resWe rstdiscussthequantumsphere.AsrepresentationofUq,wecanconsider
resD 1F(n)=Uq·(t1)n Sq,C(5.1)
insteadof(4.1).However,notalltheseF(n)shouldbeconsideredas eldsonthe“real”quantumsphere,onlythosewhichareunitaryrepresentationsofthecompactform
Xi±=Xi , Hi =Hi,(5.2)
fin(so(D)),withthenaturalinnerproductdiscussedabove.Itwasprovedin[37]thatofUqthiscertainlyholdsforthoseLres(kλ1)=Lfin(kλ1)withk≤kS(4.2).Heretheassumption(2.24)isimportant.
Hencewecouldde neaHilbertspaceoffunctionsontherealquantumspheretobethedirectsumofallF(k)withk≤kS.Inordertoobtainasimplede nitionofposition
15
An algebra of functions on q-deformed Anti-de Sitter space AdS_q^D is defined which is covariant under U_q(so(2,D-1)), for q a root of unity. The star-structure is studied in detail. The scalar fields have an intrinsic high-energy cutoff, and arise most na
operatorsinSection5.5,weimposetheslightlystrongerboundk<kS,andde netheHilbertspaceoffunctionsontherealquantumspheretobe
finD 1Uq·(t1)n0.(5.3)F(n0)=Sq:=
0≤n0<kS0≤n0<kS
ThepositionoperatorswillbediscussedinSection5.5.ItsgeneratorsareessentiallythetiasinSection2;theywillhavetobeslightlymodi edinordertoaccountforthecuto .TheirstarstructureisdetermineduniquelybytheadjointontheHilbertspace,andwillbe
D 1givenexplicitlyinSection5.5.SqisatruncatedversionoftheclassicalsphereSD 1=
⊕n≥0L(nλ1),whichisrecoveredinthelimitM→∞,orq→1.
Nowconsidermoregenerallyn=n0+kMSwith0≤n0<kS,sothatF(n)=F(n0) F(kMS)accordingto(4.9).SinceF(kMS)～=Lres(kMSλ1)isa nite–dimensional
,ithasastandardpositive–de niteinvariantinnerirreduciblerepresentationoftheclassicalg
product.WejustdiscussedtheinnerproductonF(n0).Thissuggestsanaturalpositivede niteinnerproductonF(n)asthetensorproduct,sothat
(fρ(k),f′ρ′(k)):=(f,f′)(ρ(k),ρ′(k))(5.4)
wheref,f′∈F(n0)andρ(k),ρ′(k)∈F(kMS).Thisleadstoaninterestingphysicalinterpre-tation,aswewillsee.Wewillalwaysuseanorthonormalbasiscorrespondingtothisinnerproductfromnowon.
resWehaveseenintheprevioussectionthatallweightsofL(kMSλ1)havetheformλz= fin(n0λ1+λz)iziMiΛi.ThereforeF(n)isthedirectsumofirreduciblerepresentationsL
finofUqforvariousλz;ifthemultiplicityofacertainLfin(n0λ1+λz)inF(n)islargerthanone,thenthecorrespondingsubspacecanbedecomposedasanorthogonalsumofirreduciblecomponents.HenceF(n)isadirectorthogonalsumofHilbertspacesoftypeLfin(n0λ1+λz),withtheinducedinnerproduct(5.4),andthedi erentcomponentsarerelatedbytheaction ±(2.36).OnecannowcalculatetheadjointofthegeneratorsofoftheclassicalgeneratorsXifinfinUqonL(n0λ1+λz)withrespecttothisinnerproduct.Theresultis(see[37],Theorem
5.1) Hi =Hi,Xi±=siXi ,wheresi=( 1)zi.(5.5)
jfinIfπi(u)istherepresentationofu∈UqonLfin(n0λ1+λz)withrespecttoanorthonormalbasis,thismeansthatji (π(u) )j(5.6)i=πj(u)=πi(u)
asin(3.3);here reallydependsonzasin(5.5),whichwillhoweverbesuppressedinthefollowing.HenceLfin(n0λ1+λz)isaunitaryrepresentationofacertainrealformofthetype
fin(3.4)ofUq(so(D,C)),andthe“sectors”withdi erentλzbutthesamesiareunitarywith
16
An algebra of functions on q-deformed Anti-de Sitter space AdS_q^D is defined which is covariant under U_q(so(2,D-1)), for q a root of unity. The star-structure is studied in detail. The scalar fields have an intrinsic high-energy cutoff, and arise most na
paringwithSection3.1,weconcludethatifallziareeven,thenLfin(n0λ1+λz)isaunitaryrepresentationofthecompactform(3.6)andhence
D 1ascalar eldonSq.Iftheziaresuchthatsi=( 1)ziisasin(3.8),thenitisaunitary
finrepresentationoftheAnti–deSittergroupUq(so(2,D 2)),andshouldinfactbeviewed
D 1asascalar eldonAdSq,correspondingtoasquare–integrablescalar eldintheclassical
case.Tounderstandthis,considertheLfin((2MS k)λ1)aslowest–weightrepresentations
finLfin(kλ1)ofUqwithlowestweightkλ1.Forlowenergies,theyhavethesamestructure
asthescalar eldsontheclassicalAdSspace,whichareirreducibleunitarylowest–weightrepresentationsofSO(2,D 2)realizedintermsofsquare–integrablefunctions.Itisveryremarkablethattheyarerealizedhereintermsofpolynomialsinthecoordinatefunctions.Thereforewede netheHilbertspaceoffunctionsontherealquantumAnti–deSitterspacetobe
D 1finnD 1Lfin(nλ1)=Sq(t1)MS.(5.7)Uq·(t1)=AdSq:=
MS≤n<MS+kSMS≤n<MS+kS
MS nMSfinfinIntermsoflowest–weightrepresentationsLfin(nλ1)=Uq·(tDt1)ofUq,thiscanbe
writtenas D 1Lfin(nλ1).(5.8)AdSq=
(D 2)/2<n≤MS
ForenergieslessthanMS,thestatesintheHilbertspacesarethesameasclassically,and
fintheactionofUqapproachestheclassicaloneforanygivenweightasM→∞.Theenergy
ofallstatesislessthan2MS.Theprecisede nitionofthepositionoperatorswillbegiveninSection5.5.Intheclassicalcase,thelowerbound(D 2)/2canbeseenbyasimpledimensionalargument;howeveritcanbeviolatedslightly.M (D 4)/2MSfinForevenD>4,theirreduciblequotientofUq·(tDSt1)isthescalarsingletonLfin((D 4)/2λ1),withlowestweight(D 4)/2λ1;wewillnotconsideritanymorehere.