J.Phys.A:Math.Gen.39(2006)13783–13806
JOURNALOFPHYSICSA:MATHEMATICALANDGENERAL
doi:10.1088/0305-4470/39/44/012
Completenessofnon-normalizablemodes
PhilipDMannheimandIonelSimbotin
DepartmentofPhysics,UniversityofConnecticut,Storrs,CT06269,USAE-mail:philip.mannheim@uconn.eduandsimbotin@phys.uconn.edu
Received13July2006Published17October2006
Onlineatstacks.iop.org/JPhysA/39/13783
Abstract
Weestablishthecompletenessofsomecharacteristicsetsofnon-normalizablemodesbyconstructingfullylocalizedsquarestepsoutofthem,witheachsuchconstructionexpresslydisplayingtheGibbsphenomenonassociatedwithtryingtouseacompletebasisofmodestofitfunctionswithdiscontinuousedges.Aswellasbeingofinterestinandofitself,ourstudyisalsoofinteresttotherecentlyintroducedlargeextradimensionbrane-localizedgravityprogramofRandallandSundrum,sincetheparticularnon-normalizablemodebasesthatweconsider(specificallytheirregularBesselfunctionsandtheassociatedLegendrefunctionsofthesecondkind)areassociatedwiththetensorgravitationalfluctuationswhichoccurinthosespecificbraneworldsinwhichtheembeddingofamaximallyfour-symmetricbraneinafive-dimensionalanti-deSitterbulkleadstoawarpfactorwhichisdivergent.Sincethebrane-worldmasslessfour-dimensionalgravitonhasadivergentwavefunctionintheseparticularcases,itsresultinglackofnormalizabilityisthusnotseentobeanyimpedimenttoitsbelongingtoacompletebasisofmodes,andconsequentlyitslackofnormalizabilityshouldnotbeseenasacriterionfornotincludingitinthespectrumofobservablemodes.Moreover,becausethedivergentmodesweconsiderformcompletebases,wecanevenconstructpropagatorsoutoftheminwhichthesemodesappearaspoleswithresidueswhichareexpresslyfinite.Thus,eventhoughnormalizablemodesappearinpropagatorswithresidueswhicharegivenastheirfinitenormalizationconstants,non-normalizablemodescanjustasequallyappearinpropagatorswithfiniteresiduestoo—itisjustthatsuchresidueswillnotbeassociatedwithbilinearintegralsofthemodes.PACSnumbers:02.30.Gp,03.65.Ge,04.50.Th,11.25.−w
1.Introduction
Inconstructingcompletebasesofmodesolutionstowaveequationsitisveryconvenienttoworkwithmodeswhicharenormalizablesincetheyobeyaclosurerelation.Specifically,if
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onehassomecompleteorthonormalbasisofmodesfm(w)witheigenvalueslabelledbymandorthonormalityrelation
∞
dwe−2A(w)fm(w)fm(w)=δm,m,(1)
−∞
wheree−2A(w)isanappropriatenormalizationmeasure,thecompletenessofthebasiswill
thenrequirethatanylocalizedfunctionbeexpandableintermsofthebasismodesas
ψ(w)=amfm(w)(2)
m
withcoefficientswhicharegivenas
∞
am=dwe−2A(w)ψ(w)fm(w).
−∞
(3)
Withinsertionofthesecoefficientsbackintoequation(2)yielding
∞
ψ(w)=dwδ(w−w)ψ(w)
−∞
∞
dwe−2A(w)ψ(w)fm(w)fm(w),=
m
−∞
(4)
thearbitrarinessofthechoiceofψ(w)willthenrequirethatthebasismodesobeyaclosure
relationoftheform
fm(w)fm(w)=e2A(w)δ(w−w).(5)
m
Withequation(5)beingrecognizedasbeingaspecialcaseofequation(2)(namelythe
expansionoftheextremelylocalizedδ(w)inacompletebasisoffm(w)withcoefficientsam=fm(0)),thenotionsofcompletenessandclosureareoftentreatedinterchangeablyintheliterature,withequation(5)notonlyoftenbeingreferredtoasbeingacompletenessrelation,butwithitevenbeingregardedasbeinganessentialrequirementforabasistobecompleteinthefirstplace.
Itisthepurposeofthispapertoshowthatthisneednotinfactbethecaseandthatmodescanbecompleteevenwhentheydonotobeyequation(5)atall.Indeed,thestepswhichleadfromequation(1)toequation(5)onlyholdwhenthebasisisoneforwhichtheintegralsontheleft-handsideofequation(1)doinfactexist.Withbothequations(1)and(5)involvingbilinearfunctionsofthebasismodes,butwithequation(2)onlybeingalinearfunctionofthemodes,itisstillpossibleforthesummationinequation(2)tobewelldefinedevenwhenthebilinearexpressionswhichappearinequations(1)and(5)arenot.Moreover,thewaveequationsforwhichfm(w)arethemodesolutionsarethemselvesonlylinearfunctionsoffm(w),anditshouldthusbeimmaterialtothecompletenessoftheirsolutionsastowhetherornotbilinearintegralsofthemodesexist.Ingeneral,thencompletenessofabasishastobeunderstoodasbeingtherequirementthatforlocalizedfunctionsψ(w)thereexistsanexpansionoftheformofequation(2)withfinitecoefficientsamregardlessofwhetherornottheintegralsontheleft-handsideofequation(1)actuallyexist.Non-normalizablemodeswhosebehaviourissobadastocausethesebilinearintegralstodivergecanstillbecompleteinthesenseofequation(2),withtheamcoefficientsbeingsuchastoleadtototaldestructiveinterferencebetweenfm(w)intheregionswherefm(w)diverge.Itisthusequation(2)whichhastoberecognizedasbeingthegeneralstatementofcompleteness,andinthispaperweshallconfirmthisbyexplicitlyconstructinglocalizedsquarestepsassumsoversomecharacteristicbasesofdivergentmodes.Whiletheexistenceornotofthenormalizationintegralsof
Completenessofnon-normalizablemodes13785
equation(1)isimmaterialtoadifferentialwaveequation,ifthesolutionstothewaveequationarerequiredtobelongtoaHilbertspaceonecanrestricttosquareintegrablefunctionsalone,thoughotherwisethereisnoreasontodiscardanynon-normalizablesolutions1.SincewaveequationsinclassicalphysicsdonotactinaHilbertspace,inclassicalphysicsoneisnotfreetodiscardnon-normalizablemodes,andsinceclassicalphysicswaveequationsplayaprominentroleinclassicalgravitywheretheyareassociatedwithclassicalgravitationalfluctuationsaroundclassicalgravitybackgrounds,itistoclassicalgravitythatweshalllookforexamplesinwhichtotestwhethernon-normalizablemodescanbecomplete.2.Waveequationsforgravitationalfluctuations
Thewaveequationsweshallexplicitlyexploreareassociatedwiththerecentlyintroducedbrane-localizedgravityprogramofRandallandSundrum[2,3].Asintroduced,thebranegravityprogramprovidesforthepossibilitythatourfour-dimensionaluniversecouldbeembeddedinsomeinfinitelysizedbulkspaceandyetnotconflictwiththefactthatthereisnoapparentsignofanysuchhigherdimensionalbulk.Specifically,bytakingthehigherdimensionalbulktopossessaveryspecialgeometry,namelythefive-dimensionalanti-deSittergeometryAdS5,andbytakingourfour-dimensionaluniversetobeabrane(i.e.membrane)embeddedinit,RandallandSundrumfoundthatundercertaincircumstancesitwasthenpossibleforgravitationalsignalstolocalizearoundthebraneandnotpenetrateveryfarintothebulk,withAdS5actingasasortofrefractivemediumwhichrapidlyattenuatesanysignalswhichtrytopropagateinit.WithintheRandall–Sundrumbraneworldtherearesixfullysolubleset-ups(technicallyAdS5bulkswithembeddedMinkowski,deSitteroranti-deSitter
±
,dS±braneseachwitheitherpositiveornegativetensionλ—tobereferredtoastheM44and±
AdS4braneworldsinthefollowing),withallsixofthemhavingbackgroundswhichcanbedescribedbythegenericfive-dimensionalmetric
ds2=dw2+e2A(|w|)qµν(xλ)dxµdxν
(6)
wherethew-independentqµνisthefour-dimensionalmetricandtheso-calledwarpfactore2A(|w|)istakentobeafunctionof|w|wherewisthefifthcoordinate.WiththecurvatureofAdS5beingtakentobegivenas−b2,inthevariouscasestheexplicitbackgroundmetricsaregivenas±
=dw2+e−2(λ)b|w|[dx2+dy2+dz2−dt2],ds2M4(7)2bH2
ds2dS±−(λ)b|w|[e2Ht(dx2+dy2+dz2)−dt2],sinh2arcsinh4=dw+2bH
(8)
and
bH2±222
−(λ)b|w|[dx2+e2Hx(dy2+dz2−dt2)],dsAdS4=dw+2cosharccosh
bH
(9)
EveninquantummechanicswenotethattheSchr¨odingerequationH|ψ=E|ψisanoperatorequationwhichactslinearlyontheketvector|ψ,withitsexistencebeingindependentofwhatparticulardualvectorbraψ|mightbeusedtoconstructthebilinearnormψ|ψ.Thereisthusfreedomavailableinchoosingthedualspacevectors,withchoicesforthemotherthansimplyastheconjugatesoftheketshavingbeenfoundtoleadtoasensibleprobabilityinterpretationinthecaseoftheorieswithanon-Hermitianpotential(thefirstpartof[1])oranindefinitemetric(thefourth-orderoscillatortheorydiscussedinthesecondpartof[1]).Eveninquantummechanicsthen,imposingthefinitenessoftheψ|ψnormisnotthemostgeneralrequirementthatonecanconsider.
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±
where(λ)isthesignofλ.(TheM4backgroundmetricsaregivenin[2,3],thedS±4
±
backgroundmetricsaregivenin[4,5]andtheAdS4backgroundmetricsaregivenin[4].)
Forthebraneworldthegravitationalfluctuationsaroundthesesixbackgroundsaremostreadilytreatedintheaxialgaugewherethetransverse-tracelesstensorfluctuationmodeshTTµνthenallobeythegenericwaveequation(see,e.g.,[6]wherefullderivationsandrelevantcitationsaregiven)
∂2dA2
˜α∇˜αhTT−4+e−2A∇(10)µν=0,2∂|w|d|w|
assubjecttotheconstraint(technicallytheIsraeljunctioncondition)
∂dA
δ(w)−2hTT=0
∂|w|d|w|µν
(11)
˜αindicatethat˜α∇atabranewhichislocatedatw=0.Inequation(10),thetildesin∇
theseparticularcovariantderivativesaretobeevaluatedinthegeometryassociatedwiththefour-dimensionalqµν.Andwiththefour-dimensionalsectorofthetheorybeingseparableaccordingto
2TT˜α−2kH2]hTT˜α∇(12)[∇µν=mhµν,asdefinedheresothattensorfluctuationswithm2=0propagateontheappropriatedS4,M4
orAdS4lightcones(k=1,0,−1,respectively),aseparationofthemodesintotheform
λ
hTTµν=fm(|w|)eµν(x,m)thenrequiresthatfm(|w|)obey
222
ddAdA
+e−2Am2fm(|w|)=0−4−2(13)22d|w|d|w|d|w|(ineachofthesixbackgroundcasesofinteresttoustheidentityd2A/d|w|2=−kH2e−2A
holds),assubjecttotheconstraint
dAd
−2fm(|w|)=0.(14)δ(w)
d|w|d|w|
Ourtaskisthustoexplorethecompletenessofsolutionstoequations(13)and(14),andareaderunfamiliarwiththephysicsofthebraneworldcanstartatthispointasnoneoftheanalysiswhichensueswilldependonhowequations(13)and(14)werefirstarrivedat.Whatwillmatterinthefollowingisonlythattheseequationsadmitofexactsolutions,solutionswhoselarge|w|behaviourcanthenexplicitlybemonitored.
Beforeactuallyidentifyingexplicitsolutionstoequations(13)and(14)forthespecificchoicesofAand(λ)ofinterest,wenotethatviamanipulationofequation(13)wefindthateverypairofitssolutionshavetoobey
2dddAdAd−2A2
fm1−2fm2−fm2−2fm1,m1−m2fm1fm2=e
d|w|d|w|d|w|d|w|d|w|
(15)whichwithequation(14)thenrequiresthemodestoobey
∞2
d|w|e−2Afm1fm2m1−m22
0
dddAdA
−2−2=limfm1(16)fm2−fm2fm1.
|w|→∞d|w|d|w|d|w|d|w|
Orthogonalityofmodeswithdifferentseparationconstantsisthusachievedwhenthemodesarewell-enoughbehavedat|w|=∞tocausetheright-handsideofequation(16)tovanish
Completenessofnon-normalizablemodes13787
(withtheorthogonalitymeasurethenbeingpreciselytheoneweintroducedinequation(1)),withmodeswhichdivergebadlyenoughatinfinitycausingtheintegralontheleft-handsidetonotexist.Whileonecouldnowproceedtodeterminethemodesolutionsandidentifyforwhichparticularonestheintegralontheleft-handsideofequation(16)convergesordiverges,beforedoingsoitisinstructivetorecallthatviaasequenceoftransformationsitispossibletobringequation(13)toamorefamiliarform.Specifically,ifwechangevariablesfromwtoz
ˆm,fˆmwillthenobey[3]bysettingdz=e−A(w)dwanddefinefm=eA(z)/2f
d29dA23d2Aˆm=0,−2++−m2f(17)2dz4dz2dzwhileatthesametimethenormalizationintegralwillchangeas∞z[∞]
ˆm1(z)fˆm2(z).d|w|e−2Afm1(|w|)fm2(|w|)→dzf
0
z[0]
(18)
Whilewethusrecognizeequation(17)asbeinginthefamiliarformofaone-dimensional
Schr¨odingerequationandequation(18)asbeingintheformofitsconventionalquantum-mechanicalnormalizationintegral,nonetheless,asnotedabove,sinceinthecaseswhichare
ˆmmodestobelongtoaHilbertspace,weshouldofinteresttousherewearenotrequiringthef
notdiscardthenon-normalizablesolutionstoequation(17).2Andhavingnowrecognizedtherationalefornotdiscardingnon-normalizablesolutions,wereturntoequations(13)and(14)toactuallyfindandthenexplorethem.
3.CompletenesstestsfortheMinkowskibranecases3.1.Positivetensioncase
+
FortheM4casewhereA=−b|w|,thesolutionstoequation(13)arereadilyobtainedbysettingy=meb|w|/basthistransformationbringsequation(13)totheBesselequationform
2
4d1d
+1−2fm(y)=0.+(19)
dy2ydyy
Modesolutionswithanypositivem2arethusgivenby
fm(y)=αmJ2(y)+βmY2(y)
(20)
whereαmandβmarey-independentcoefficients,withthosesolutionswithm2=0being
givendirectlyfromequation(13)as
f0(y)=α0e−2b|w|+β0e2b|w|.
(21)
Tosatisfythejunctionconditionofequation(14)thenrequiresthatthevariousmodecoefficientsobey
αmJ1(m/b)+βmY1(m/b)=0,
β0=0,
(22)
withthecontinuumofm2>0modesthussatisfyingthejunctionconditionviaaninterplayofthetwotypesofBesselfunction,andthem2=0modef0(y)=α0e−2b|w|satisfyingitallon
Eveninquantummechanicsonedoesnotdiscardplanewavemodeseventhoughtheycausetheintegralontheright-handsideofequation(18)todiverge,sincedivergentastheymaybe,onecanstillconstructlocalizedwavepacketsoutofthem.Inthisrespectthen,thepointofthispaperwillbetoconstructlocalizedconfigurationsoutofbasisvectorswhichdivergeevenmorerapidlythanplanewaves.Andwhileweshallrestrictthestudyofthispapertotheclassical-mechanicalcontext,wenotethatwithinaquantum-mechanicalcontextsuchlocalizedconfigurationscouldstillbelongtoaHilbertspaceevenifthebasisvectorsthemselvesoutofwhichtheyarebuiltdonot.
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itsown.Inthebraneworldthem2>0modesareknownastheKK(Kaluza–Klein)modes,whilethem2=0modeservesasamasslessgraviton.Atlargey,thesesolutionsbehaveas
1/2
2α0m2
fm→f0→22.(23)[αmcos(y−5π/4)+βmsin(y−5π/4)],
πybyWithallofthesemodeshavingwavefunctionswhichfallveryfastin|w|aswegoawayfrom
thebrane,thegravitationalfluctuationmodesarethuslocalizedaroundit,thisbeingthekeyresultof[3].Withthemeasureofthenormalizationintegralbeingrewriteableas
∞∞
2b|w|
d|w|e=bdxx(24)
0
1/b
onsettingx=eb|w|/b,weseethatthemasslessgravitonwavefunctionisboundstate
normalizableandthattheKKmodespossessthesamecontinuumnormalizationasflatspaceBesselfunctions.Consequently,thetotalityofmasslessgravitonplusKKcontinuummodesiscompleteinexactlythesamewayasplanewaves,withbothofequations(1)and(5)beingsatisfied(thesummationinequation(5)isunderstoodtocontainbothdiscreteandcontinuousindices).Whilewethusseethatthereisnoneedtoperformanyexplicitcompletenesstestfor
+
aseverythingisstandard,aquitedifferentsituationwillemergewhenwethemodesofM4
−
considerM4.
3.2.Negativetensioncase
−
FortheM4casewhereA=+b|w|,them2>0andthem2=0solutionstoequation(13)aregivenby
fm(y)=αmJ2(y)+βmY2(y),
and
f0(y)=α0e−2b|w|+β0e2b|w|,
(25)(26)
wherenowy=me−b|w|/b,whiletosatisfythejunctionconditionofequation(14)thistimerequires
αmJ1(m/b)+βmY1(m/b)=0,
α0=0.
(27)
+
UnliketheM4casethistimeygoestozeroas|w|goestoinfinity,withlarge|w|asymptoticsnowbeingcontrolledbythebehaviourofBesselfunctionsatsmallargumentratherthanlarge,withthesolutionsbehavingatsmallyas
4βmαmy2−,fm→8πy2β0m2
f0→22
by
(28)
(Y2(y)behaveirregularlyatsmallargument).Withthemeasureofthenormalizationintegralnowbeinggivenas
1/b∞
−2b|w|
d|w|e=bdxx(29)
0
0
onsettingsetx=e−b|w|/b,thistimeweseethatitisonlytheJ2(y)modeswhichare
normalizable,andthatthemasslessgravitonwavefunctionandalltheY2(y)modesarenotonlynon-normalizable,theydivergefartooviolentlytoevenbeplanewavenormalizable.Inordertobeabletosatisfythejunctionconditionofequation(27)withnormalizablemodesalone,theconvergentJ2(y)modeswouldhavetosatisfyequation(27)allbythemselves,withthemodesthenneedingtoobeyJ1(m/b)=0.Solutionstothisconditionexistandaregiven
Completenessofnon-normalizablemodes13789
asthezeros,ji,oftheBesselfunctionJ1.Thissetofzerosisdiscreteandinfinite,withthe
−
braneworldthenbeinggivenasmodeswithmassesmi=bji.normalizablemodesoftheM4
Similarly,thedivergentY2(y)modescansatisfythejunctionconditionallontheirowniftheirmassesobeymi=byi,whereyiarethezerosoftheBesselfunctionY1,toyieldanotherinfinitesetofdiscretemodes.Withthedivergentmasslessgravitonmodewithwavefunctionβ0e2b|w|alsosatisfyingthejunctionconditiononitsown,wethusrecognizetwoclassesof
−
braneworld,theconvergentJ2(jie−b|w|),andthedivergente2b|w|andbasismodesintheM4
Y2(yie−b|w|).Andwhileourobjectiveistoapplyacompletenesstesttothedivergentmode
−
modebasis,itwillbeinstructivetoactuallyapplyacompletenesstesttotheconvergentM4
basisfirst.
−
modes4.CompletenesstestforconvergentM4
Totestforcompletenessofabasis,weneedtodeterminewhetheritispossibletoexpandtheˆ,αe−b|w|/bβ,VJ=0otherwiseintermsofthetypicallocalizedsquarestepVJ=V
modesofthebasis,namelyweseektofindasetofVmfromwhichwecanreconstructthesquarestepaccordingto
VJ(|w|)=VmJ2(me−b|w|/b).(30)∞
TodeterminetheneededcoefficientsVm,weapply0d|w|e−2b|w|J2(me−b|w|/b)toequation(30)andusetheorthogonalityrelationsthattheasymptoticallywell-behavedJ2(me−b|w|/b)modesobey.Specifically,withtheright-handsideofequation(16)vanishingforthesemodes,themodeswillthenobey
∞
d|w|e−2b|w|J2(me−b|w|/b)J2(me−b|w|/b)=0(31)whenmisnotequaltom,withuseofsomestandardpropertiesofBesselfunctionsobliging
themtoobey
1/b∞
22
d|w|e−2b|w|J2(me−b|w|/b)=bdxxJ2(mx)
21/bx(m/b)J22J2(mx)−J1(mx)J3(mx)0=(32)=b
22b
whenmandmareequalandmissuchthatJ1(m/b)iszero.Armedwithequations(31)and(32)wethusfindthatVJ(|w|)istobegivenby
2bBm
VJ(|w|)=(33)J2(me−b|w|/b),2J(m/b)2m0
0
2
0
m
wherethecoefficientsBmaregivenby
∞
−2b|w|−b|w|ˆBm=d|w|eVJ(|w|)J2(me/b)=−bV
0
β
ˆmβˆbVbV
=−2[2J1(x)−xJ0(x)]=2[2J0(x)+xJ1(x)]|mβmα
mmαmˆˆbVbV
=2[2J0(mβ)+mβJ1(mβ)]−2[2J0(mα)+mβJ1(mα)].(34)mm
Witheveryquantitywhichappearsinequation(33)nowbeingknown,VJ(|w|)canreadilybeplotted,andwedisplayitinfigure1asevaluated3throughtheuseofthefirst1000modesin
3
α
xdxJ2(mx)
Whileequation(33)isgiveninclosedform,theactualsumovermodesisitselfdonenumerically.
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PDMannheimandISimbotin
VJ(w)0.5
1
00.00.20.40.60.81.00.10.20.30.4e
-w
Figure1.TheleftpanelshowsareconstructionofthesquarestepVJ(|w|)=1,1<|w|<
−
2,VJ=0otherwiseviatheM4discreteJ2(jie−b|w|)modebasis,withtheparameterbbeingsetequaltoone.Therightpanelshowsablow-upoftheregionnearthetopofthestep.
thesum4.Aswesee,thebasisisindeedcapableofgeneratingthesquaresteptoveryhighaccuracy,withitscompletenessthusbeingconfirmed.
Withregardtotheplotinfigure1,ascanbeseenfromtheblow-upoftheregionnearthetopofthestep,themodesumexpresslydisplaystheGibbsphenomenonassociatedwithtryingtofitadiscontinuitywithacompletebasis,withtherebeinganovershoot(tonearVJ=1.1inthefigure)atthetopofthediscontinuityandanaccompanyingundershootatthebottom,anovershootandundershootwhichasrequiredoftheGibbsphenomenonwereexplicitlyfoundtogetnarrower(in|w|)asthenumberofmodesinthesumwasincreased,butnottoshorteninheight,alwaysreachingclosetoVJ=1.1inthefigure.WeregardtherecoveringoftheGibbsphenomenonasaverygoodindicatorofthereliabilityofourconstruction,andtogetherwiththequalityoftheoverallfititself,asprovidingverygoodevidenceforcompletenessof
−
modebasis.theconvergentM4
−
5.CompletenesstestfordivergentM4modes
TotestforcompletenessofthedivergentY2(yie−b|w|)pluse2b|w|modebasis,wetrytoreconstructthesquarestepviatheexpansion
VnY2(ne−b|w|/b)+V0e2b|w|.(35)VY(|w|)=
n
(Inequation(35)weusentodenotetheyizerosofY1(y),andshallusemtodenotetheji
zerosofJ1(y).)Nowwhilesuchareconstructionmightatfirstbethoughtunlikelytosucceedsinceeverytermontheright-handsideofequation(35)divergesbadlyinthelarge|w|regionwhereweneedthesummationtovanish,thevarioustermsinequation(35)arenotdivergingarbitrarilybut,ascanbeseenfromequation(28),areactuallyalldiverginginexactlythesamee2b|w|manner.Inconsequenceofthis,wearethereforeabletoadjustthevariouscoefficientsinequation(35)soastoexpresslycanceloutthedivergentpart.However,inordertogetVY(|w|)toactuallyvanishratherthanmerelynotdivergeoutsidethestep,wewillalsoneed
ThisparticularcompletenesstestwascarriedoutincollaborationwithDrAHGuth,DrDIKaiserandDrANayeri,andgrewoutofastudyofbrane-worldfluctuationsinwhichtheywereengagedwithoneofus(PDM).
4
Completenessofnon-normalizablemodes13791
tocancelthefinitepartthereaswell.Thus,witheachY2(y)havingaleadingbehaviouroftheform−4/πy2−1/πatsmallargument,i.e.withequation(35)behavingas
2Vn4b1
Vn(36)VY(|w|)→e2b|w|V0−−
πnn2πnatlarge|w|,weneedtoimposethetwoconditions4b2Vn
=V0,Vn=0
πnn2
n
(37)
onthecoefficients,withthetwoleadinglarge|w|termsthenbeingcancelled.
Havingthustakencareoftheleadingbehaviouratlarge|w|,wenowtrytoproceedaswithouranalysisexpansionofVJ(|w|)inconvergentmodes.However,wecannot∞ofthe−2b|w|
Y2(ne−b|w|/b)toequation(35)aseveryoverlapintegralwouldsimplyapply0d|w|e
∞
diverge.However,wehavefounditveryconvenienttoapply0d|w|e−2b|w|J2(me−b|w|/b)toequation(35)instead,wherewetakem/btobethejizerosofJ1(y).WithnoneoftheJ1(m/b)zeroscoincidingwithanyofthezerosofY1(n/b),5theneededoverlapintegralsaregiven(onsettingx=e−b|w|/b)by
1/b∞
−2b|w|−b|w|−b|w|
d|w|eJ2(me/b)Y2(ne/b)=bdxxJ2(mx)Y2(nx)
0
and∞
1/b
nY1(nx)J2(mx)−mJ1(mx)Y2(nx)2bm2
=bx=πn2(n2−m2),(m2−n2)0
d|w|e
−2b|w|
0
(38)
dx
J2(mx)x00
111/bdJ1(mx)=,(39)=−dx
b0dxmx2b
overlapintegralswhichdespitethebadlydivergentbehaviourofY2(y)ande2b|w|arenonethelessactuallyfiniteduetothecompensatingconvergentbehaviourofJ2(y).On∞
thusapplying0d|w|e−2b|w|J2(me−b|w|/b)toequation(35),wefindthatforthesquarestep
ˆ,αe−b|w|/bβ,VY(|w|)=0otherwise,theexpansioncoefficientsmustVY(|w|)=V
thusobey
1V02b1m2V02b
+=+−Vn22Vn2bπnn(n−m2)2bπn(n2−m2)n2
2bVn
=Bm=(40)
πn(n2−m2)
J2(me
−b|w|
/b)e
2b|w|
1=b
1/b
forallm,wheretheBmcoefficientsaregivenby
β
ˆbVˆBm=−bVdxxJ2(mx)=2[2J0(mx)+mxJ1(mx)]|βαmα
ˆˆbVbV
=2[2J0(mβ)+mβJ1(mβ)]−2[2J0(mα)+mαJ1(mα)].mm
5
(41)
ThezerosofJ1(y)andY1(y)aresimple,discreteoneswhichinterlaceeachother,withfirstthreepositivezeros
ofJ1(y)forinstanceoccurringat3.832,7.016and10.173,andwiththenthpositivezerobeingwellapproximatedbyjn≈(n+1/4)πwhennislarge;whilethefirstthreepositivezerosofY1(y)occurat2.197,5.430and8.596,withthenthpositivezerobeingwellapproximatedbyyn≈(n−1/4)πwhennislarge.Whiletheseparticularapproximationsdonotholdatsmalln,theparameternwhichappearsinthejn≈(n+1/4)πandyn≈(n−1/4)πexpressionsdoesdenotethenumberofthezero(countingthefirstpositivezeroasn=1),sothatfornlargeorsmalltheseexpressionsgiveacorrectcountingofthenumberofzeros.
13792
1.1
1
PDMannheimandISimbotin
VY(w)0.5
1
00.00.20.4e-w
0.60.81.00.10.20.30.4Figure2.TheleftpanelshowsareconstructionofthesquarestepVY(|w|)=1,1<|w|<
−
2,VY=0otherwiseviatheM4discreteY2(yie−b|w|)pluse2b|w|modebasis,withtheparameterbbeingsetequaltoone.Therightpanelshowsablow-upoftheregionnearthetopofthestep.
WithBmbeinggiveninclosedform,equation(40)isthusasetofNequationsforNunknownsandcanbeviewedasaneigenvalueequationforVn.(WhiletheJ2(me−b|w|/b)Y2(ne−b|w|/b)overlapintegralsofequation(38)arefinite,theJ2(me−b|w|/b)andY2(ne−b|w|/b)modesarenotorthogonal,withequation(40),unlikeequation(33),thusnotbeingdiagonalinitsindices.)TheVncoefficientscanthusbedeterminedand,onbeingfoundtobefiniteandrapidlyoscillatinginsign,lead,forthecaseofthefirst1000modesinthebasis,totheplotdisplayedinfigure2(i.e.,werestricttothefirst1000yiandthefirst1000jiinequation(40)).Asfigure2thusindicates,andquitespectacularlyso,thedivergentmodebasisiseverybitascapableofreconstructingthesquarestepastheconvergentoneandeverybitascapableofrecoveringtheGibbsphenomenon,andisthuseverybitascomplete6.Itisthusinvalidtousenormalizabilityasacriterionfordiscardingmodesasnon-normalizablemodesarefullycapableofservingasacompletebasisforconstructinglocalizedpackets7.Asafinalcomment,werecallthatfortheharmonicoscillatorwaveequationtherearetwosetsofsolutions,thesinesandthecosines,andbothsetsarecomplete.Itishenceperfectlyreasonabletoexpectothersecond-orderwaveequationstoalsohavetwocompletesetsofbasesevenifoneofthemconsistsentirelyofdivergentmodes.
6.Completenesstestsfortheanti-deSitterbranecases6.1.Thebasismodes
A(|w|)
ForAdS+=Hcosh(σ−b|w|)/bwherecoshσ=b/H,4braneworldwithwarpfactore
thetransformationy=tanh(b|w|−σ)bringsequation(13)totheform
2
4dd
+ν(ν+1)−(42)(1−y2)2−2yfm(y)=0.
dydy(1−y2)
wherewehaveintroducedtheconvenientparameterνdefinedby
1/2
m219m2
+2−,=(ν−1)(ν+2).(43)ν=
4H2H2
Thereconstructionofthesquarestepusingthedivergentmodebasisissogoodthattheonlyperceptibledifference
betweenfigures1and2isthatintheregionclosetoe−w=0,theJ2(me−b|w|/b)contributioniseversoslightlythicker.(Theconstraintsofequation(37)forceamorerapidconvergenceontheY2(ne−b|w|/b)modesum.)7FortheM−braneworldthisisjustaswell,sinceitcouldotherwisenotcontainanymasslessgraviton.
4
6
Completenessofnon-normalizablemodes13793
Equation(42)isrecognizedasanassociatedLegendreequation,withitssolutionsbeingtheassociateLegendrefunctionsofthefirstandsecondkinds,sothatform=0(namelyν=1)wecanset
fm(y)=αmPν2(y)+βmQ2ν(y).
(44)
Thissolutionalsoappliestooneofthem=0solutionsaswell,namelyQ21(y),aquantity
22
whichcanbewrittenintermsofthewarpfactorasQ1(y)=2/(1−y)=2cosh2(b|w|−σ)=
2
(y)iskinematicallyzero.Thissecond2b2e2A(|w|)/H2,butmissesoneothersolutionsinceP1
m=0solutioncanbefoundbysettingν=1inequation(42)andsolvingitdirectly,toyield
2
f0(y)=α0(45)−y+β0Q21(y).(1+y)
Requiringthemodestoalsoobeythejunctionconditionofequation(14)thenrestrictsthemaccordingto
αmPν1(−tanhσ)+βmQ1ν(−tanhσ)=0,
α0=0,
(46)
tothusdefinetheAdS+4braneworldbasismodes.
2
Asfunctions,allofthefunctionsPν1(y),Pν2(y),Q1ν(y)andQν(y)possessacutinthecomplexyplanewhichcanbelocatedtorunfromy=−∞toy=1.FortheAdS+4braneworldtheparametery=tanh(b|w|−σ)liesintherange−tanhσy1,andsointhisrangetheLegendrefunctionshavetobeevaluatedonthecut(astherealPνµ(y)=(1/2)[eiπµ/2Pνµ(y+
−iπµiπµ/2µ
i)+e−iπµ/2Pνµ(y−i)],Qµ/2)[e−iπµ/2QµQν(y−i)])whereν(y)=(eν(y+i)+e
theycanthenbepowerseriesexpandableviatheirrelationtohypergeometricfunctionstoyield
(−1)m(ν+m+1)m
(1−y2)m/2F(ν+m+1,−ν+m;m+1;(1−y)/2)Pν(y)=m
2m!(ν−m+1)
(ν+m+1)(−ν+m)(1−y)(−1)m(ν+m+1)
(1−y2)m/21+=m
2m!(ν−m+1)(m+1)1!2
(ν+m+1)(ν+m+2)(−ν+m)(−ν+m+1)(1−y)2++···,
(m+1)(m+2)2!22
eimπ2ν(ν+1)(ν+m+1)m
Qν(y)=F(ν−m+1,ν+1;2ν+2;2/(1+y))
(2ν+2)(1+y)ν+1−m/2(1−y)m/2
(m)eimπ2ν(ν+1)(ν+m+1)
=
(1+y)ν+1−m/2(1−y)m/2(ν+1)(ν+m+1)(ν−m+1)n(ν+1)n(y−1)n(−1)m(y−1)m
+×n(1−m)n!(y+1)(ν−m+1)(ν+1)(y+1)mnn=0
∞(ν+1)n(ν+m+1)n(y−1)n
ψ(n+1)+ψ(n+m+1)×n(n+m)!n!(y+1)n=0
1−y
−ψ(ν+1+n)−ψ(ν+m+1+n)−log(47)
1+y
whenµisageneralpositiveintegerm.(Inequation(47),ψ(y)denotes(d(y)/dy)/(y)and(a)ndenotes(a+n)/(a).)Fromequation(47)weseethatinthe−tanhσy1rangeofinterestthePν2(y)functionsarewellbehaved,behavingasyapproachesonefrombelow(namelyas|w|→∞)as
22
(ν+ν−3)(1−y)
(48)Pν2(y→1)→P(ν)(1−y)−
6
m−1
13794PDMannheimandISimbotin
where
ν(ν2−1)(ν+2)
,(49)P(ν)=
4
tothusbefullynormalizableandhavefinitenormalization
∞∞
222222b1−2A−2A
Nν=Pν(|w|)=2Pν(|w|)=2dwed|w|edyPν2(y).
H−tanhσ−∞0
(50)However,unlikethePν2(y),theQ2ν(y)areallfoundtodivergeaty=1,behavingthereas
Q2ν(y→1)→
(ν2+ν−1)1
++O((1−y)ln(1−y)),
(1−y)2
(51)
22
andthusintheAdS+4braneworldnoneofQν(y),andparticularlythemasslessQ1(y)graviton,arenormalizable.Weshallthusseektoconstructcompletebasesinboththenormalizableandnon-normalizablesectors.
6.2.CompletenesstestforconvergentAdS+4modes
ToconstructacompletebasisoutofnormalizablemodesalonerequiresthatthenormalizablePν2(y)satisfyequation(46)allontheirown,withtheeigenmodesthenneedingtosatisfy
Pν1(−tanhσ)=Pν1(−(1−H2/b2)1/2)=0.
(52)
Forarbitraryσthesolutionstoequation(52)cannotbewritteninaclosedform,butonnotingthatforoneparticularvalueofσ,namelyσ=0(i.e.,H=b),Pν1(0)isknowninclosedformas
2π1/21
,(53)Pν(0)=
(ν/2+1/2)(−ν/2)
tothusbezeroatν=2,4,6,...,weseethatonsolvingforanarbitrarygivenσnumericallyaninfinitediscretesetofallowedνvalueswillthenbefoundtoensue8.ThenormalizablemodesectorofAdS+4isthusdiscreteandinfinite,aresultfirstobtainedin[7]bydirectlynumericallysolvingequation(13).
TotestforcompletenessofthenormalizableAdS+4modebasis,weneedtofindasetofcoefficientsVmforwhichtheexpansion
VP=VmPν2(y)(54)
m
ˆwhen|w1|<|w|<|w2|,VP=0otherwise.WithreproducesthesquarestepVP=V
thePν2(y)modesbeingorthogonal,thecoefficientsarereadilygivenasVm=Bm/NνwhereNνisthenormalizationfactorgiveninequation(50),wheremandνarerelatedasinequation(43),andwheresomestandardpropertiesoftheassociatedLegendrefunctionsallowtheBmcoefficientstobewrittenas
|w2|2ˆby2ˆby2VV−2A222dPν(y)ˆd|w|ePν(|w|)=2dyPν(y)=2dy(1−y)Bm=V
Hy1Hy1dy2|w1|
ˆby2dV
=2dy[(2−ν)yPν+νPν−1]−2Pν
Hy1dy
Thetypicalcaseoftanhσ=0.9(namelyH/b=0.436)yieldsν=1.088,2.216and3.362asthe
1(−tanhσ)=0,withthenthpositivezerobeingwellapproximatedbythreelowestpositivesolutionstoPν
νn≈(n+1/4)π/arccos(−tanhσ)−1/2whennislarge.
8
Completenessofnon-normalizablemodes
1.1
1
13795
VP(w)0.5
1
0
0.0-0.50.5tanh[w - arctanh(.9)]
1.00.10.2Figure3.TheleftpanelshowsareconstructionofthesquarestepVP(w)=1,0.1
2tanh(b|w|−arctanh(0.9))0.2,VP(w)=0otherwise,viatheAdS+4discretePν(tanh(b|w|−σ))
modebasisinthetypicalcasewheretanhσ=0.9,H/b=0.436andb=1.Therightpanelshowsablow-upoftheregionnearthetopofthestep.
ˆby2Vd(2−ν)2
=2dy[(ν+1)Pν+1+νPν−1]+νPν−1−(Pν+1−Pν−1)Hy1dy(2ν+1)(2ν+1)
ˆb(ν+1)(ν+2)Pν−1−ν(ν−1)Pν+1y2V.=2(55)H2ν+1y1
Witheveryquantitywhichappearsinequation(54)nowbeingknown,VP(|w|)canreadilybeplotted,andwedisplayitinfigure3asevaluatedthroughtheuseofthefirst1000modesinthesum.Aswesee,thebasisisindeedcapableofgeneratingthesquaresteptoveryhighaccuracy,andwithitexpresslydisplayingtheGibbsphenomenon9,itscompletenessisthusconfirmed.
6.3.CompletenesstestfordivergentAdS+4modes
2WiththemasslessAdS+4gravitonwithdivergentwarpfactorwavefunctionf0(y)=β0Q1(y)=2β0/(1−y2)obeyingthejunctioncondition,itcouldalsobelongtoacompletebasisofdivergentQ2ν(y)modes(modeswhichaccordingtoequation(51)actuallydivergeinpreciselythesame1/(1−y)wayneary=1asthemasslessgravitonitself)iftheQ2ν(y)modesweretosatisfythejunctionconditionontheirown,i.e.iftheyweretoobey
1221/2
)=0.Q1ν(−tanhσ)=Qν(−(1−H/b)
(56)
Withequation(56)beingfoundtopossessaninfinitesetofdiscretesolutionsforthe
ˆwhenarbitraryσ,10weshallthusseektoexpandthelocalizedsquarestepVQ=V
|w1||w||w2|,VQ=0otherwise,intermsofthesesolutionsas
V0
VnQ2.(57)VQ=ν(y)+21−yn
ItispossiblethatthismightperhapsbethefirsttimethattheGibbsphenomenonhasexplicitlybeendemonstrated
forassociatedLegendrefunctions,andespeciallyforthedivergentQ2ν(y)modeswhichweshowbelow.
10Thetypicalcaseoftanhσ=0.9yieldsν=0.536,1.649and2.788asthethreelowestpositivesolutionstoQ1ν(−tanhσ)=0,withthenthpositivezerobeingwellapproximatedbyνn≈(n−1/4)π/arccos(−tanhσ)−1/2
1(−tanhσ)=0andQ1(−tanhσ)=0thusinterlacingeachother.Asregardswhennislarge,withthezerosofPνν
2theQ1ν(−tanhσ)=0solutions,wenoteadditionallythatthelowestpositiveoneactuallycorrespondstoanm<0
tachyonsinceithasν<1.
9
13796PDMannheimandISimbotin
(Forclarityweusen2heretodenotethesquaredmassesn2/H2=(ν−1)(ν+2)ofthe
22
Q2ν(y)sectormodes,andusemforthePν(y)sector.)Giventheasymptoticlimitexhibitedinequation(51),inordertofirstcancelboththeleading1/(1−y)termandthenexttoleadingO(1)termfromtheright-handsideofequation(57),wemustconstraintheVncoefficientsaccordingto
1V0V0
=0,=0,(58)Vn+Vn(ν2+ν−1)+224nntothusenableustoreexpressthesquarestepexpansionas
2
VQ=,VnQ2ν(y)−21−yn
(59)
assubjecttotheconstraintn22
Vn[ν+ν−2]=Vn2=0.(60)
Hnn
∞
Whilewecannotapply0d|w|e−2AQ2overlapintegralν(y)toequation(59)asevery∞
woulddiverge,finiteoverlapintegralsareobtainedifweinsteadapply0d|w|e−2APν2(y),whereweuseνtolabelthePν2(y)sectorsothatitssquaredmassesaregivenbym2/H2=(ν−1)(ν+2).WithnoneofthePν1(−tanhσ)andQ1ν(−tanhσ)zerosbeingfoundtocoincide,viaequations(16),(48)and(51),theneededoverlapintegralsarefoundtobeoftheform
∞
4bP(ν)−2A22
(61)d|w|ePν(y)Qν(y)=
(m2−n2)0
(P(ν)isgiveninequation(49)),andareindeedfinite,justasrequired.Withtheoverlapintegralwhichinvolvesthemasslessgravitonmodebeinggivenby
∞2
2bP(ν)−2APν(y)d|w|e,(62)=
(1−y2)m20
∞
theapplicationof0d|w|e−2APν2(y)toequation(59)thusyields
b(m2+2H2)n211
4bVnP(ν)Vn42(63)−2==Bm,2−n2)2)(mmH(m−nnnwhereBmisthesamefunctionthatwasalreadygivenearlierinequation(55).
Givenequation(63),theVncoefficientscannowbefoundnumerically,andlead,forthecaseofthefirst1000modesinthebasis,totheplotdisplayedinfigure4(i.e.,werestricttothefirst1000Pν1(−tanhσ)zerosandthefirst1000Q1ν(−tanhσ)zeros).Asfigure4thusindicates,thedivergentmodebasisiseverybitascapableofreconstructingthesquarestepastheconvergentoneandeverybitascapableofrecoveringtheGibbsphenomenon,andisthuseverybitascomplete11.Onceagainthenweseethatitisinvalidtousenormalizabilityasacriterionfordiscardingmodes,andinthisregardwedifferfromtheviewof[7]thatitispermissibletodiscardmodessuchasthemasslessAdS+4gravitonsimplybecausetheyarenot
12
normalizable.
11
Theconstructionissogoodthattheonlyperceptibledifferencebetweenfigures3and4isthatintheregionsclosetotheedgesofthestepstheGibbsphenomenonovershoot,asshowninfigure3blow-upiseversoslightlycloserto1.1thantheoneshownintheblow-upoffigure4.
12SincethenegativetensionAdS−braneworldwithdivergentwarpfactoreA(|w|)=Hcosh(σ+b|w|)/balsohas
4
2(y)anddivergentQ2(y)modes(wherenowy=tanh(b|w|+σ)withrangetanhσy1),itsconvergentPνν
structureisanalogoustothatofthedivergentwarpfactorAdS+4world,andsowedonotseekcompletenesstestsforithere.
Completenessofnon-normalizablemodes
1.1
1
13797
VQ(w)0.5
1
0
0.0-0.50.5tanh[w - arctanh(.9)]
1.00.10.2Figure4.TheleftpanelshowsareconstructionofthesquarestepVQ(w)=1,0.1
2tanh(b|w|−arctanh(0.9))0.2,VQ(w)=0otherwise,viatheAdS+4discreteQν(tanh(b|w|−σ))
pluscosh2(tanh(b|w|−σ))modebasisinthetypicalcasewheretanhσ=0.9,H/b=0.436andb=1.Therightpanelshowsablow-upoftheregionnearthetopofthestep.
7.CompletenesstestsforthedeSitterbranecases7.1.Thebasismodes
A(|w|)
FordS±=Hsinh(σ∓b|w|)/bwheresinhσ=b/H,4braneworldswithwarpfactore
thetransformationy=coth(σ∓b|w|)bringsequation(13)totheform
2
4dd
+ν(ν+1)−fm(y)=0,(1−y2)2−2y(64)
dydy(1−y2)
wherewehaveintroducedtheconvenientparameterνdefinedby
1/29m2m21−−,=(1−ν)(ν+2).(65)ν=
4H22H2
Recognizingequation(64)tobethepreviouslydiscussedassociatedLegendreequation,itsm=0(namelyν=1)solutionsaregivenas
fm(y)=αmPν2(y)+βmQ2ν(y),
(66)
whileitsν=1solutionsareoftheform
2
−y+β0Q2(67)f0(y)=α01(y).(1+y)
Requiringthemodestoalsoobeythejunctionconditionofequation(14)thenrestrictsthemaccordingto
αmPν1(cothσ)+βmQ1ν(cothσ)=0,
α0=0,
(68)
tothusdefinethedS±4brane-worldbasismodes.+
WhilethedS4anddS−4basismodesarequitesimilartoeachotherintheirgenericstructure,theydifferfromeachothersignificantlyinonecrucialregard.Specifically,unlike
A(|w|)
=Hsinh(σ+b|w|)/bwhichnevervanishes(σhavingbeenthedS−4warpfactore
A(|w|)
definedtobepositive),thedS+=Hsinh(σ−b|w|)/bhasazeroat4warpfactore
b|w|=σ.Withanullsignaltakinganinfiniteamountoftimetotravelfromthebranetothelocationofthiszero,thiszeroservesasahorizonforanobserveronthebrane[8],withthebraneobserveronlybeingsensitivetofluctuationmodesintheσb|w|0region.WiththedS+4parametery=coth(σ−b|w|)lyingintherangecothσy∞,wesee
13798PDMannheimandISimbotin
thatyisinfiniteatthedS+4horizon.Then,withtheassociatedLegendrefunctionsbehaving
2ν−ν−1
)asy→∞,theν=1masslessdS+asPν(y)→O(y)+O(y−ν−1),Q2ν(y)→O(y4
221/2
gravitonandalldS+modeswithcomplexν=−1/2±i(m/H−9/4)willbenormalizable4
withinthehorizon13.Withthemasslessgravitonandamassivecontinuumofmodeswithm2/H29/4whichsatisfythejunctionconditionofequation(68)byaninterplay(oftherealPν2(y)andtherealpartofQ2ν(y))thusprovidingaconventionalcontinuumnormalized
+
braneworld,inthedS+completebasisinthesenseofequations(1)–(5),aswiththeM44brane
worldthereisnoneedtotestexplicitlyforcompleteness.
However,fordS−4thesituationisquitedifferentsincethereisnownovanishingofthewarpfactorandnohorizon,withthecoordinate|w|nowextendingallthewaytoinfinity,andwiththeparametery=coth(σ+b|w|)insteadnowlyinginthe1ycothσ=(1+H2/b2)1/2range.UnlikethepreviouslydiscussedAdS+4braneworldcasewhereyapproachedonefrom
−
belowas|w|wenttoinfinity,inthedS4caseyinsteadapproachesonefromaboveinthelarge|w|limit,withequations(48)and(51)havingtobereplacedbythelimits
22
(ν+ν−3)(y−1)
Pν2(y→1)→P(ν)(y−1)+
6
(69)2
(ν+ν−1)1
−+O((y−1)ln(y−1)),Q2ν(y→1)→(y−1)2whereP(ν)=ν(ν2−1)(ν+2)/4isasgiveninequation(49).SincethePν2(y)arewell
+
behavedaty=1,whileQ2ν(y)divergethere,aswiththeAdS4case,thenormalizablesectorwillconsistofallPν2(coth(σ+b|w|))modeswhichsatisfythejunctionconditionontheirownaccordingto
Pν1(cothσ)=Pν1((1+H2/b2)1/2)=0,
(70)
whilethenon-normalizablesectorwillconsistofthedivergentwarpfactorwavefunction
2
Q21(coth(σ+b|w|))(=2/(y−1)iny>1)masslessgravitonandallmassive
WiththearbitraryhypergeometricfunctionF(a,b;c,z)beingequaltoonewhenitsargumentzistakentobezero,
2(y)andQ2(y)arereadilyobtainedfromtheir|y|>1hypergeometricfunctionrepresentationsthelargeylimitsofPνν
µ
oftheformPν(y)=2ν+1(−2ν−1)−1(−ν)−1(−ν−µ)(y+1)µ/2−ν−1(y−1)−µ/2F(ν+1,ν−µ+1;2ν+2,2/(1+y))+2−ν(2ν+1)−1(ν+1)−1(ν−µ+1)(y+1)µ/2+ν(y−1)−µ/2F(−ν,−ν−µ;−2ν,2/(1+y)),µ
Qν(y)=eiµπ2−ν−1π1/2(ν+µ+1)−1(ν+3/2)y−ν−µ−1(y2−1)µ/2F(ν/2+µ/2+1,ν/2+µ/2+1/2;ν+3/2,1/y2).
µµ
WhiletheserepresentationsshowthatPν(y)andQν(y)willingeneralbecomplexinthe|y|>1region,theform
µ
forPν(y)showsthatitwillactuallyberealwhenyandµarerealandtheparameterνtakesthevalueν=−1/2+iλwhereλisreal,avalueforwhichthequantityν(ν+1)=(ν+1/2)2−1/4whichappearsinthedefiningequationfortheassociatedLegendrefunctionsofequation(64)isthengivenastherealν(ν+1)=−λ2−1/4.With
2(y)andtherealandimaginaryequation(64)remainingrealatν=−1/2+iλ,forsuchvaluesofνthethenrealPν
2partsofQν(y)willallseparatelyobeyit.However,sinceequation(64)canonlyhavetwoindependentsolutions,
itmustbethecasethatoneofthesethreeclassesofsolutionsisredundant.Onnotingthatnomatterwhatthe
2valueofν,thedivergentpartofQ2ν(y)aty=1isrealwhilePν(y)iswellbehavedthere,wethusanticipatethat
whenyisrealandgreaterthanone,itmustbethe(thuswellbehavedaty=1)imaginarypartofQ2−1/2+iλ(y)
2whichmustcoincidewiththerealP−1/2+iλ(y);andsinceitisnotimmediatelyobvioushowonemayexplicitlycheck
suchaconnectionanalytically,wehaveinsteadconfirmeditnumerically.Inthefollowing,thenwecanrestrict
−+22thediscussiontotheuseofP−1/2+iλ(y)andRe[Q−1/2+iλ(y)]asbasismodes(inboththedS4andthedS4brane
µ
worlds).AswellasenablingustoshowthatPν(y)isrealforrealy,realµandcomplexν=−1/2+iλ,the
µµ
aboverepresentationsofthePν(y)andQν(y)arealsoofuseforactualcomputationalpurposeswhenyisgreaterthanone,sinceforargument|z|<1ahypergeometricfunctioncanberepresentedastheabsolutelyconvergent∞
powerseriesF(a,b;c,z)=[(c)/(a)(b)]n=0(a+n)(b+n)zn/[(c+n)n!].Moreover,forlargevalues
µµµ
oftheparameterλ,thefunctionsP−1/2+iλ(y)andRe[Q−1/2+iλ(y)]canevenbeapproximatedbyP−1/2+iλ(coshθ)=λµ−1/2(2/πsinhθ)1/2cos(λθ+µπ/2−π/4)−λµ−3/2(1/2πsinhθ)1/2(µ−1/2)(µ+1/2)cothθsin(λθ+µπ/2−π/4)
µ
andRe[Q−1/2+iλ(coshθ)]=λµ−1/2(π/2sinhθ)1/2cos(λθ+µπ/2+π/4)−λµ−3/2(π/8sinhθ)1/2(µ−1/2)(µ+1/2)cothθsin(λθ+µπ/2+π/4).(Itisnecessarytocarrythefirstnon-leadingtermsheresincetheoscillatoryleadingtermscanvanishatsomespecificθvalues.)
13
Completenessofnon-normalizablemodes13799
Q2ν(coth(σ+b|w|))modeswhichobey
1221/2
Q1)=0.ν(cothσ)=Qν((1+H/b)
(71)
−
WhilethispatternisthusquitesimilartothesituationfoundintheAdS+4case,thedS4braneworlddiffersfromitinonekeyregard,namelythattheparameteryisrequiredtobegreaterorequaltooneratherthanlessthanorequaltoit,andthusthecompletenessofitsmodebasesrequiresindependenttesting.
7.2.CompletenesstestforconvergentdS−4modes
Withthegeneralequation(16)takingtheform2cothσm1m2
−2dyfm1(y)fm2(y)2HH21
(y)(y)dfdfm1m2
−(y2−1)fm1(y)=lim(y2−1)fm2(y)
y→1dydy
(72)
22
inthedS−4case,andwiththePν(y)modesbehavingneary=1asinequation(69),thePν(y)modesformanorthonormalbasis,andwecanthusnormalizethusthemaccordingto
∞
222bcothσ22−2A
Pν(|w|)=2dwedyPν(y).(73)Nν=
H1−∞
WiththecothσargumentofPν1(cothσ)inequation(70)beinggreaterthanone,the
Pν1(cothσ)=0conditionhasnosolutionswithrealν.Rather,allofitssolutionsareoftheformν=−1/2+iλwhereλisrealanddiscrete14.Accordingtoequation(65),forsuchsolutionstheassociatedsquaredmassesobeym2/H2=9/4+λ2andarethusnicelypositive.Additionally,asnotedpreviously,fortheparticularchoiceofν=−1/2+iλ,thePν2(y)modewavefunctionsthemselvesarereal.
HavingnowexplicitlyidentifiedthedS−4normalizablemodebasis,totestforcompletenessweneedtofindasetofcoefficientsVmforwhichtheexpansion
ˆVP=VmPν2(y)(74)
m
ˆP=Vˆwhen|w1|<|w|<|w2|,VˆP=0otherwise.WithreproducesthesquarestepV
thePν2(y)modesbeingorthogonal,thecoefficientsarereadilygivenasVm=Bm/NνwhereNνisthenormalizationfactorgiveninequation(73),wheremandνarerelatedasinequation(65),andwheretheBmaregivenas
|w2|
ˆby2ˆby2VVd2Pν(y)−2A222ˆBm=Vd|w|ePν(|w|)=−2dyPν(y)=−2dy(y−1)HHdy2y1y1|w1|
ˆby2dV
=−2dy[(ν−2)yPν−νPν−1]+2Pν
Hy1dy
WiththedS−4braneworldrangeforybeingrestrictedtothefiniterange1ycothσ,incasesinwhichwerestrict
2(y),P1(y)andtocothσ<3,weareactuallyabletouseanextremelycompactrepresentationforevaluationofPνν
m2m/2mPν(y),namelytheformPν(y)=(y−1)(ν+m+1)F(−ν+m,ν+m+1,m+1,(1−y)/2)/[2m!(ν−m+1))]
whichholdsforanypositiveintegerm,andthelimitingformPν(y)=F(−ν,ν+1,1,(1−y)/2))whichholdswhenm=0,aseachofthesehypergeometricfunctionrepresentationscanbewrittenasapowerserieswhichisabsolutelyconvergentovertheentire1y3range.Fromtheserepresentationwefindinatypicalcasewithcothσ=1.1
1thatthethreelowestpositiveλsolutionstoP−1/2+iλ(cothσ)=0aregivenasλ=8.624,15.808and22.930,with
thenthpositivesolutionbeingwellapproximatedbyλn≈(n+1/4)π/arccosh(cothσ)whennislarge.
14
13800
1.1
1
PDMannheimandISimbotin
VP(w)0.5
1
^01.001.021.041.061.08coth[arccoth(1.1) + w]
1.101.051.06ˆP(w)=1,1.05Figure5.TheleftpanelshowsareconstructionofthesquarestepV
−2(coth(σ+b|w|))ˆcoth(arccoth(1.1)+b|w|)1.06,VP(w)=0otherwise,viathedS4discretePν
modebasisinthetypicalcasewherecothσ=1.1,H/b=0.458andb=1.Therightpanelshowsablow-upoftheregionnearthetopofthestep.
ˆby2d(ν−2)V2
=−2[(ν+1)Pν+1+νPν−1]−νPν−1+(Pν+1−Pν−1)
Hy1dy(2ν+1)(2ν+1)
ˆb[ν(ν−1)Pν+1−(ν+1)(ν+2)Pν−1]y2V.=−2(75)H2ν+1y1
ˆP(|w|)canreadilybeplotted,andwedisplayitinfigure5asevaluatedGivenequation(75),V
throughtheuseofthefirst500modesinthesum.Aswesee,thebasisisindeedcapableofgeneratingthesquaresteptoveryhighaccuracy,andwithitalsonicelydisplayingtheGibbsphenomenon,itscompletenessisthusconfirmed.7.3.CompletenesstestfordivergentdS−4modes
AswiththePν1(cothσ)=0condition,thesolutionstoQ1ν(cothσ)=0arealsoalloftheformν=−1/2+iλwhereλisagainrealanddiscrete,withthesolutionstoPν1(cothσ)=0
15
andQ1ν(cothσ)=0beingfoundtointerlaceeachother.WithitbeingonlytherealpartsoftheQ2ν(y)wavefunctionswithν=−1/2+iλandyrealwhichareindependentofthereal2
Pν(y),thenon-normalizabledS−4braneworldmodebasisconsistsofthemasslessgravitonwithitsrealwarpfactorwavefunctionplustherealpartsoftheQ2ν(y)wavefunctionswiththeappropriateν=−1/2+iλ.Then,withthey→1limitofequation(69)holdingfor
2
thegeneralQ2ν(y)witharbitraryν,weseethattherealpartsoftheQν(y)wavefunctionsallhavethesame1/(y−1)leadingbehaviouraty=1asthemasslessgravitonitself,withthenon-normalizablemodesalldivergingaty=1at2oneandthesamerate.
InordertotestforcompletenessintheReQν(y)plusmasslessgravitonsector,weneed
ˆQ=Vˆwhen|w1||w||w2|,VˆQ=0otherwise,intoexpandthelocalizedsquarestepV
termsofthesesolutionsas
V0
ˆVQ=.(76)VnReQ2(y)+ν2−1yn
Thetypicalcaseofcothσ=1.1yieldsλ=4.928,12.231and19.373asthethreelowestpositiveλ
solutionstoRe[Q1−1/2+iλ(cothσ)]=0,withthenthpositivesolutionbeingwellapproximatedbyλn≈(n−1/4)π/arccosh(cothσ)whennislarge.
15
Completenessofnon-normalizablemodes13801
(Aspreviously,forclarityweusen2heretodenotethesquaredmassesoftheQ2ν(y)
22
sectormodes,andusemforthePν(y)sector.)Giventheasymptoticlimitexhibitedinequation(69),inordertocancelboththeleading1/(y−1)termandthenexttoleadingO(1)termfromtheright-handsideofequation(76),wemustconstraintheVncoefficientsaccordingto
1V0V0
=0,=0,(77)Vn+Vn(ν2+ν−1)+224nntothusenableustoreexpressthesquarestepexpansionas
22ˆQ=V,VnReQν(y)−2
y−1n
(78)
assubjecttotheconstraint
n22
Vn[ν+ν−2]=−Vn2=0.(79)
Hnn
∞cothσ
Onnowapplying0d|w|e−2APν2(|w|)=(b/H2)1dyPν2(y)toequation(78)whereν2+ν−2=−m2/H2,useoftherelations
1cothσcothσ
dyPν2(y)Re
24H2P(ν)
,Qν(y)=
(m2−n2)
(80)(81)
1
Pν2(y)2H2P(ν)
=dy2,
(y−1)m2
whichfollowfromequations(69)and(72)(withP(ν)=ν(ν2−1)(ν+2)/4nowbeing
givenbym2(m2−2H2)/4H4)thenyields
1b(m2−2H2)n21
−2==Bm,VnP(ν)Vn(82)4b2−n2)4(m2−n2)(mmHnnwhereBmisthesamefunctionthatwasalreadygivenearlierinequation(75).
Givenequation(82),theVncoefficientscannowbefoundnumerically,andlead,forthecaseofthefirst500modesinthebasis,totheplotdisplayedinfigure6(i.e.,werestricttothefirst500Pν1(cothσ)zerosandthefirst500Re[Q1ν(cothσ)]zeros).Asfigure6thusindicates,thedivergentmodebasisiseverybitascapableofreconstructingthesquarestepastheconvergentoneandeverybitascapableofrecoveringtheGibbsphenomenon,andisthuseverybitascomplete.Aswithourearlierexamplesthen,weonceagainconfirmthatcompletenessisnotatalltiedtonormalizability.8.Finalcomments
Inthiswork,wehaveshownthatinandofitselftherequirementofnormalizabilityofbasismodesisnotatallneededforcompleteness,andthatonecanconstructlocalizedstepsoutofbaseswhosemodesarenotnormalizableatall.Sincethelocalizedstepsthatwehaveconstructedoutofnon-normalizablebasesinvolveexpansioncoefficientsVnwhichareexplicitlyfoundtobefinite,thissuggeststhatweshouldbeabletoconstructpropagatorsinvolvingthemodesinwhichthesemodesappearaspoleswhichhaveresidueswhicharethemselvesfinite.Thus,insharpcontrasttothesituationinwhichpropagatorsarebuiltoutofnormalizablemodes,forpropagatorswhicharebuiltoutmodesofwhicharenotnormalizable,
13802PDMannheimandISimbotin
1.1
1
VQ(w)0.5
1
^01.001.021.041.081.06coth[arccoth(1.1) + w]
1.101.051.06ˆQ(w)=1,1.05Figure6.TheleftpanelshowsareconstructionofthesquarestepV−ˆcoth(arccoth(1.1)+b|w|)1.06,VQ(w)=0otherwise,viathedS4discreteRe[Q2ν(coth(σ+
2
b|w|))]plussinh(coth(σ+b|w|))modebasisinthetypicalcasewherecothσ=1.1,H/b=0.458andb=1.Therightpanelshowsablow-upoftheregionnearthetopofthestep.
theseresiduesmustthennotberelatedtonormalizationconstantsortoanybilinearintegralsofthemodesatallforthatmatter.
Toexplicitlyconstructsuchdivergentmodebasedpropagators,wemustfirstintroduceexplicitsourceterms.Forthecaseofinteresttothebraneworld,thesourceistypicallytaken
TT
whichisconfinedtothebraneattobeatransverse-tracelessenergy–momentumtensorSµν
w=0,withequations(10)and(11)beingreplacedby(see,e.g.,[6])
22
∂dA˜α∇˜αhTT−4+e−2A∇(83)µν=0,2∂|w|d|w|
dA∂2TT−2hTTδ(w)(84)µν=−κ5δ(w)Sµν,∂|w|d|w|
2
whereκ5isthebrane-worldgravitationalconstant.
+
braneworldwhereequations(83)andForthecasefirstoftheconvergentwarpfactorM4
(84)reduceto
2
∂22b|w|αβTT
h−4b+eη∂∂(85)αβµν=0,∂|w|2
∂2TT
(86)δ(w)+2bhTTµν=−κ5δ(w)Sµν,∂|w|
onrecallingthattheBesselfunctionsobey
b|w|b|w|dqeqe
+2bαqJ2+βqY2
d|w|bb
b|w|b|w|
qeqeb|w|
αqJ1+βqY1,(87)=qe
bb
anexplicitsolutiontoequations(85)and(86)canreadilybegiven,namely[9]
2b|w|κ/b)+βqY2(qeb|w|/b)]TT5TT44ip·(x−x)[αqJ2(qehµν(x,|w|)=−Sµν(x)dxdpe(2π)4q[αqJ1(q/b)+βqY1(q/b)]
2+TTˆTT(x,x,w,0;αq,βq,M4=−2κ5d4xG)Sµν(x),(88)
Completenessofnon-normalizablemodes13803
¯2(qbeingunderstoodtohavethesamesignasp0here),andαqandβqwhereq2=(p0)2−p
arearbitraryconstants.
−
braneworldwhereThegeneralizationofthissolutiontothedivergentwarpfactorM4
wehave
2
∂
−4b2+e−2b|w|ηαβ∂α∂βhTT(89)µν=0,2∂|w|
∂2TT
−2bhTTδ(w)(90)µν=−κ5δ(w)Sµν,∂|w|and
−b|w|
qe−b|w|qeαqJ2+βqY2
bb−b|w|−b|w|
qeqe
+βqY1,=−qe−b|w|αqJ1
bb
ip·(x−x)[αqJ2(q
d
−2bd|w|
(91)
isoftheform[6]
2κ5TT
hµν(x,|w|)=
(2π)4
dxdpe
44
e−b|w|/b)+βqY2(qe−b|w|/b)]TT
Sµν(x)
q[αqJ1(q/b)+βqY1(q/b)]
(92)
=2−2κ5TT
ˆTT(x,x,w,0;αq,βq,M−)Sµνd4xG(x),4
withαqandβqagainbeingarbitraryconstants.Thatthesolutionofequation(92)satisfies
equation(89)followsdirectly,sincebothJ2(qe−b|w|/b)andY2(qe−b|w|/b)separatelysatisfytheBesselfunctionequationgivenasequation(19)withybeinggivenbyy=qe−b|w|/b;andthatthesolutionsatisfiesequation(90)followsfromequation(91).Forthissolutionwenotethatitistherequirementthatequation(92)obeyequation(90)(technicallytheIsraeljunctionconditioninthepresenceofthesource)whichfixestheoverallnormalizationoftheintegrandinequation(92),withnoneoftheαqorβqcoefficientsneedingtobeinfinite.Infactthesame
+
braneworldpropagatorasitsoverallnormalizationisfixedbythejunctionistrueoftheM4
−+
solutionofequation(88)andtheM4conditionofequation(86),withthesimilarityoftheM4
solutionofequation(92)essentiallyshowingcompleteinsensitivitytothenormalizabilityorlackthereofofbasismodes.
Inordertobeabletomakecontactwiththevariousbasesweusedinourconstruction
−
braneworld,weneedtomakespecificoflocalizedstepsinthedivergentwarpfactorM4
choicesfortheαqandβqcoefficientswhichappearinequation(92).TomakecontactwiththeconvergentJ2(qe−b|w|/b)modes,werecallthataTaylorseriesexpansionofJ1(q/b)aroundanyjizeroofJ1isoftheform
J(j)qqq
1i
−jiJ1(ji)=−ji−jiJ2(ji).−J2(ji)=−(93)J1(q/b)=bbjibThusonsettingαq=1,βq=0andrecallingthateachjizeroofJ1(ji)isalsoazeroof
J1(−ji),weseethatthepropagatorofequation(92)containsasetofisolatedpolesatthezerosofJ1(apoleatq=bj1whenp0ispositiveandapoleatq=−bj1whenp0isnegative),withap0planecontourintegrationyieldinganetpolecontributiontothepropagatoroftheform
¯·x¯d3peip−TTˆfi(|w|)fi(0)[e−iEit−eiEit],G(x,0,w,0;αq=1,βq=0,M4)=−i3(2π)2Ei
i
(94)
13804PDMannheimandISimbotin
where
1/22b1/2J2(jie−b|w|)
¯+b2ji2,(95)fi(|w|)=,Ei=p
J2(ji)
andwherethesummationinequation(94)onlyneedsextendovertheji>0modes.Finally,recallingequation(32),namely
∞2
(m/b)J2−2b|w|2−b|w|
,(96)d|w|eJ2(me/b)=
2b0
weseethatthefi(|w|)basismodespreciselyobeyequations(1)and(5),withthepolestructure
−ˆTT(x,0,w,0;αq=1,βq=0,M−)nicelyrecoveringthebrane-worldpropagatorGoftheM44
2−b|w|
/b)sectorbasismodes.orthonormalityandclosurestructureofthenormalizableJ2(me
−
Inordertomakecontactwiththenon-normalizableM4modesector,weneedtotakeβqto
−
propagator.RecallingthatJ1(y),J2(y),Y1(y)andY2(y)respectivelybenon-zerointheM4
2
behaveasy/2,y/8,−2/πy+O(y)and−4/πy2−1/πneary=0,weseethatonceβqisnon-zero,theintegrand[αqJ2(qe−b|w|/b)+βqY2(qe−b|w|/b)]/q[αqJ1(q/b)+βqY1(q/b)]willbehaveas2be2b|w|/q2nearq2=0independentoftheactualvaluesofαqandβq,tothusgiverisetoamasslessgravitonpoletermcontributionoftheform
¯·x¯
d3peip−TT2b|w|−i|p|ti|p|tˆG(x,0,w,0;αq,βq=0,M4,graviton)=ibe[e−e].(97)(2π)32|p|
−
Non-normalizableastheM4brane-worldgravitonmightbe,aswesee,itnonethelessappearsinthepropagatorwithafiniteresidue16.
−
braneworlddivergentY2(qe−b|w|/b)modeswesetαq=0TomakecontactwiththeM4
ˆTT(x,0,w,0;αq,βq,M−),andwhileweimmediatelythenobtainpolesatthezerosofinG4
Y1(q/b),sincebothY2(qe−b|w|/b)andY1(q/b)havebranchpointsatq=0,wealsoobtainacutdiscontinuity,withthefullsingulartermevaluatingto[6]
¯·x¯
d3peip−TT2b|w|ˆG(x,0,w,0;αq=0,βq=0,M4)=ibe[e−i|p|t−ei|p|t]3(2π)2|p|
¯·x¯d3peip
˜˜−ifi(|w|)fi(0)[e−iEit−eiEit]3(2π)2Eii
¯·x¯ip
J2(me−b|w|/b)i3e−iEptiEpt
[e−e]dm1−2idp+
(2π)32EpY1(m/b)
−b|w|−b|w|
[Y1(m/b)J2(me/b)−J1(m/b)Y2(me/b)]
2×,(98)2π4J1(m/b)+Y1(mb)
where
b1/2Y2(yie−b|w|)˜,fi(|w|)=
Y2(yi)
1/22
¯+b2yi2.Ei=p
(99)
Asweagainsee,despitethelackofnormalizabilityofY2(me−b|w|/b)modes,alltheterms
whichappearinequation(98)dosowithcoefficientswhicharenonethelessfinite.
−
DespitethefactthatthenegativetensionM4braneworldpossessesamasslessgravitonwhoseresidueisfinite,wenotethatitsresidueappearswithanoverallminussign(namelynegativesignature)comparedtotheotherwise
+braneworld(compareidenticalinstructurepositivesignaturemasslessgravitonresidueofthepositivetensionM4
thefirstformsgivenforhTTµν(x,|w|)giveninequations(88)and(92)whichdifferbyanoverallminussignoccasionedbytheoveralldifferenceinsignbetweentheright-handsidesofequations(87)and(91)).Suchnegativesignatureis
−−
thoughttoindicateaninstabilityoftheM4braneworld.Nonetheless,eventhoughtheM4braneworldmightthusnotbeofdirectphysicalinterest,itcanstillserveasausefulmathematicallaboratoryforexploringthecompletenesspropertiesofbasesbuiltoutofnon-normalizablemodes.16
Completenessofnon-normalizablemodes13805
Furtherexamplesofthisphenomenonmaybefoundintheotherdivergentwarpfactorbraneworldswehavebeenconsidering.However,unliketheexactpropagatorsolutionsofequations(88)and(92),fortheAdS4anddS4basedbraneworldssofarsuchpropagatorshaveonlybeenconstructedinloworder.Specifically,fortheAdS+4braneworldforinstancewherethebackgroundmetricofequation(6)takestheexplicitform
ds2=dw2+e2A(|w|)[dx2+e2Hx(dy2+dz2−dt2)]
(100)
withtheAdS+4warpfactorA(|w|)beinggiveninequation(9)andλbeingpositive,tolowestorderinHtheappropriateAdS+4propagatorisgivenas[6]
ˆTTx,x,w,0;αˆν,AdS+ˆν,βG4
∞∞
1023ˆν)=dpdpdpdp1p1Bν(tanh(b|w|−σ),αˆν,β32H(2π)−∞0
×eHx/2eHx/2e−ip
0
(t−t)+ip2(y−y)+ip3(z−z)
Jτ(ke−Hx/H)Jτ(ke−Hx/H)(101)
wherekisgivenbyk=[(p0)2−(p2)2−(p3)2]1/2,τandνaregivenbyτ=ν+1/2=
ˆν)isgivenbyˆν,β[9/4+k2/H2−(p1)2/H2]1/2,andthequantityBν(tanh(b|w|−σ),αˆν)ˆν,βBν(tanh(b|w|−σ),α
=
1
H(ν−1)(ν+2)
αˆνPν2(tanh(b|w|−σ))
αˆνPν1(−tanhσ)
2ˆ+βνQν(tanh(b|w|−σ))
.(102)
ˆνQ1+β(−tanhσ)ν
ˆν)obeysˆν,βAsconstructedthequantityBν(tanh(b|w|−σ),α
dAdˆν)=δ(w),−2Bν(tanh(b|w|−σ),αδ(w)ˆν,β
d|w|d|w|
+
andhasasmallHlimitintotheanalogousM4integrand,namely
(103)
[αqJ2(qeb|w|/b)+βqY2(qeb|w|/b)]ˆ,ˆν,βν)→Bν(tanh(b|w|−σ),α
q[αqJ1(q/b)+βqY1(q/b)]
(104)
ˆν=(2/π)[−αqsin(νπ)+βqcos(νπ)].InthesmallHwhereαˆν=αqcos(νπ)+βqsin(νπ),β+ˆTT(x,x,w,0;αˆν,AdS4)obeysˆν,βlimitG
dA2dA∂2−2A˜˜αTT+ˆˆGδ(w)+e∇∇x,x−4−4,w,0;αˆ,β,AdSανν4
∂w2d|w|d|w|
=eHxδ(x−x)δ(t−t)δ(y−y)δ(z−z)δ(w),
withthefluctuationhTTµν(x,|w|)
=
2
−2κ5
(105)
TT
ˆTTx,x,w,0;αˆν,AdS+d4xe−HxGˆν,β4Sµν(x)
(106)
+
thusbeinganexactAdS+4braneworldsmallHsolutiontotheAdS4variantofequations(83)
TT
(x)sourceonthebrane.and(84)foranarbitrarySµν
22
AsregardspoletermsintheAdS+4brane-worldpropagator,since(ν−1)(ν+2)=q/H,
ˆν)generatesamasslessν=1gravitonthe(ν−1)(ν+2)terminBν(tanh(b|w|−σ),αˆν,β
poleinthepropagatorwhichisfoundtobeoftheform[6]
ˆTTx,x,w,0;αˆν,AdS+ˆν,β,gravitonG4
ˆS(x,x,m=0)be2AD
,(107)=
ˆ1)(H2/b2)+(1−H2/b2)1/2+(H2/b2)arccosh(b/H)][−(αˆ1/β
13806PDMannheimandISimbotin
ˆS(x,x,m)isthepureAdS4spacetimepropagatorwhichobeyswhereD222ˆS(x,x,m)=eHxδ4(x−x).∂x−H∂x+e−2Hx∂y+∂z−∂t2−2H2−m2D
(108)
Aswesee,despitethelackofnormalizabilityofthegravitonwavefunction,theresidueat
17
theAdS+ˆν=0in4masslessgravitonpoleisnonethelessfinite.Similarly,ifwesetαequation(101)wewillimmediatelygeneratethedivergentQ2ν(tanh(b|w|−σ))modesas
1
polesassociatedwiththezerosofQν(−tanhσ),withthesepoletermsalsopossessingfiniteresidues.Consequently,inthebraneworlddivergentmodesarefullycapableofappearingwithfiniteresiduesinpropagatorsandtheirlackofnormalizabilityshouldnotbetakenasbeing
+ˆTT(x,x,w,0;αq,βq,M+)propagatorGacriterionforexcludingthem.Infact,withtheM44
ofequation(88)beingcausalwhenwesetαq=1,βq=i[9,10](sothatitisthen
ˆν)ˆν,βbasedonoutgoingHankelfunctions),giventhesmallHlimitofBν(tanh(b|w|−σ),α
ˆTT(x,x,w,0;αˆν,AdS+ˆν,βexhibitedinequation(104),itwillbetheG4)propagatorwith
+iπνˆiπν
αˆν=e,βν=(2i/π)ewhichwillbetheAdS4analogueoftheoutgoingHankelfunction
+
brane-worldpropagator,withthisparticularAdS+basedcausalM44braneworldpropagator
+
explicitlybeingfoundtobecausal[6].Assuch,thecausalAdS4brane-worldpropagator
ˆν=(2i/π)eiπνpossessesanexplicitmasslessgravitonpolewhoseresiduewithαˆν=eiπν,β
isfinite,withtherethusbeingnojustificationforexcludingit18.Acknowledgments
TheauthorswouldliketothankDrAHGuth,DrDIKaiserandDrANayerifortheiractiveparticipationinthiswork,andfortheirmanyhelpfulcomments.References
[1]BenderCM,BrodyDCandJonesHF2003Am.J.Phys.711095
MannheimPDandDavidsonA2005Phys.Rev.A71042110[2]RandallLandSundrumR1999Phys.Rev.Lett.833370[3]RandallLandSundrumR1999Phys.Rev.Lett.834690
[4]DeWolfeO,FreedmanDZ,GubserSSandKarchA2000Phys.Rev.D62046008[5]KimHBandKimHD2000Phys.Rev.D61064003
[6]MannheimPD2005Brane-LocalizedGravity(Hackensack,NJ:WorldScientific)[7]KarchAandRandallL2001J.HighEnergyPhys.JHEP05(2001)008[8]GarrigaJandSasakiM2000Phys.Rev.D62043523
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−
UnlikethemasslessgravitonofthenegativetensionM4braneworld,themasslessgravitonofthepositivetension+
AdS4braneworldhasaresiduewithaperfectlyacceptablepositivesignature—asindeeditmustsinceitsresidue
+braneworldinthelimitincontinuesintothatofthepositivesignaturemasslessgravitonofthepositivetensionM4
whichHistakentozero.
18FromtheperspectiveofthepossiblephysicalviabilityoftheAdS+braneworld,theneedtoincludenon-normalizable
4
modesisactuallysomewhatunfortunatesincetheyleadtoagravitywhichisnotatalllocalizedtothebrane.ItwasthefactthatarestrictiontonormalizedmodesdidleadtogravitationalfluctuationmodeswhichwerelocalizedtothebranewhichpromptedtheAdS+4studyof[7].17
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