
by
π(b+)e2k = p2(k+1)e2k+1,
π(b−)e2k = p2(2µ+k−1)e2k−1, (11)
π(b+)e2k+1 = p2(2µ+k)e2k+2,
π(b−)e2k+1 = p2(k+1)e2k.
Theactionsoftheothergeneratorsfollowimmediatelyfromtheserelationsand
areleftforthereadertocalculate.
Proof: For δ = µ, we get the first representation, which is a lowest weight
−
representationsinceπ(b−)e0 = 0. Itisclearthatµmustbestrictlypositiveso
thatallthegivenactionsarewelldefined.Thecaseµ=0isexcludedtobesure
thatπ(b+)e2k differsfromzero.
In the case of the second representation, for δ = µ 1, we must add the
conditionµ > 12 toguaranteethatπ(b+)e−1 iswelldefi−nedanddifferentfrom
zero.Weendupwiththedesiredclassification. (cid:3)
Notethatifweweretochooseλ = λ2 inthediscussionprecedingLemma
4,wewouldfindexactlythesameclassofirreducible -representations.Indeed,
∗
thesetworepresentationswouldpopupforthechoices δ 1= µor δ 1=
− − − − −
µ 1.Itimmediatelyfollowsthattheotheractionsremainthesameinthiscase.
−
Finally, we notice an equivalence between both representation classes in
Proposition 6. Thus, we end up with only one class of irreducible represen-
tationsofosp(12).
|
Theorem7 Theonlyclassofirreducible -representationsofosp(12)isadi-
∗ |
rect sum oftwo positive discrete series representationsof su(1,1), determined
by a parameter µ > 0. The actions of the generators on the basis vectors
e k =0,1,2,... oftherepresentationspacearedeterminedby(10).
k
{ | }
Proof: Forµ > 12,definee¯k = e¯k−1 fork = 0,1,2,.... Thentheactions(11)
provetobeequivalentto(10)forµ¯ = µ 1. Hence, bothrepresentationsare
equivalent. − 2 (cid:3)
4 Conclusions and further results
Inthistextwehaveobtainedaclassificationofallirreducible -representations
∗
of osp(12). The latter Lie superalgebra showed up naturally in the Wigner
|
quantizationof the consideredHamiltonianH = xp. Our main concernhow-
ever, was to investigate the spectrum of the operators Hˆ and xˆ. Since these
operatorsarewrittenintermsofgeneratorsofosp(12)we felttheneedtoex-
|
plorerepresentationsof thisLiesuperalgebra. Theyprovideuswitha suitable
frameworkinwhichweknowhowthecrucialoperatorsact.
9
Results about the spectrum of Hˆ and xˆ have already been found and the
detailswillbepublishedinasubsequentpaper,butitisinterestingtosummarize
theresultshere.
In order to find all eigenvalues of one of the operators, one defines a formal
eigenvectorforaspecificeigenvaluet,
∞
v(t)= Xαn(t)en,
n=0
wherethee aretheeigenvectorsoftheosp(12)representationspaceV andthe
n
|
α (t)areunknowncoefficientsdependingontheeigenvaluet. Demandingthat
n
v(t)isaneigenvectoroftheoperatorinquestionwillgivesusathreetermrecur-
rence relation for the coefficientsα (t). These coefficientsare then identified
n
withtheorthogonalpolynomialsthatcomplywiththesamerecurrencerelation.
Thespectrumoftheoperatoristhenequaltothesupportoftheweightfunction
ofthistypeoforthogonalpolynomials.
ConcretelywehavethatthespectrumofHˆ isrelatedtoMeixner-Pollaczekpoly-
nomialsandis equalto R with multiplicitytwo. GeneralizedHermitepolyno-
mialsareconnectedwiththespectrumofxˆ,whichissimplyR.
Recall that Wigner quantization is a somewhat more general approach than
canonicalquantization.Thismeansthatoneshouldbeabletorecoverthecanon-
icalcasefromtheresultsafterWignerquantization.Indeed,ourresultsproveto
becompatiblewiththewell-knowncanonicalcasefortherepresentationparam-
eterµ= 1.
4
Acknowledgments
G. Regniers was supported by project P6/02 of the Interuniversity Attraction
PolesProgramme(BelgianState—BelgianSciencePolicy)
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