Real-Time Perturbation Theory in de Sitter Space

arXiv:hep-th/0308135v33Dec2003BROWN-HET-1375Real-TimePerturbationTheoryindeSitterSpaeKevinGoldstein∗andDavidA.Lowe†PhysisDepartment,BrownUniversity,Providene,R.I.02912,USAAbstratWeonsidersalareldtheoryindeSitterspaewithageneralvauuminvariantundertheontinuouslyonnetedsymmetriesofthedeSittergroup.Webeginbyreviewingapproahestodenethisasaperturbativequantumeldtheory.OneapproahleadstoFeynmandiagramswithpinhsingularitiesinthegeneralase,whihrendersthetheoryperturbativelyill-dened.Anotherapproahleadstowell-denedperturbativeorrelationfuntionsontheimaginarytimeontinuationofdeSitterspae.Whenontinuedtoreal-time,apathintegralwithanon-loalationgeneratesthetime-orderedorrelators.Curiously,observablesbuiltoutofloalprodutsoftheeldsshownosignofthisnon-loality.Howeveroneoneouplestogravity,weshowaausaleetsareunavoidableandpresumablymakethetheoryill-dened.TheBunh-DaviesvauumstateistheuniquedeSitterinvariantstatethatavoidstheseproblems.∗Eletroniaddress:kevinhet.brown.edu†Eletroniaddress:lowebrown.edu1I.INTRODUCTIONTherehasbeenmuhreentdebateaboutwhetherquantumeldtheoryindeSitterspaehasauniquevauuminvariantunderalltheontinuouslyonnetedsymmetriesofthespae.Theresolutionofthisquestionisruialtotheunderstandingofpossibletrans-Plankianeetsonthepreditionsofination[1℄,andobservableeetstodaysuhasultrahighenergyosmirayprodution[2,3℄.Thesequestionsareallthemorepressinggivenreentexperimentalresultsonrminggeneralpreditionsofinationfortheosmimirowavebakground[4℄,andofsupernovaobservationsonsistentwithapositiveosmo-logialonstanttoday[5℄.Attheleveloffreeeldtheory,deSitterspaehasaone-omplexparameterfamilyofvaua,dubbedtheα-vaua[6,7,8,9℄.Itwasbeenarguedutoversionsoftheseanberelevantduringination,whereαparameterizestheeetsoftrans-Plankianphysis[10,11,12,13,14,15,16℄.Othershavearguedtheα-vauasuerfrominonsistenies[17,18,19,20℄oneinterationsareinluded,andthattheBunh-Davies/Eulideanvauumstateistheuniqueonsistentstate.Inthispaperwereviewexistingapproahestothisissue,andelaborateontheonnetionsbetweenthem.Themoststraightforwardapproah,whereonetreatsthevauumstateasasqueezedstatefailsduetotheappearaneofpinhsingularities,whihrenderstheperturbationtheoryill-dened[18℄.Weemphasizethisisnotaproblemwiththeultra-violetstrutureofthetheory,butratherFeynmanintegralsbeomeill-denedwhenpropagatorsoninternallinesarenullseparated.Apotentiallymorepromisingapproahbasedonanimaginarytimeformulation[21℄leadstoasensibleperturbationtheory,andpropagatorsthatagreewiththeimaginarytimeontinuationsofthefreepropagatorsof[8,9℄.Thisperturbativeexpansionanbeontinuedtoreal-timeandwrittenintermsofapathintegralwithanon-loalkinetiterm,butloalpotentialandsoureterms.Forthepuresalareldtheory,thealgebraofobservablesbuiltoutofloalprodutsoftheeldsremainsloal.Howeveronethetheoryisoupledtogravitytheaausalitybeomesunavoidableandpresumablyrendersthetheoryill-dened,inkeepingwiththehronologyprotetiononjeture[22℄.2II.FREEPROPAGATORToestablishnotation,webeginbyreviewingtheresultsof[6,7,8,9℄forthefreevauaindeSitterspae,invariantundertheelementsofthedeSittergroupontinuouslyonnetedtotheidentity.Fieldsmaybedeomposedasmodesumsφ(x)=XnφEn(x)an+φE∗n(x)a†n=Xnφαn(x)bn+φα∗n(x)b†n.OnethendenestheBunh-Davies/Eulideanvauumasan|Ei=0andtheα-vauaasbn|αi=0.(1)Therespetivemodefuntionsarerelatedasφαn=Nα(φEn+eαφE∗n)φEn=Nα(φαn−eαφα∗n),(2)withNα=1/p1−exp(α+α∗).ThereationandannihilationoperatorsarethenrelatedbyamodenumberindependentBogoliubovtransformationbn=Nα(an−eα∗a†n).(3)Asshownin[9℄weanhoosemodefuntionssothatφEn(x)∗=φEn(¯x)(4)with¯xtheanti-podalpointtox.Thesemodefuntionsarenormalizedwithrespettothenorm(φ1,φ2)=iZΣ(φ∗1∂μφ2−φ2∂μφ∗1)dΣμwithΣanyCauhysurfae.3TheWightmanfuntionisGα(x,y)=hα|φ(x)φ(y)|αi=Pnφαn(x)φα∗n(y)=N2αGE(x,y)+eαGE(¯x,y)+eα∗GE(x,¯y)+|eα|2GE(¯x,¯y)
.(5)Thestate|αianbethoughtofasasqueezedstatewithrespettotheEulideanvauum|αi=U|EiwiththeunitaryoperatorUdenedasU=expXnβaE†n
2−β∗aEn
2!,β=14logtanh−Reα2e−iImα.Itisthennaturaltoonstrut[21℄˜φ(x)=U†φ(x)U=NαXnφEn(x)+eαφEn(¯x)
an+φEn(x)+eαφEn(¯x)
∗a†n(6)whihsuggeststhatfromtheEulideanvauumviewpoint,reationofapartileintheα-vauumanbethoughtofasreatingapartilewithrespettotheEulideanvauumatxtogetherwithapartileattheanti-podalpoint¯x.A.Real-timeorderingHavingdisussedtheWightmanfuntions,wenowneedtodisussmorearefullytime-orderingpresriptions.FirstletusrepresentdeSitterspaeasahyperboloidinatR5withmetriηab=diag(−1,1,1,1,1)andoordinatesXawitha=1···5XaXbηab=H−2.Following[9℄wedenethesignedgeodesidistanebetweenpointsas˜d(x,y)=H−1arccos˜Z(x,y)where˜Z(x,y)=H2ηabXa(x)Xb(y)+iǫ,ifxtothefutureofyH2ηabXa(x)Xb(y)−iǫ,ifxtothepastofy.4Withthisdenition˜d(x,y)=−˜d(¯x,¯y).NotthatalthoughonlypointswithZ≥−1areonnetedbygeodesis,˜d(x,y)anbedenedbyanalytiontinuationforZ−1.TheEulideanvauumWightmanfuntionisgivenbyGE(x,y)=c2F1(h+,h−;2;1+˜Z2)(7)whereh±≡32±iμμ≡sm2−3H22c≡Γ(h+)Γ(h−)(4π)2.Unlessotherwisestated,weonsidertheasem3H/2inthispaper.Someofthepropertiesofthisfuntionareasfollows:•apolewhenpointsoinide(˜Z=1)•abranhutrunningalong˜Z=(1,∞),wheretheimaginaryparthangess