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Quantum Gravitational Corrections to Vacuum Polarization During Inflation

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Title:
Quantum Gravitational Corrections to Vacuum Polarization During Inflation
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1 online resource (143 p.)
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english
Creator:
Leonard, Katie E
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University of Florida
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Gainesville, Fla.
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
Woodard, Richard P
Committee Members:
Sikivie, Pierre
Ramond, Pierre
Detweiler, Steven L
Tan, Jonathan Charles

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Subjects / Keywords:
graviton -- inflation -- polarization -- vacuum
Physics -- Dissertations, Academic -- UF
Genre:
Physics thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Abstract:
I compute the one-loop vacuum polarization from gravitons in a locally de Sitter background. Via dimensional regularization I obtain a fully renormalized result in the BPHZ sense. The inevitable application of this result will be to quantum correct Maxwell's equation; therefore, the Hartree approximation is presented here in anticipation of that result. As supplementary material I reproduce my work on the flat space equivalent of this problem, a discussion on covariant and noncovariant representations, and a formalism for converting between formalisms.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Katie E Leonard.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Woodard, Richard P.

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UFRGP
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Applicable rights reserved.
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lcc - LD1780 2013
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MISSING IMAGE

Material Information

Title:
Quantum Gravitational Corrections to Vacuum Polarization During Inflation
Physical Description:
1 online resource (143 p.)
Language:
english
Creator:
Leonard, Katie E
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
Woodard, Richard P
Committee Members:
Sikivie, Pierre
Ramond, Pierre
Detweiler, Steven L
Tan, Jonathan Charles

Subjects

Subjects / Keywords:
graviton -- inflation -- polarization -- vacuum
Physics -- Dissertations, Academic -- UF
Genre:
Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
I compute the one-loop vacuum polarization from gravitons in a locally de Sitter background. Via dimensional regularization I obtain a fully renormalized result in the BPHZ sense. The inevitable application of this result will be to quantum correct Maxwell's equation; therefore, the Hartree approximation is presented here in anticipation of that result. As supplementary material I reproduce my work on the flat space equivalent of this problem, a discussion on covariant and noncovariant representations, and a formalism for converting between formalisms.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Katie E Leonard.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Woodard, Richard P.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045836:00001


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QUANTUMGRAVITATIONALCORRECTIONSTOVACUUMPOLARIZATIONDURINGINFLATIONByKATIEELIZABETHLEONARDADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013KatieElizabethLeonard 2

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Tomyparents 3

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ACKNOWLEDGMENTS Iwouldrstliketothankmyadvisor,ProfessorRichardWoodard,thetrueinventoroftheso-calledBesselfunctions.Hisunendingpatienceandgenuineconcernforstudents,impressiveintellect,andstrongworkethicattributetoaremarkablecharacterthatIstrivetoemulate.ThroughmyyearsasagraduatestudentIhaveconsideredmyselfincreasinglyluckytohavehimasanadvisor.Hefullysupportedallofmyacademicinterestsandwasfundamentalinorchestratingmyresearchabroad;Icouldneverhavecompletedthisprojectwithouthim.TherearemanypeopleIwouldliketothankfromanacademicperspective.MostimportantlyProfessorsTomislavProkopecandL.RaulAbramo,bothofwhomIconsiderhonoraryadvisors.Whilevisitingtheirrespectiveinstitutionstheywenttogreatpersonallengthstoensuremyvisitwasenjoyable.Theyareextraordinaryteachersandresearchers,andIamforeverindebtedtothemfortheircontributionstomyacademicsuccess.AlsoProfessorsLawrenceRudnick,LiliyaWilliams,andMarcoPelosofortheirguidanceduringmyundergraduate;myfellowmentees:Prof.EmreKahya,Prof.Shun-PeiMiao,Dr.SohyunPark,andPedroMora;andmycommitteemembers:ProfessorsStevenDetweiler,JonathanTan,PierreSikivie,andPierreRamond.OnapersonalnoteIwouldliketothankmydearestfriends:ChrisMueller,ErinKorbel,SierraBurris,TessaVernstrom,Jimmy,Dr.GaurabSarangi,andDr.ManojSrivastava.Icherishtheirfriendship,andtheiradviceandhelphavesavedmecountlesstimes.Lastly,Iwouldliketothankmyparents,brothers,andgrandparentsfortheirneverendingsupportandlove.IfitwerenotfortheirencouragementImayneverhavebeeninspiredtoenterphysics[ 1 2 ].Thankyouall. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 10 1.1Ination ..................................... 10 1.2UncertaintyPrincipleDuringInation ..................... 11 1.3ConformalInvariance ............................. 12 1.4ParticleProductionDuringInation ...................... 14 1.5UsingQuantumGravity ............................ 16 2THEFLATSPACEEQUIVALENT .......................... 19 2.1OneLoopVacuumPolarization ........................ 19 2.1.1FeynmanRules ............................. 20 2.1.2DimensionallyRegulatedResult .................... 22 2.1.3Renormalization ............................ 23 2.2QuantumCorrectedMaxwellEquations ................... 24 2.2.1Schwinger-KeldyshFormalism .................... 25 2.2.2Photons ................................. 29 2.2.3InstantaneouslyCreatingAPointDipole ............... 31 2.2.4AnAlternatingPointDipole ...................... 35 2.2.5AStaticPointCharge .......................... 36 2.3GaugeDependence .............................. 38 2.3.1GeneralConsiderations ........................ 38 2.3.2GeneralCovariantGauges ...................... 39 2.3.3GaugeDependentProportionalityConstant ............. 41 2.4Discussion ................................... 44 3COVARIANTVACUUMPOLARIZATIONSONDESITTERBACKGROUND .. 47 3.1AMassless,MinimallyCoupledScalar .................... 50 3.1.1ThePrimitiveResult .......................... 50 3.1.2GeneralRepresentations ....................... 52 3.1.3RepresentationforthisSystem .................... 55 3.2AMassiveScalar ................................ 58 3.2.1ThePrimitiveResult .......................... 58 3.2.2The(y)Representation ........................ 60 5

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3.2.3Renormalization ............................ 62 3.3UsingthedeSitterInvariantForm ...................... 65 3.3.1AParadoxfromtheEffectiveFieldEquations ............ 65 3.3.2ResolutionusingGreen's2ndIdentity ................ 68 3.4Discussion ................................... 71 4REPRESENTINGTHEVACUUMPOLARIZATIONONDESITTER ....... 75 4.1KnownResultsonTensorStructures ..................... 78 4.2TheDerivation ................................. 80 4.3Correspondence ................................ 82 4.4Discussion ................................... 85 5GRAVITONCORRECTIONSTOVACUUMPOLARIZATIONDURINGINFLATION 87 5.1FeynmanRules ................................. 89 5.1.1OurdeSitterBackground ....................... 90 5.1.2OurPrimitiveDiagrams ........................ 91 5.1.3OurPropagators ............................ 93 5.1.4OurStructureFunctions ........................ 97 5.1.5OurCounterterms ........................... 98 5.2The4-PointContribution ............................ 101 5.2.1NaiveContractions ........................... 101 5.2.2SubstitutionofGravitonPropagator .................. 102 5.2.3Findingthe4-PointStructureFunctions ............... 103 5.3The3-PointContribution ............................ 107 5.3.1Naivecontractions ........................... 107 5.3.2GravitonandPhotonPropagatorSubstitutions ............ 108 5.3.3Findingthe3-PointStructureFunctions ............... 113 5.4Renormalization ................................ 119 5.4.1ConvertingtoFunctionsofx ..................... 120 5.4.2FindingtheFiniteandDivergentPartsofFandG .......... 121 5.4.3FullResult ................................ 124 5.5HartreeApproximation ............................. 125 5.6Discussion ................................... 128 6CONCLUSION .................................... 133 REFERENCES ....................................... 135 BIOGRAPHICALSKETCH ................................ 143 6

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LISTOFTABLES Table page 2-1Coefcientfunctionsandtensordifferentialoperators ............... 46 2-2CoefcientsAI,BIandCI .............................. 46 3-1ResultforexpandingDD0(@@0[y)(@0]@y)f1(y,u,v). ............. 73 3-2ResultforexpandingDD0(@[y)(@]@0[y)(@0]y)f2(y,u,v). .......... 74 5-1Partsof4-pointvertexfunction ........................... 130 5-2Termsof4-pointcontributionafternaiveindexcontractions. ........... 130 5-3Variousgravitontensorfactorcontractions. .................... 130 5-44-pointcontributionscomingfromtheconformalpartofthegravitonpropagator. ............................................. 131 5-54-pointcontributionscomingfromtheA-typepartofthegravitonpropagator. 131 5-64-pointcontributionscomingfromtheC-typepartofthegravitonpropagator. 131 5-7Piecesof3-pointvertexfunction. ......................... 132 5-83-pointresultsafternaiveindexcontractions. ................... 132 7

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LISTOFFIGURES Figure page 2-1Gravitoncontributionstotheoneloopvacuumpolarization. ........... 45 2-2Vertexcorrection. ................................... 46 3-1Oneloopcontributiontothevacuumpolarizationfromthe4-point(seagull)interaction. ...................................... 72 3-2Oneloopcontributiontothevacuumpolarizationfromthe3-pointinteraction. 73 3-3Photoneldstrengthrenormalizationcounterterm. ................ 73 5-1Gravitoncontributionstotheoneloopvacuumpolarization. ........... 130 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyQUANTUMGRAVITATIONALCORRECTIONSTOVACUUMPOLARIZATIONDURINGINFLATIONByKatieElizabethLeonardAugust2013Chair:RichardP.WoodardMajor:PhysicsIcomputetheone-loopvacuumpolarizationfromgravitonsinalocallydeSitterbackground.ViadimensionalregularizationIobtainafullyrenormalizedresultintheBPHZsense.TheinevitableapplicationofthisresultwillbetoquantumcorrectMaxwell'sequation;therefore,theHartreeapproximationispresentedhereinanticipationofthatresult.AssupplementarymaterialIreproducemyworkontheatspaceequivalentofthisproblem,adiscussiononcovariantandnoncovariantrepresentations,andaformalismforconvertingbetweenformalisms. 9

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CHAPTER1INTRODUCTIONMyresearchhasfocusedonhowquantumgravityaffectsvacuumpolarizationintheregimeofprimordialination,whichcanbemodeledasalocallydeSitterbackgroundgeometry.Thischapterwilldiscussthekeyissuesfundamentaltosuchresearch.Iwillreviewthebasicdenitionofaninatinguniverse,howtheuncertaintyprincipleismodiedduringination,conformalinvariance,particleproductionduringination,andtheultimateissueofusinganincompletetheoryofquantumgravity.Iendthischapterwithanoverviewofmyresearchdescribedinsubsequentchapters. 1.1InationThemostbasicdenitionofaninatinguniverseissimplyoneinwhichthereisacceleratedexpansionofsaiduniverse.Primordialinationreferstoaperiodofextremeacceleratedexpansionwhentheuniversewasveryyoung,approximately1030)]TJ /F10 7.97 Tf 6.58 0 Td[(33secondsold,ascurrentdataandtheorysuggest[ 3 4 ].Toputthisinmathematicaltermswemustconsiderthephysicalrealityoftheuniverseweliveinandspeculatewhatpropertiesitpossessedduringprimordialination,ifindeedsuchaperiodexisted.Oncosmologicalscalesouruniverseappearshomogeneousandisotropic;aswellashavingzerospatialcurvature[ 5 ].BasedontheseobservationswecandescribethebackgroundgeometryoftheuniverseusingtheFriedmann-Robertson-Walker(FRW)metric,withlineelement ds2=)]TJ /F5 11.955 Tf 9.3 0 Td[(dt2+a(t)2d~xd~x.(1)Heretisthephysicaltimeanda(t)isthescalefactorwhichconvertsEuclideancoordinatedistance,jj~x)]TJ /F3 11.955 Tf 11.54 .5 Td[(~yjj,tophysicaldistance,a(t)jj~x)]TJ /F3 11.955 Tf 11.53 .5 Td[(~yjj.Therearethreeimportantcosmologicalquantitiesthatcanbebuiltfroma(t);theyarethecosmologicalredshift,theHubbleparameter,andthedecelerationparameter: z(t)a0 a)]TJ /F4 11.955 Tf 11.96 0 Td[(1,H(t)_a a,q(t))]TJ /F5 11.955 Tf 23.11 8.09 Td[(aa _a2=)]TJ /F4 11.955 Tf 9.3 0 Td[(1)]TJ /F4 11.955 Tf 18.58 10.75 Td[(_H H2,(1) 10

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wherea0isthescalefactorevaluatedatthecurrenttimet0.TheHubbleparametergivestheexpansionrateoftheuniverse;therefore,aninatinguniverseisproducedbyanegativedecelerationparameterandapositiveHubbleparameter.Observationsindicatethatouruniverseiscurrentlyundergoingaperiodofination.PlanckmeasuresthecurrentHubbleparametertobeH0'67.3km=s Mpc(ref)andobservationsofTypeIasupernovaendthecurrentdecelerationparametertobeq0)]TJ /F4 11.955 Tf 22.17 0 Td[(0.6[ 7 8 ],whichagreeswithauniversemadeupofabout30%matterand70%vacuumenergy.Intheregimeofprimordialination,forwhichIaminterested,theHubbleparameterispredictedtohavebeenmuchgreater,HI'31056km=s Mpc,andadecelerationparameteronlyinnitesimallygreaterthan-1.InthenextsectionsIwillshowhowthisexaggeratedexpansiongivesrisetosomeratheruniqueprocessesandsurprisingcircumstancesthatmakecalculationsintheearlyuniversesointeresting.ItshouldbenotedthathenceforthwheneverIrefertoinationIamimplyingprimordialination. 1.2UncertaintyPrincipleDuringInationTheenergy-timeuncertaintyprincipleiscommonlywritteninnaturalunitsasEt>1.ThismeanstoresolveachangeinenergyEwehavetowaitatleastatimet;thisthenrestrictshowlongapairofvirtualparticlesmayexist.Statedanotherway,inordertonotseepaircreationoftwoparticleswithenergyEtheinequalitymustbeinverted 2Et<1.(1)ItisobviousthatanythingthatdecreasesEwillincreaset.Inatspacethisimpliesthatvirtualparticlescanonlyliveforaniteamountoftime,andmasslessparticlescanlivelongerthanmassiveparticleswiththesamemomentumbecauseE(~k)=q m2+jj~kjj2.Duringinationphotonsarestilllabeledbytheirwavenumber~k,whichisinverselyrelatedtocoordinatewavelength.Obtainingaphysicalwavenumberrequiresmultiplication 11

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bytheinversescalefactor,~k=a(t),andEquation( 1 )thenbecomesanintegral Zt+ttdt02E(t0,~k)<1,(1)wherenowE(t,~k)=q m2+jj~kjj2=a2.FormasslessparticlesEquation( 1 )simpliesto2jj~kjjZt+ttdt0 a(t0)<1.Thisintegralwillconvergeforanyscalefactorthatgrowsfastenough,evenasttendstoinnity.LetusconsidertheparticularexampleofadeSitteruniverseforwhichthescalefactor1isdenedasa(t)=aIeHt.Solvingtheintegralonends 2jj~kjj Ha(t))]TJ /F4 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(e)]TJ /F6 7.97 Tf 6.59 0 Td[(Ht<1,(1)whichleadstotheextraordinaryconclusionthatmasslessvirtualparticlescanliveforeverduringinationiftheyemergewithjj~kjj
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whereFistheelectromagneticeldstrengthdenedintheunusualway,F@A)]TJ /F3 11.955 Tf 12.04 0 Td[(@A,andgisthemetric(wedonotneedtobemorespecicaboutitsformfornow).Theconformaltransformationisdenedbysettingg0=2(x)gandA0=A;appliedto( 1 )wend L0=FF)]TJ /F10 7.97 Tf 6.58 0 Td[(2g)]TJ /F10 7.97 Tf 6.59 0 Td[(2gDp )]TJ /F5 11.955 Tf 9.3 0 Td[(g=D)]TJ /F10 7.97 Tf 6.59 0 Td[(4L.(1)Therefore,inD=4dimensions,theelectromagneticLagrangiandensityisconformallyinvariant.Howdoesconformalinvarianceaffectvirtualparticlesemergingduringination?Toanswerthiswemustlookattherateatwhichtheyemergefromthevacuum.Itisusefultorewrite( 1 )nowinconformalcoordinates ds2=a2(t)()]TJ /F5 11.955 Tf 9.3 0 Td[(d2+d~xd~x),(1)whered=dt=a(t),tisstillthephysicaltime,andistheconfromaltime.Inconformalcoordinates,conformallyinvarianttheoriesarelocallyidenticaltoatspacetheories.Thustherateatwhichvirtualparticlesemergefromthevacuuminconformaltimemustbethesamerate()]TJ /F1 11.955 Tf 6.77 0 Td[()atwhichtheyemergefromthevacuuminatspace.Hence,theemergencerateinphysicaltimeis dN dt=dN dd dt=)]TJ ET q .478 w 263.48 -451.27 m 284.78 -451.27 l S Q BT /F5 11.955 Tf 263.48 -462.46 Td[(a(t).(1)Itisnowclearthatthisconformalinvariancemanifestsinsuppressingtherateofemergencebyafactorof1=a(whichrecallisextremelysmallduringination).Thepointbeing,thatalthoughtheuncertaintyprincipleallowsmasslessvirtualparticlestoliveforever,itisnotlikelytheywillemergefromthevacuumatall.Luckilyforustherearetwoexceptionstothis:gravitonsandmasslessminimallycoupled(MMC)scalars,neitherofwhichareconformallyinvariant. 13

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1.4ParticleProductionDuringInationAlthoughmycalculationconcernsgravitions,IwilluseaMMCscalartondtherateofparticleproductionduringinationfornonconformallyinvariantparticles.ThetwosenarioswereshowntobeequivalentbyGrishchuck[ 9 ].WebeginwiththeLagrangiandensityofaMMCscalar LMMC=)]TJ /F4 11.955 Tf 10.49 8.08 Td[(1 2@@gp )]TJ /F5 11.955 Tf 9.3 0 Td[(g.(1)Usingg=a(t),whereisthetime-likeatspacemetric,andintegratingtondtheLagrangianwehave LZd3xL=1 2a3Zd3x_(t,~x)_(t,~x))]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2aZd3x~r~r.(1)ToputthisinmomentumspaceweapplyaFouriertransformEquationusingParseval'stheorem Z1dDxf(~x)g(~x)=Z1dDk (2)D~f(k)~g()]TJ /F5 11.955 Tf 9.3 0 Td[(k),(1)andtherealitycondition~f()]TJ /F5 11.955 Tf 9.3 0 Td[(k)=~f(k),iff(~x)isreal,revealing L=1 2a3Zd3k (2)3_~(t,~k)_~(t,~k))]TJ /F4 11.955 Tf 13.15 8.08 Td[(1 2aZd3k (2)3k2~(t,~k)~(t,~k).(1)Thereisnocouplingbetweendifferent~k'ssowecanconsidereachmodeindependently.Ifwechooseonemodedescribedbyaspecic~kandcallitq(t)wecanrewriteEquation( 1 )as L=1 2a3_q2)]TJ /F4 11.955 Tf 13.16 8.09 Td[(1 2ak2q2,(1)whichcanimmediatelyberecongnizedastheequationforaharmonicoscillatorwithm(t)=a3(t)and!(t)=k=a(t).Sincemand!aretimedependenttherearenostationarystates,butwecanstillwritetheenergyofeachmodeintheformofaHamiltonian H=_q@L @_q)]TJ /F5 11.955 Tf 11.95 0 Td[(L=1 2m(t)_q2+1 2!2(t)q2,(1) 14

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withequationofmotion q+3H_q+k2 a2q=0.(1)Thisisnotaneasyequationtosolveforagenerala(t),butweareonlyinterestedindeSitterspaceandcanthereforeseta(t)=eHtand_H=0.Wethenndasolutionintheformofmodefunctions q(t)=u(t)+u(t)y,u(t,k)=H p 2k31)]TJ /F5 11.955 Tf 23.25 8.09 Td[(ik Ha(t)eik Ha(t),(1)whereyandaretheusualcreationandannihilationoperators,cononicallynormalizedsuchthat[,y]=1andji=0.ThestatejiisBunch-Daviesvacuum,denedtohaveminimumenergyinthedistantpast.TondthenumberofparticlesproducedwithwavenumberkweneedtotaketheexpectationvalueoftheenergyhjH(t)ji=1 2a3(t)hj_q2(t)ji+1 2a(t)k2hjq2(t)ji (1)=k a"1 2+Ha 2k2#k a1 2+\N(t)".Thenumberofnonconformallyinvariantvirtualparticlesproducedduringinationis N(t,k)=Ha(t) 2k2.(1)Thisisafantasticresultbecauseitcangrowverylargeforwavenumbersk
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recentbodyofworkconcerningone-loopvacuumpolarization;theeffectsinducedbyscalarshasbeenwelldocumented[ 12 ],butconsideringgravitonsiscompletelynew. 1.5UsingQuantumGravityQuantumgravityisnotperturbativelyrenormalizable[ 13 15 ]evenatonelooporder,andthisposesaproblemformakingreliablepredictionsinquantumgeneralrelativity(GR).Thefactthatperturbativegeneralrelativitydoesnotworkmeansthatwedonotpossessafulltheoryofquantumgravity.Thisisknownastheultravioletcompletionofquantumgravity.Inordertoxitweeitherneedanewtheoryofgravityorsomenewunperturbativetechniqueforquantumgravity.NonrenormalizabilitymayseemlikeanunavoidableobstaclethatwouldrenderanyquantumGRpredictionuseless,butallhopeisnotlost.Wecanstillmakereliablepredictionsaboutlongrangequantumgravitationalphenomenausingpertrubativegeneralrelativityasalowenergyeffectiveeldtheory.Sincemycalculationisconcernedwithquantumgravitationaleffectsoncosmologicalscales,theperturbativenonrenormalizabilityofquantumgeneralrelativitywillnotbeaproblem.Thesolutionisthattheultravioletcompletionmanifestsinourlackofknowledgeaboutthenitepartsofthelocalcounterterms.Butinthelongdistanceregimethesetermsaredominatedbyoneloopeffectsfrommasslessparticles,sowedon'tneedtoknowtheniteparts.Morespecically,theBogoliubov,Parasiuk,HeppandZimmerman(BPHZ)theoremallowsustoconstructlocalcountertermsthatareneededtoabsorbtheultravioletdivergencesofanyquantumeldtheoryatanyxedorderintheloopexpansion[ 16 ].Thisincludesquantumgeneralrelativity+quantumelectrodynamics(QED)andmycounterterms.However,inquantumgeneralrelativitythesecountertermsarenotpresentintheoriginalLagrangianandneedtobeputinbyhand.Wecantreatthesecountertermsperturbativelyandregardthemasproxiesofthefulltheorythatisunknown. 16

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Theproblemarisesthatwestilldonotknowtheirniteparts.Thedivergentpartswecanxbydemandingthattheyexactlycancelthedivergencesarisingfromloopcorrections.Butwehavenothingtoxtheniteparts,andtheycantakeonanyarbitraryvalue,whichcanaffectphysicalresults.Therefore,theonlyreliablepredictionswecanmakearewhenthenitepartsareunimportant.Suchpredictionsexistbecauseoftwokeyfacts: 1. BPHZcountertermsarelocalcorrectionstotheeffectiveeldtheory,and 2. masslessparticlesmakenonlocalcorrectionstotheeffectiveeldequations.Toseethisexplicitly,considerthecorrectionstothequantumgravityeffectiveactionquadradicinA(Forsimplicitywewillworkinaatspacebackgroundanddroptheindices).Thentheoneloopcountertermslooklike )]TJ /F10 7.97 Tf 6.77 5.44 Td[(1loopctermsZd4x@2A@2A.(1)Wedonotknowthenumericalcoefcientsoftheseterms.Ontheotherhand,theoneloopeffectsfrommasslessparticleslookslike )]TJ /F10 7.97 Tf 6.77 5.45 Td[(1loopniteZd4x@2Aln()]TJ /F3 11.955 Tf 9.3 0 Td[(@2)@2A.(1)Wedoknowthecoefcientsofthesetermsfromquantumgeneralrelativity.Itisclearthat( 1 )willdominate( 1 )inthelongdistanceregime,since( 1 )hasaln()]TJ /F3 11.955 Tf 9.3 0 Td[(@2)enhancementthatdivergesintheinfrared.Inmomentumspace@2!)]TJ /F5 11.955 Tf 24.69 0 Td[(p2,andinthelongwavelengthregimep20.Thenthelocalcountertermeffectsgolikep4,andthenonlocalprimitiveeffectsgolikep4ln(p2).Forsmallenoughp2thenonlocaleffectswillalwaysbethedominantterms.Inatspaceclassicalresultsgolikep2,andquantumeffectsarenegligible,butindeSitterspacequantumeffectsbecomemuchlarger.Myresearchexploitsthepoweroflowenergyeffectiveeldtheory,butindeSitterspace.Mostimportantly,workinginthelowenergyregime,thenitequantumcorrectionsfromtheoriginalLagrangiancanbe 17

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distinguishedfromthelocalcounterterms,andthusreliablepredictionscanbemade.Itisexcitingthatthesequantumeffectsbecomesignicantduringinationthankstotheirenhancednumberdensityandcosmologicalwavelengths.Moreover,theseeffectscanstillbecalculatedreliablywithoutknowingtheultravioletcompletionofquantumgravity.Theseresultswillalwaysbevalid,evenifacompletetheoryofquantumgravityisfound.Theideaofusinganeffectiveeldtheorytoderivevalidquantumeffectsinthelongdistanceregimehasbeenusedextensively.BlockandNordsieckin1937famouslyshowedthattheinfrareddivergenceproblemcouldberesolvedregardlessofultravioletdivergences[ 17 ].Weinbergsolvedthegravitationalinfrareddivergenceproblemonatspacebackgroundwithoutasatisfactoryquantumtheoryofgravity[ 18 ].AndrecentlyDonoghuehasappliedtheprinciplesoflowenergyeffectiveeldtheorytocomputegravitoncorrectionstothelongrangegravitationalforce[ 19 20 ].Furtherexamplesabound[ 21 ]provingthatalthoughmakingpredictionswithoutacompletetheoryofquantumgravitymayseemappallingatrst,itisactuallyquitesound. 18

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CHAPTER2THEFLATSPACEEQUIVALENTAsapreludetothefulldeSitterspacecalculationIwillreviewtheatspaceequivalent.Thisworkservedasauseful,andcertainlyeasier,exampleofwhatthedeSitterspacecalculationwouldentail;Ireproduceithereforthesamereasonsofenlightenment.Detailsofthecalculationaredrawnfrom[ 22 ].Whetherduringprimordialinationoronatspacebackground,thepropervehicleforstudyingquantumdistortionsofelectrodynamicsisthequantum-correctedMaxwellequation.Onegetsthisbyrstcomputingthevacuumpolarizationi[](x;x0),whichistheone-particle-irreducible(1PI)2-pointfunctionforthephoton.Thisisthenusedtoquantum-correctMaxwell'sequation, @hp )]TJ /F5 11.955 Tf 9.3 0 Td[(gggF(x)i+Zd4x0hi(x;x0)A(x0)=J(x).(2)Thisframeworkhasbeenemployedtoinfertheeffectsofinationarychargedscalarproductiononphotons[ 12 ],andonelectrodynamicforces[ 51 ].Ourgoalwastofacilitateasimilarstudyoftheeffectsofinationarygravitonproductionintheatspacecorrespondencelimit.Thequantumgravitationalcontributiontotheoneloopvacuumpolarizationisderivedinsection 2.1 .Section 2.2 solvesthequantum-correctedMaxwellequation( 2 )forphotons,theinstantaneouscreationofapointdipole,analternatingpointdipole,andastaticpointcharge.Theissueofgaugedependenceisdiscussedinsection 2.3 ,andsomenalremarksonthesubjectcomprisesection 2.4 2.1OneLoopVacuumPolarizationThepurposeofthissectionistocomputetherenormalized,oneloopcontributiontothevacuumpolarizationfromquantumgravityonatspacebackground.WeusetheFeynmanrulespresentedinsection 2.1.1 tocomputethedimensionallyregulatedresult.Byaprocessofsuccessivepartialintegrationsthisisexpressedasadivergent, 19

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localtermwhichiscanceledbyaBPHZcountertermplusthenite,nonlocalcontributionwhichconstitutestherenormalizedresult. 2.1.1FeynmanRulesOurtotalLagrangiancontainsthreeparts, L=LGR+LEM+LBPHZ.(2)Theseare,respectively,theLagrangiansofgeneralrelativity,electromagnetismandtheBPHZcountertermrequiredforthiscomputation, LGR=1 16GRp )]TJ /F5 11.955 Tf 9.3 0 Td[(g, (2) LEM=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 4FFggp )]TJ /F5 11.955 Tf 9.3 0 Td[(g, (2) LBPHZ=C4DFDFgggp )]TJ /F5 11.955 Tf 9.3 0 Td[(g. (2) WeemployaD-dimensional,spacelikemetricg,withinverseganddeterminantg=det(g).OurafneconnectionandRiemanntensorare, )]TJ /F7 7.97 Tf 6.77 4.93 Td[(1 2gh@g+@g)]TJ /F3 11.955 Tf 9.96 0 Td[(@gi, (2) R@)]TJ /F7 7.97 Tf 6.77 4.94 Td[()]TJ /F3 11.955 Tf 9.97 0 Td[(@)]TJ /F7 7.97 Tf 6.77 4.94 Td[(+)]TJ /F7 7.97 Tf 16.74 4.94 Td[()]TJ /F7 7.97 Tf 6.78 4.94 Td[()]TJ /F4 11.955 Tf 9.96 0 Td[()]TJ /F7 7.97 Tf 6.78 4.94 Td[()]TJ /F7 7.97 Tf 6.78 4.94 Td[(. (2) OurRiccitensorisRRandtheassociatedRicciscalarisRgR.Theelectromagneticeldstrengthtensoranditsrstcovariantderivativeare, F@A)]TJ /F3 11.955 Tf 9.96 0 Td[(@A, (2) DF@F)]TJ /F4 11.955 Tf 9.96 0 Td[()]TJ /F7 7.97 Tf 6.78 4.94 Td[(F)]TJ /F4 11.955 Tf 9.96 0 Td[()]TJ /F7 7.97 Tf 6.78 4.94 Td[(F. (2) Wedenethegravitoneldh(x)asthedifferencebetweenthefullmetricanditsMinkowskibackgroundvalue, g(x)+h(x),(2) 20

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where216Gistheloopcountingparameterofquantumgravity.Wefollowtheusualconventionswherebyacommadenotesordinarydifferentiation,thetraceofthegravitoneldishh,andgravitonindicesareraisedandloweredusingtheMinkowskimetric,hhandhh.AfterextractingasurfacetermthegravitationalLagrangiancanbewrittenas, LGR)]TJ /F4 11.955 Tf 11.95 0 Td[(Surface=p )]TJ /F5 11.955 Tf 9.3 0 Td[(gggg(1 2h,h,)]TJ /F4 11.955 Tf 11.16 8.09 Td[(1 2h,h,+1 4h,h,)]TJ /F4 11.955 Tf 11.16 8.09 Td[(1 4h,h,). (2) ThequadraticpartoftheinvariantLagrangianis, L(2)GR=1 2h,h,)]TJ /F4 11.955 Tf 11.16 8.09 Td[(1 2h,h,+1 4h,h,)]TJ /F4 11.955 Tf 11.16 8.09 Td[(1 4h,h,.(2)Wexthegaugebyadding, LGRx=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 2FF,Fh,)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2h,.(2)Theresultinggravitonpropagatorcanbeexpressedintermsofthemasslessscalarpropagatori(x;x0), ihi(x;x0)=h2())]TJ /F4 11.955 Tf 21.37 8.09 Td[(2 D)]TJ /F4 11.955 Tf 9.97 0 Td[(2ii(x;x0).(2)ThespacetimedependenceofthescalarpropagatorderivesfromtheLorentzintervalx2(x;x0), x2(x;x0)~x)]TJ /F3 11.955 Tf 9.43 .5 Td[(~x02)]TJ /F9 11.955 Tf 9.96 13.27 Td[(jt)]TJ /F5 11.955 Tf 9.96 0 Td[(t0j)]TJ /F5 11.955 Tf 13.95 0 Td[(i"2=)i(x;x0)=\(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1) 4D 21 x2D 2)]TJ /F10 7.97 Tf 6.59 0 Td[(1.(2)Thequadraticpartoftheelectromagneticactionis, LEM=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 2@A@A+1 2(@A)2.(2)Wexthegaugebyadding, LEMx=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(1 2(@A)2.(2) 21

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Theassociatedphotonpropagatoris, ihi(x;x0)=i(x;x0).(2)Electromagneticinteractionverticesdescendfromthesecondvariationalderivativeoftheaction, 2SEM A(x)A(x0)=@(p )]TJ /F5 11.955 Tf 9.3 0 Td[(g(x)hg(x)g(x))]TJ /F5 11.955 Tf 9.96 0 Td[(g(x)g(x)i@D(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0)).(2)Thenecessaryvertexfunctionsareobtainedbyexpandingthemetricfactors, p )]TJ /F5 11.955 Tf 9.3 0 Td[(ggg)]TJ /F5 11.955 Tf 9.96 0 Td[(gg)]TJ /F3 11.955 Tf 9.96 0 Td[(+Vh+2Uhh+O(3). (2) The3-pointand4-pointverticesare, V=[]+4)[][](, (2) U=h1 4)]TJ /F4 11.955 Tf 11.16 8.09 Td[(1 2()i[]+)[][](+)[][](+()[]()+()[]()+()()[]+()()[]+[]()()+[]()(). (2) Notethatparenthesizedindicesaresymmetrized,whereasindicesenclosedinsquarebracketsareantisymmetrized. 2.1.2DimensionallyRegulatedResultThethreeoneloopdiagramswhichcontributetothevacuumpolarizationaredepictedinFig. 2-1 .Theycaneachbeexpressedusingthenotationexplainedinsection 2.1.1 .Thelefthanddiagramis, ih3pti(x;x0)=(i)2@@0(Vihi(x;x0)V@@0ihi(x;x0)).(2) 22

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Substitutingexpressions( 2 ),( 2 )and( 2 ),actingtheinnerderivativesandperformingtheinnercontractionsgives, ih3pti(x;x0)=(i)2)]TJ /F10 7.97 Tf 6.77 4.34 Td[(2(D 2)]TJ /F4 11.955 Tf 9.97 0 Td[(1) 16D)]TJ /F4 11.955 Tf 21.25 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(3)D@@(2[)]TJ /F3 11.955 Tf 9.97 0 Td[(] x2D)]TJ /F10 7.97 Tf 6.58 0 Td[(2+D[xx)]TJ /F4 11.955 Tf 9.97 0 Td[(xx] x2D)]TJ /F5 11.955 Tf 13.15 8.09 Td[(D[xx)]TJ /F4 11.955 Tf 9.96 0 Td[(xx] x2D). (2) Thenextstepistoacttheouterderivatives,atwhichpointwecanextractamanifestlytransverseform, ih3pti(x;x0)=)]TJ /F3 11.955 Tf 10.49 8.79 Td[(2)]TJ /F10 7.97 Tf 6.78 4.34 Td[(2(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1) 16D(D)]TJ /F4 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F4 11.955 Tf 9.97 0 Td[(2)2D((D+1) x2D)]TJ /F4 11.955 Tf 11.15 8.09 Td[(2Dxx x2D+2), (2) =)]TJ /F3 11.955 Tf 10.49 8.78 Td[(2)]TJ /F10 7.97 Tf 6.78 4.34 Td[(2(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1) 16D(D)]TJ /F4 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)2D 2(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)h@2)]TJ /F3 11.955 Tf 9.97 0 Td[(@@i1 x2D)]TJ /F10 7.97 Tf 6.59 0 Td[(2. (2) ThemiddlediagramofFig.1is, ih4pti(x;x0)=i2@(Uihi(x;x)@D(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0)).(2)Thisdiagramvanishesbecausethecoincidencelimitofthemasslessscalarpropagatorinatspaceiszeroindimensionalregularization,i(x;x)=0.ThediagramontherightofFig.1is, ihctmi(x;x0)=i4C4@2)]TJ /F3 11.955 Tf 9.96 0 Td[(@@@2D(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0).(2) 2.1.3RenormalizationTorenormalize( 2 )wemustrstlocalizetheultravioletdivergencesothatitcanbesubtractedbythecounterterm( 2 ).Thisprocessoflocalizationisaccomplishedbyrstpartiallyintegratingthefactorof1=x2D)]TJ /F10 7.97 Tf 6.59 0 Td[(2in( 2 )untiltheremainderisintegrable[ 55 ].Indimensionalregularizationthestepsare[ 56 ], 1 x2D)]TJ /F10 7.97 Tf 6.59 0 Td[(2=@2 2(D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)21 x2D)]TJ /F10 7.97 Tf 6.59 0 Td[(4=@4 4(D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)2(D)]TJ /F4 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F4 11.955 Tf 9.96 0 Td[(4)1 x2D)]TJ /F10 7.97 Tf 6.59 0 Td[(6.(2) 23

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Nextweaddzerointheform[ 56 ], @21 xD)]TJ /F10 7.97 Tf 6.58 0 Td[(2=i4D 2 \(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1)D(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0).(2)Adding( 2 )tothekeypartof( 2 )inadimensionallyconsistentwaygives, @2 D)]TJ /F4 11.955 Tf 9.96 0 Td[(4(1 x2D)]TJ /F10 7.97 Tf 6.58 0 Td[(6)=i4D 2 \(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1)D)]TJ /F10 7.97 Tf 6.58 0 Td[(4D(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0) D)]TJ /F4 11.955 Tf 9.96 0 Td[(4+@2 D)]TJ /F4 11.955 Tf 9.97 0 Td[(4(1 x2D)]TJ /F10 7.97 Tf 6.58 0 Td[(6)]TJ /F3 11.955 Tf 15.15 8.09 Td[(D)]TJ /F10 7.97 Tf 6.58 0 Td[(4 xD)]TJ /F10 7.97 Tf 6.59 0 Td[(2), (2) =i4D 2 \(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1)D)]TJ /F10 7.97 Tf 6.58 0 Td[(4D(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0) D)]TJ /F4 11.955 Tf 9.96 0 Td[(4)]TJ /F3 11.955 Tf 13.15 8.09 Td[(@2 2(ln(2x2) x2)+O(D)]TJ /F4 11.955 Tf 9.96 0 Td[(4). (2) Substituting( 2 )and( 2 )into( 2 )resultsinthedesiredlocalizeddivergence, ih3pti(x;x0)=)]TJ /F5 11.955 Tf 10.5 8.78 Td[(i2\(D 2)]TJ /F4 11.955 Tf 9.97 0 Td[(1) 4D 2D 8(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)(D)]TJ /F4 11.955 Tf 9.96 0 Td[(4)h@2)]TJ /F3 11.955 Tf 9.96 0 Td[(@@i@2D(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0)+2 1924h@2)]TJ /F3 11.955 Tf 9.96 0 Td[(@@i@4(ln(2x2) x2)+O(D)]TJ /F4 11.955 Tf 9.97 0 Td[(4). (2) Thelocaldivergenceofexpression( 2 )willbecompletelycanceledbythecounterterm( 2 )ifwemakethechoice, C4=2\(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1) 16D 2D 8(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)(D)]TJ /F4 11.955 Tf 9.97 0 Td[(4).(2)Wecanthentaketheunregulatedlimit(D!4)toobtainthefullyrenormalizedgravitoncontributiontotheoneloopvacuumpolarization, hreni(x;x0)=)]TJ /F5 11.955 Tf 18.12 8.08 Td[(i2 1924h@2)]TJ /F3 11.955 Tf 9.96 0 Td[(@@i@4(ln(2x2) x2).(2)Notethattheambiguityregardingthenitepartofthecountertermisreectedinthedimensionalregularizationscale. 2.2QuantumCorrectedMaxwellEquationsThepurposeofthissectionistouseouroneresult( 2 )fortheoneloopvacuumpolarizationtoquantumcorrectMaxwell'sequations,andtheninferquantum 24

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gravitationalcorrectionstoelectrodynamicsbysolvingtheseequations.WebeginbyderivingthecausaleffectiveeldequationsoftheSchwinger-Keldyshformalism.Subsequentsubsectionssolvetheseequationsperturbativelyforthespecialcasesoffreephotons,apointdipolepulse,analternatingpointdipole,andastaticpointcharge. 2.2.1Schwinger-KeldyshFormalismWecomenowtothequestionofwhattouseforthevacuumpolarization[](x;x0)inthequantumcorrectedMaxwellequation( 2 ).Itmightseemobviousthatthein-outresult( 2 )wehavejustderivedshouldbeused,butthatwouldleadtotwoproblems: CausalityThein-outvacuumpolarization( 2 )isnonzeroforpointsx0whichlieinthefutureofx,oratspacelikeseparationfromit;and RealityThein-outvacuumpolarization( 2 )isnotreal.Onecangettherightresultforastaticpotentialbysimplyignoringtheimaginarypart[ 44 45 ],butcircumventingthelimitationsofthein-outformalismbecomesmoreandmoredifcultastimedependentsourcesandhigherordercorrectionsareincluded,andthesetechniquesbreakdownentirelyforthecaseofcosmologyinwhichtheremaynotevenbeasymptoticvacua.Notethatthereisnothingwrongwiththein-outvacuumpolarization( 2 );itisexactlytherightthingtocorrectthephotonpropagatorforasymptoticscatteringcomputationsinatspace.Thepointisratherthatemploying( 2 )inequation( 2 )failstoprovideasetofeldequationswiththesamescopeandpowerastheclassicalMaxwell'sequations.ThemoreappropriateeldequationsarethoseoftheSchwinger-Keldyshformalism.ThistechniqueprovidesawayofcomputingtrueexpectationvaluesthatisalmostassimpleastheFeynmandiagramswhichproducein-outmatrixelements[ 57 ].WeshalldeveloptheSchwinger-Keldyshrulesinthecontextofascalareld'(x)whoseLagrangian(thespaceintegralofitsLagrangiandensity)attimetisL['(t)].SupposewearegivenaHeisenbergstatejiwhosewavefunctionalintermsoftheoperatoreigenketsattimet0is['(t0)],andwewishtotaketheexpectationvalue, 25

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inthepresenceofthisstate,ofaproductoftwofunctionalsoftheeldoperator:A['],whichisanti-time-ordered,andB['],whichistime-ordered.TheSchwinger-Keldyshfunctionalintegralforthisis[ 36 ], DA[']B[']E=%&[d'+][d')]TJ /F4 11.955 Tf 6.26 1.8 Td[(]h')]TJ /F4 11.955 Tf 6.25 1.8 Td[((t1))]TJ /F3 11.955 Tf 9.97 0 Td[('+(t1)iA[')]TJ /F4 11.955 Tf 6.25 1.79 Td[(]B['+][')]TJ /F4 11.955 Tf 6.25 1.79 Td[((t0)]eiRt1t0dtnL['+(t)])]TJ /F6 7.97 Tf 6.59 0 Td[(L[')]TJ /F10 7.97 Tf 6.25 1.08 Td[((t)]o['+(t0)]. (2) Thetimet1>t0isarbitraryaslongasnooperatorineitherA[']orB[']isevaluatedatalatertime.TheSchwinger-Keldyshrulescanbereadofffromitsfunctionalrepresentation( 2 ).Becausethesameeldoperatorisrepresentedbytwodifferentdummyfunctionalvariables,'(x),theendpointsoflinescarryapolarity.Externallinesassociatedwiththeanti-time-orderedoperatorA[']havethe)]TJ /F1 11.955 Tf 12.62 0 Td[(polaritywhereasthoseassociatedwiththetime-orderedoperatorB[']havethe+polarity.Interactionverticesareeitherall+orall)]TJ /F1 11.955 Tf 9.3 0 Td[(.Verticeswith+polarityarethesameasintheusualFeynmanruleswhereasverticeswiththe)]TJ /F1 11.955 Tf 12.62 0 Td[(polarityhaveanadditionalminussign.Ifthestatejiissomethingotherthanfreevacuumthenitcontributesadditionalinteractionverticesontheinitialvaluesurface[ 58 ].Propagatorscanbe++,+)]TJ /F1 11.955 Tf 9.3 0 Td[(,)]TJ /F4 11.955 Tf 9.3 0 Td[(+,or\000.Allfourpolarityvariationscanbereadofffromthefundamentalrelation( 2 )whenthefreeLagrangianissubstitutedforthefullone.Itisusefultodenotecanonicalexpectationvaluesinthefreetheorywithasubscript0.Withthisconventionweseethatthe++propagatorisjusttheordinaryFeynmanpropagator, i++(x;x0)=DT'(x)'(x0)E0=i(x;x0),(2) 26

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whereTstandsfortime-orderingand Tdenotesanti-time-ordering.TheotherpolarityvariationsaresimpletoreadoffandtorelatetotheFeynmanpropagator, i)]TJ /F15 5.978 Tf 5.75 0 Td[(+(x;x0)=D'(x)'(x0)E0=(t)]TJ /F5 11.955 Tf 9.96 0 Td[(t0)i(x;x0)+(t0)]TJ /F5 11.955 Tf 9.96 0 Td[(t)hi(x;x0)i, (2) i+)]TJ /F4 11.955 Tf 6.25 1.8 Td[((x;x0)=D'(x0)'(x)E0=(t)]TJ /F5 11.955 Tf 9.96 0 Td[(t0)hi(x;x0)i+(t0)]TJ /F5 11.955 Tf 9.96 0 Td[(t)i(x;x0), (2) i\000(x;x0)=D T'(x)'(x0)E0=hi(x;x0)i. (2) Inourcase,boththephotonandthegravitonpropagatorsdependuponthemasslessscalarpropagator( 2 ),whichisafunctionoftheLorentzintervalx2(x;x0).Itfollowsfromrelations( 2 2 )thatthevariousSchwinger-KeldyshpropagatorscanbeobtainedbymakingsimplereplacementsfortheLorentzinterval, x2++(x;x0)~x)]TJ /F3 11.955 Tf 9.44 .5 Td[(~x02)]TJ /F5 11.955 Tf 11.95 0 Td[(c2jt)]TJ /F5 11.955 Tf 9.97 0 Td[(t0j)]TJ /F5 11.955 Tf 13.94 0 Td[(i2, (2) x2+)]TJ /F4 11.955 Tf 6.26 2.95 Td[((x;x0)~x)]TJ /F3 11.955 Tf 9.44 .5 Td[(~x02)]TJ /F5 11.955 Tf 11.95 0 Td[(c2t)]TJ /F5 11.955 Tf 9.97 0 Td[(t0+i2, (2) x2)]TJ /F15 5.978 Tf 5.76 0 Td[(+(x;x0)~x)]TJ /F3 11.955 Tf 9.44 .5 Td[(~x02)]TJ /F5 11.955 Tf 11.95 0 Td[(c2t)]TJ /F5 11.955 Tf 9.97 0 Td[(t0)]TJ /F5 11.955 Tf 9.96 0 Td[(i2, (2) x2\000(x;x0)~x)]TJ /F3 11.955 Tf 9.44 .5 Td[(~x02)]TJ /F5 11.955 Tf 11.95 0 Td[(c2jt)]TJ /F5 11.955 Tf 9.97 0 Td[(t0j+i2. (2) Becauseeachexternallinecanbeeither+or)]TJ /F1 11.955 Tf 12.62 0 Td[(intheSchwinger-Keldyshformalism,every1PIN-pointfunctionofthein-outformalismgivesriseto2N1PIN-pointfunctionsintheSchwinger-Keldyshformalism.Foreveryclassicaleld(x)ofanin-outeffectiveaction,thecorrespondingSchwinger-Keldysheffectiveactionmustdependupontwoeldscallthem+(x)and)]TJ /F4 11.955 Tf 6.25 1.8 Td[((x)inordertoaccesstheappropriate1PIfunction[ 59 ].Forthescalarparadigmwehavebeenconsideringthe1PI2-pointfunctionasthescalarself-mass-squared,M2(x;x0),andtheeffectiveactiontakestheform, [+,)]TJ /F4 11.955 Tf 6.26 1.79 Td[(]=S[+])]TJ /F5 11.955 Tf 11.96 0 Td[(S[)]TJ /F4 11.955 Tf 6.25 1.79 Td[(])]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2Zd4xZd4x08><>:+(x)M2++(x;x0)+(x0)++(x)M2+)]TJ /F4 11.955 Tf 4.26 2.95 Td[((x;x0))]TJ /F4 11.955 Tf 6.26 1.79 Td[((x0)+)]TJ /F4 11.955 Tf 6.26 1.8 Td[((x)M2)]TJ /F15 5.978 Tf 5.76 0 Td[(+(x;x0)+(x0)+)]TJ /F4 11.955 Tf 6.26 1.8 Td[((x)M2\000(x;x0))]TJ /F4 11.955 Tf 6.25 1.8 Td[((x0)9>=>;+O(3), (2) 27

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whereSistheclassicalaction.Theeffectiveeldequationsareobtainedbyvaryingwithrespectto+andthensettingbotheldsequal[ 59 ], [+,)]TJ /F4 11.955 Tf 6.26 1.79 Td[(] +(x)==h@2)]TJ /F5 11.955 Tf 11.95 0 Td[(m2i(x))]TJ /F9 11.955 Tf 9.96 16.28 Td[(Zd4x0hM2++(x;x0)+M2+)]TJ /F4 11.955 Tf 4.26 2.95 Td[((x;x0)i(x0)+O(2).(2)Thetwo1PI2-pointfunctionswewouldneedtoquantumcorrectthelinearizedscalareldequationareM2++(x;x0)andM2+)]TJ /F4 11.955 Tf 4.26 2.96 Td[((x;x0).Theirsumin( 2 )giveseffectiveeldequationswhicharecausalinthesensethatthetwo1PIfunctionscancelunlessx0liesonorwithinthepastlight-coneofx.Theirsumisalsoreal,whichneither1PIfunctionisseparately.Fromtheprecedingdiscussionitisapparentthatwewishtomakethefollowingsubstitutioninequation( 2 ), hi(x;x0))167(!h++i(x;x0)+h+)]TJ /F9 11.955 Tf 6.26 16.22 Td[(i(x;x0),(2)wherewecanreadofftheappropriateSchwinger-Keldyshvacuumpolarizationfromexpression( 2 ), hi(x;x0)=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(()()i2 1924h@2)]TJ /F3 11.955 Tf 9.96 0 Td[(@@i@4(ln(2x2) x2), (2) =)]TJ /F4 11.955 Tf 10.5 8.09 Td[(()()i2 15364h@2)]TJ /F3 11.955 Tf 9.96 0 Td[(@@i@6nln2(2x2))]TJ /F4 11.955 Tf 9.96 0 Td[(2ln(2x2)o. (2) Nowdenethetemporalandspatialintervalsas, tt)]TJ /F5 11.955 Tf 9.96 0 Td[(t0,rk~x)]TJ /F3 11.955 Tf 9.44 .5 Td[(~x0k.(2)Itisapparentfromexpressions( 2 2 )thatdifferencesoflogarithmsofthethe++and+)]TJ /F1 11.955 Tf 12.62 0 Td[(intervalsgive, ln(2x2++))]TJ /F4 11.955 Tf 11.96 0 Td[(ln(2x2+)]TJ /F4 11.955 Tf 6.25 2.95 Td[()=2it)]TJ /F4 11.955 Tf 9.96 0 Td[(r, (2) ln2(2x2++))]TJ /F4 11.955 Tf 11.96 0 Td[(ln2(2x2+)]TJ /F4 11.955 Tf 6.25 2.96 Td[()=4it)]TJ /F4 11.955 Tf 9.96 0 Td[(rlnh2(t2)]TJ /F4 11.955 Tf 9.96 0 Td[(r2)i. (2) 28

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Hencethevacuumpolarizationwhichbelongsinequation( 2 )is, hi(x;x0)=2 3843h@2)]TJ /F3 11.955 Tf 9.97 0 Td[(@@i@6(t)]TJ /F4 11.955 Tf 9.96 0 Td[(r"lnh2t2)]TJ /F4 11.955 Tf 9.96 0 Td[(r2)i)]TJ /F4 11.955 Tf 9.97 0 Td[(1#)+O(4). (2) 2.2.2PhotonsExpression( 2 )givesalltheoneloopcontributionswhichderiveexclusivelyfrominteractionvertices,buttherearealsocontributionsfromperturbativecorrectionstotheinitialstatewavefunctionals.Inthescalarfunctionalintegral( 2 )thesewavefunctionalsare['+(t0)]and[')]TJ /F4 11.955 Tf 6.25 1.79 Td[((t0)];forgravitypluselectromagnetismtheywouldbefunctionalsofAandh,evaluatedattheinitialtime.Eachstatewavefunctionalcanbeexpressedasthewavefunctionaloffreevacuumtimesaseriesofperturtbativecorrections, [A,h]=0[A,h](1+OhA2).(2)Itisstraightforwardtoshowthatthefreevacuumcontributioniswhatxestherealpartofthepropagatorinthefunctionalformalism[ 60 ].Iftherewerenoperturbativestatecorrectionsthenmerelyemployingthecorrectpropagatorswouldcompletelyaccountforthestatewavefunctionals.However,theremustbeperturbativestatecorrectionsbecausefreevacuumcannotbethetruevacuumstateofaninteractingquantumeldtheory.Perturbativestatecorrectionsmanifestasnewinteractionsontheinitialvaluesurface[ 36 ].Whentheinitialvaluesurfaceisintheasymptoticpast(ortheasymptoticpastandfutureforin-outmatrixelements)theseinteractionshavenoeffectonoperatorsatnitetimes.However,theycanbeimportantwhentheinitialvaluesurfaceisatanitetime,asitmustbeincosmology.Therstcorrectionrelevantforamassless,minimallycoupled4theoryhasrecentlybeenworkedoutondeSitterbackground[ 58 ].In 29

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thiscasetheinitialstatecorrectionisnecessarytomakethelinearizedeffectiveeldequationwelldenedattheinitialtime[ 61 ],andtoeliminateaninniteseriesofrapidlyredshiftingtermsfromthetwoloopexpectationvalueofthestresstensor[ 56 ].Weshallassumethatthemissingstatecorrectionsexactlycancelthesurfacetermswhicharisewhen( 2 )ispartiallyintegrated.Toseewhatthisentails,rstnotethatallordersofthepure-vertexpartofthevacuumpolarizationtakethemanifestlytransverseform, hi(x;x0)=@2)]TJ /F3 11.955 Tf 9.96 0 Td[(@@(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0).(2)ThepartialintegrationwehaveinmindconcernsthequantumcorrectiontoMaxwell'sequation, Zd4x0@2)]TJ /F3 11.955 Tf 9.96 0 Td[(@@(x)]TJ /F5 11.955 Tf 9.97 0 Td[(x0)A(x0)=Zd4x0(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0)@0F(x0)+SurfaceTerms. (2) IntheSchwinger-Keldyshformalismthe++and+)]TJ /F1 11.955 Tf 12.62 0 Td[(contributionsexactlycancelonthefuturetemporalsurface,aswellasonthesurfaceatspatialinnity.Hencetheonlysurfacetermscomefromtheinitialtime.Ofcoursethisisalsotrueofperturbativestatecorrections.Weassumethatthetwocontributionsexactlycancel,sothatthefull,quantum-correctedMaxwellequationis, @F(x)+Zd4x0(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0)@0F(x0)=J(x).(2)Wearenallyreadytoconsiderthecaseoffreephotons,whichcorrespondstoJ(x)=0.Notefromequation( 2 )thattheseobey@F(x)=0,thesameasintheclassicaltheory.Onemightworryaboutthepotentialforsolutionsoftheform@F(x)=S(x),whereS(x)obeystheintegralequation, S(x)+Zd4x0(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0)S(x0)=0.(2) 30

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However,aneffectiveeldequationsuchas( 2 )canonlybeusedtoperturbativelycorrectclassicalsolutions[ 62 ],whichmeanswemustexcludeanysuchsolutions.Henceweconcludethatquantumgravityonatspacebackgroundmakesnocorrectiontofreephotonsatanyorder,exceptforpossibleeldstrengthrenormalization. 2.2.3InstantaneouslyCreatingAPointDipoleThechargedensityofastaticpointelectricdipole~pattheoriginis=)]TJ /F3 11.955 Tf 8.73 .5 Td[(~p~r3(~x).Wemightimaginecreatingsuchadipoleattheinstantt=0byseparatingthechargesinaverysmall,neutralparticlesuchasaneutron.Theconserved4-currentassociatedwiththiseventis, J0(t,~x)=)]TJ /F3 11.955 Tf 9.3 0 Td[((t)~p~r3(~x),Ji(t,~x)=pi(t)3(~x).(2)Theresponseofthemagneticeldprovidesagoodperturbativeillustrationofthesmearingofthelight-conewhichwasconjecturedsolongago[ 39 ].Beforeproceedingitisdesirabletoreorganizeequation( 2 )intwoways.Thersthastodowiththelimitationinherentinonlypossessingtherstordertermintheloopexpansionof(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0), (x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0)=(1)(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0)+(2)(x)]TJ /F5 11.955 Tf 9.97 0 Td[(x0)+O(6).(2)Ofcoursethismeanswecanonlyinfertheoneloopcorrectiontotheeldstrength,sowemayaswellexpandit, F(x)=F(0)(x)+F(1)(x)+F(2)(x)+O(6).(2)Substituting( 2 )and( 2 )inthequantum-correctedMaxwellequation( 2 )andsegregatingdifferentordersof2producesthehierarchy, @F(0)(x)=J(x), (2) @F(1)(x)=)]TJ /F9 11.955 Tf 11.29 16.27 Td[(Zd4x0(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0)J(x0)J(1)(x), (2) 31

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andsoon.NotetheclassicalsourceJ(x)is0thorder.Thesecondreorganizationconcernsderivingtheeldstrengthdirectly,withoutconstructingthevectorpotential.ConsidertakingthecurloftheclassicalMaxwellequation, @@F=@2@A=@J=)@2F=@J)]TJ /F3 11.955 Tf 11.95 0 Td[(@J.(2)Combiningthiswith( 2 2 )implies, @2F(0)=@J)]TJ /F3 11.955 Tf 11.95 0 Td[(@J, (2) @2F(1)=@J(1))]TJ /F3 11.955 Tf 11.95 0 Td[(@J(1). (2) Nowrecallthatouroneloopcurrentdensitycanbeexpressedasthed'Alembertianofsomething, J(1)(x))]TJ /F9 11.955 Tf 30.55 16.27 Td[(Zd4x0(1)(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0)J(x0), (2) =i2@4 1924Zd4x0(ln(2x2++) x2++)]TJ /F4 11.955 Tf 11.16 8.45 Td[(ln(2x2+)]TJ /F4 11.955 Tf 6.26 2.96 Td[() x2+)]TJ /F9 11.955 Tf 22.75 31.6 Td[()J(x0), (2) =2@6 3843Zd4x0t)]TJ /F4 11.955 Tf 9.97 0 Td[(r(lnh2t2)]TJ /F4 11.955 Tf 9.96 0 Td[(r2i)]TJ /F4 11.955 Tf 9.96 0 Td[(1)J(x0). (2) Comparisonof( 2 )with( 2 )or( 2 )impliesaresultfortheoneloopeldstrength,uptopossiblehomogeneousterms, F(1)(x)=i2@2 19242@[Zd4x0(ln(2x2++) x2++)]TJ /F4 11.955 Tf 11.16 8.44 Td[(ln(2x2+)]TJ /F4 11.955 Tf 6.26 2.95 Td[() x2+)]TJ /F9 11.955 Tf 22.75 31.6 Td[()J](x0), (2) =2@4 38432@[Zd4x0t)]TJ /F4 11.955 Tf 9.97 0 Td[(r(lnh2t2)]TJ /F4 11.955 Tf 9.96 0 Td[(r2i)]TJ /F4 11.955 Tf 9.96 0 Td[(1)J](x0). (2) Wearenowreadytospecializetothecurrentdensity( 2 )ofaninstantaneouslycreateddipole.Substitutingin( 2 )andspecializingtopurelyspatialindicesgives, @2F(0)ij(x)=@ipj)]TJ /F3 11.955 Tf 9.96 0 Td[(@jpi4(x).(2) 32

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ThesolutioncanbeexpressedinaconvenientformbynotingtheD=4dimensionalversionofrelation( 2 ), @2(1 x2++)=42i4(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0),@2(1 x2+)]TJ /F9 11.955 Tf 7.45 31.6 Td[()=0.(2)Hencewehave, F(0)ij(x)=)]TJ /F5 11.955 Tf 17.69 8.09 Td[(i 42@ipj)]TJ /F3 11.955 Tf 9.96 0 Td[(@jpi(1 x2++)]TJ /F4 11.955 Tf 20.55 8.09 Td[(1 x2+)]TJ /F9 11.955 Tf 7.45 31.6 Td[(),(2)wherex0=0isunderstood.Nowwriteoutthetwointervals, x2++=r2)]TJ /F5 11.955 Tf 11.95 0 Td[(t2+2+2jtji, (2) x2+)]TJ /F4 11.955 Tf 16.22 2.95 Td[(=r2)]TJ /F5 11.955 Tf 11.95 0 Td[(t2+2)]TJ /F4 11.955 Tf 11.96 0 Td[(2ti. (2) CombiningtheserelationswiththeDiracidentityresultsinthefamiliarformfortheLienard-Wiechertpotential, F(0)ij(x)=)]TJ /F4 11.955 Tf 14.03 8.09 Td[(1 2@ipj)]TJ /F3 11.955 Tf 9.96 0 Td[(@jpi(t)(r2)]TJ /F5 11.955 Tf 9.96 0 Td[(t2).(2)Themostconvenientformfortheoneloopcorrectionis( 2 ), F(1)ij(x)=@ipj)]TJ /F3 11.955 Tf 9.96 0 Td[(@jpii2@2 1924(ln(2x2++) x2++)]TJ /F4 11.955 Tf 11.16 8.45 Td[(ln(2x2+)]TJ /F4 11.955 Tf 6.26 2.95 Td[() x2+)]TJ /F9 11.955 Tf 22.75 31.6 Td[(), (2) =@ipj)]TJ /F3 11.955 Tf 9.96 0 Td[(@jpii2@2 484(1 x4++)]TJ /F4 11.955 Tf 20.54 8.09 Td[(1 x4+)]TJ /F9 11.955 Tf 7.45 31.6 Td[(), (2) =@ipj)]TJ /F3 11.955 Tf 9.96 0 Td[(@jpi2 122@ @r2((t)(r2)]TJ /F5 11.955 Tf 9.97 0 Td[(t2) 2). (2) Addingtheoneloopmagneticeld( 2 )tothetreeone( 2 )leadstoaninterestingform, Fij(x)=)]TJ /F4 11.955 Tf 14.03 8.09 Td[(1 2@ipj)]TJ /F3 11.955 Tf 9.96 0 Td[(@jpi(t)(1)]TJ /F3 11.955 Tf 17.6 8.09 Td[(2 122@ @r2)(r2)]TJ /F5 11.955 Tf 9.96 0 Td[(t2)+O(4), (2) =)]TJ /F4 11.955 Tf 14.03 8.09 Td[(1 2@ipj)]TJ /F3 11.955 Tf 9.96 0 Td[(@jpi(t)r2)]TJ /F5 11.955 Tf 9.97 0 Td[(t2)]TJ /F3 11.955 Tf 17.6 8.09 Td[(2 122+O(4). (2) 33

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Itwouldthereforebefairtosaythat,bytimetthesignalhasreachedadistancerslightlyoutsidetheclassicallight-cone, r2=t2+2 122+O(4).(2)Althoughintriguing,thesuper-luminalitywehavejustfoundisunobservablysmall.Inparticular,itcannotserveasanysortofexplanationfortheOPERAresult[ 63 ].Italsoisn'tcumulative,solookingatcosmologicalsourcesmakestheeffectnolarger.AnotherthingoureffectfailstodoisbreakLorentzinvariance,ascouldhavebeenpredictedfromthefactthatperturbativequantumgravityprovidesnomechanismforspontaneouslybreakingthissymmetry.Insteadofsignalspropagatingalongtheclassicallight-conexx=0,theynowpropagatealongxx=4G 3.Soitisnotthatthespeedoflightorthedispersionrelationhasbeenchanged.Ofcoursetheorieswithanonlinearkineticoperatorcanshowsuper-luminalpropagationevenclassically[ 64 ].Oureffectisdifferentinthatitarisesfromquantumuctuationsofthemetricoperatorwhichsetsthelight-cone.Oneinterpretationforthenetsuper-luminalpropagationisthatthereismorevolumeoutsidetheclassicallight-conethaninside.Thismightbecheckedbyextendingthegravitonexpansionofthevolumeofthepastlight-coneoneorderhigherthanin[ 65 ]andthencomputingitsexpectationvalue.Iftheoneloopcorrectionispositivethenourconjectureisveried.Notealsothatthischeckwouldbeindependentofthechoiceofgaugebecausethevolumeofthepastlight-coneisagaugeinvariantoperator.Therehavebeenmanyclaimsofsuper-luminalpropagationfromquantumelectrodynamicsinnontrivialgeometries[ 66 67 ].Ourresultisdifferentinthatitoccursinatspace,andisduetouctuationsofthemetricoperator,ratherthanofsomemattereld.Wealsodoubtthattheearlierclaimsresultfromtruesuper-luminalpropagation.Onecannotcomputethevacuumpolarizationproducedbyfermionsinanarbitrarygeometrybecausethefermionpropagatorisnotknownforgeneralmetric.Whatwas 34

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doneinsteadisaderivativeexpansion.Thisshouldbevalidforlowenergyeffectiveeldtheory;forexample,itshouldgivecorrectresultsforthephasevelocityofsomecontinuous,lowfrequencysignal.However,demonstratingtruesuper-luminalityrequiresfollowingthepropagationofapulse,andthehighfrequencymodeswhichareessentialforthisarenotcorrectlytreatedbyderivativeexpansions.InfacttheSchwinger-Keldyshformalism[ 57 ]impliestherecannotbesuper-luminalpropagationfromthefermioniccontributiontovacuumpolarization. 2.2.4AnAlternatingPointDipoleThe4-currentassociatedwithanalternatingpointdipoleis, J0(t,~x)=)]TJ /F3 11.955 Tf 8.74 .5 Td[(~p~r3(~x)e)]TJ /F6 7.97 Tf 6.59 0 Td[(i!t,Ji(t,~x)=)]TJ /F5 11.955 Tf 9.3 0 Td[(i!pi3(~x)e)]TJ /F6 7.97 Tf 6.59 0 Td[(i!t.(2)Tondthequantumcorrectiontothecurrentweemploythesameexpansiontechniqueusedintheprevioussectionwheretherstordercorrectionisdenedas, J(1)(x)=)]TJ /F5 11.955 Tf 12 8.09 Td[(G@6 242Zd4x0(t)]TJ /F4 11.955 Tf 11.95 0 Td[(x)nlnh2(t2)]TJ /F4 11.955 Tf 9.96 0 Td[(x2)i)]TJ /F4 11.955 Tf 9.97 0 Td[(1oJ(x0).(2)Wecanevaluatethisintegralbyrewritingxandthedifferentialoperatorsasx=1 2(x+t)+1 2(x)]TJ /F5 11.955 Tf 11.36 0 Td[(t)and@2=1 x(@x)]TJ /F3 11.955 Tf 11.37 0 Td[(@t)(@x+@t)x.Thuswecometotheconvienientform, @4=1 2x@x)]TJ /F3 11.955 Tf 9.96 0 Td[(@t2@x+@t2(x+t)+1 2x@x)]TJ /F3 11.955 Tf 9.97 0 Td[(@t2@x+@t2(x)]TJ /F5 11.955 Tf 9.96 0 Td[(t).(2)Bysubsituting( 2 )for@4in( 2 )andapplyingthezerothordercurrents( 2 )wendtheoneloopcurrentstobe, J0(1)(t,~x)=@2(G~p~r 62h)]TJ /F5 11.955 Tf 10.49 8.09 Td[(i! x2+1 x3ie)]TJ /F6 7.97 Tf 6.59 0 Td[(i!(t)]TJ /F6 7.97 Tf 6.59 0 Td[(x)), (2) Ji(1)(t,~x)=@2(i!piG 62h)]TJ /F5 11.955 Tf 10.5 8.09 Td[(i! x2+1 x3ie)]TJ /F6 7.97 Tf 6.58 0 Td[(i!(t)]TJ /F6 7.97 Tf 6.59 0 Td[(x)). (2) 35

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From( 2 )weseethatthezerothordereldstrengthsforthissourceobey, @2F(0)0i(t,~x)=)]TJ /F9 11.955 Tf 9.3 13.27 Td[(h!2pi)]TJ /F3 11.955 Tf 9.96 0 Td[(@i~p~ri3(~x)e)]TJ /F6 7.97 Tf 6.59 0 Td[(i!t, (2) @2F(0)ij(t,~x=)]TJ /F5 11.955 Tf 9.3 0 Td[(i!h@ipj)]TJ /F3 11.955 Tf 9.96 0 Td[(@jpii3(~x)e)]TJ /F6 7.97 Tf 6.59 0 Td[(i!t. (2) ApplyingtheLienard-WiechertGreen'sfunctionwend, F(0)0i(t,~x)=1 4[!2pi)]TJ /F3 11.955 Tf 9.96 0 Td[(@i~p~rie)]TJ /F6 7.97 Tf 6.59 0 Td[(i!(t)]TJ /F6 7.97 Tf 6.59 0 Td[(x) x, (2) F(0)ij(t,~x)=i! 4h@ipj)]TJ /F3 11.955 Tf 9.97 0 Td[(@jpiie)]TJ /F6 7.97 Tf 6.59 0 Td[(i!(t)]TJ /F6 7.97 Tf 6.59 0 Td[(x) x. (2) From( 2 )weseethatoneloopeldstrengthsfollowbysimplydeletingthe@2from( 2 2 )andactingsomederivatives, F(1)0i(t,~x)=)]TJ /F5 11.955 Tf 10.49 8.08 Td[(i!G 62h!2pi)]TJ /F3 11.955 Tf 9.96 0 Td[(@i~p~rih1+i !xie)]TJ /F6 7.97 Tf 6.59 0 Td[(i!(t)]TJ /F6 7.97 Tf 6.59 0 Td[(x) x2, (2) F(1)ij(t,~x)=)]TJ /F5 11.955 Tf 10.49 8.08 Td[(i!G 62(i!)h@ipj)]TJ /F3 11.955 Tf 9.96 0 Td[(@jpiih1+i !xie)]TJ /F6 7.97 Tf 6.59 0 Td[(i!(t)]TJ /F6 7.97 Tf 6.58 0 Td[(x) x2. (2) Addingtheloopcorrectiontothetreeresultsgives, F0i(t,~x)=h!2pi)]TJ /F3 11.955 Tf 9.96 0 Td[(@i~p~rie)]TJ /F6 7.97 Tf 6.58 0 Td[(i!(t)]TJ /F6 7.97 Tf 6.59 0 Td[(x) 4x(1)]TJ /F4 11.955 Tf 11.15 8.09 Td[(2i!G 3xh1+i !xi+O(G2)), (2) Fij(t,~x)=i!h@ipj)]TJ /F3 11.955 Tf 9.96 0 Td[(@jpiie)]TJ /F6 7.97 Tf 6.59 0 Td[(i!(t)]TJ /F6 7.97 Tf 6.59 0 Td[(x) 4x(1)]TJ /F4 11.955 Tf 11.16 8.09 Td[(2i!G 3xh1+i !xi+O(G2)). (2) Ofcoursetheobviousconclusionisthattheoneloopcorrectionshavenoeffectinthefareldregime,andtheneareldregimeisunobservablyclosetothesource. 2.2.5AStaticPointChargeThechargedensityofastaticpointchargeqattheoriginis=q3(~x).Theconserved4-currentassociatedwiththissourceis, J(x)=q3(~x)0.(2)Becausethe=0componentdiffersfromthealternatingdipoleoftheprevioussubsectiononlybysetting!=0andreplacing)]TJ /F3 11.955 Tf 8.73 .49 Td[(~p~rwithq,wecanreadofftheone 36

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loopcurrentdensityfrom( 2 ), J0(1)(t,~x)=)]TJ /F5 11.955 Tf 14.71 8.09 Td[(Gq 2x5.(2)OfcoursethevectorcomponentsvanishsowendthecorrectiontotheCoulombpotentialis, (r)=q 4r(1+2G 3r2+O(G2)).(2)Ourresult( 2 )agreeswiththatfoundin1970byRadkowski[ 44 ].TheoneloopcorrectionthatBjerrum-Bohrinferredfromthescatteringofcharged,gravitatingscalarsdiffersfromwhatwegotbyafactorofnine[ 45 ].Partofthisdiscrepancymaybeduetodifferentsources;Bjerrum-Bohrconsideredachargedscalarwhereasweusedapointparticlewithworldline(), Lpoint=)]TJ /F5 11.955 Tf 9.3 0 Td[(mr )]TJ /F5 11.955 Tf 9.3 0 Td[(g()_()_()+q_()A().(2)However,webelievethelargestpartofthediscrepancyarisesfromBjerrum-BohrhavingimplicitlyincludedcorrectionstothecurrentdensitylikethediagramdepictedinFig. 2-2 .Wecouldhaveandshouldhavedonethis,butwewillseeinthenextsectionthatitwouldonlyhavealteredallofouroneloopeldstrengthsbyanoverallconstant.Radkowski[ 44 ],Bjerrum-Bohr[ 45 ]andweallagreethatquantumgravitystrengthenstheelectromagneticforceatoneloop.Theoppositeconclusionseemstoarisefromcomputationsofthequantumgravitationalcontributiontotheelectromagneticbetafunction[ 46 68 69 ].Theseshowthatquantumgravitydecreasestheelectromagneticcouplingconstantathighenergyscales.Thatwouldnormallybeassumedtomeanthatquantumgravityweakenstheelectromagneticforceatshortdistances,butitiswelltokeepinmindthatthebetafunctionisnotdirectlyobservable.Theobservablethingisthestrengthscatteringbetweenchargedparticles,andtheBjerrum-Bohrcomputationshowsthatoneloopquantumgravityeffectsweakenthis,ratherthanstrengtheningit. 37

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2.3GaugeDependenceThepurposeofthissectionistoexaminehowtheresultsoftheprevioussectiondependuponourchoicesofthegravitationalgaugexingterm( 2 )andtheelectromagneticgaugexingterm( 2 ).Webeginwithsomegeneralconsiderationswhichreducetheissuetoasingleproportionalityconstant.Thegravitonandphotonpropagatorsarethenworkedoutforageneral3-parameterfamilyofcovariantgauges.Althoughoneoftheseparametersdropsout,theothertwocanchangetheproportionalityconstantallthewayfromminusinnitytoplusinnity.WeclosebyexploitingthegaugeindependentresultofBjerrum-Bohrtoarguethatthisseeminggaugedependencemaycanceloutifquantumgravitationalcorrectionstothecurrentdensityareincluded. 2.3.1GeneralConsiderationsNotefromexpression( 2 )thatthevacuumpolarizationistransverseoneachofitstwoindicesasatrivialconsequenceoftheantisymmetryofthevertexfunctiononitsrstandthirdindices, V=)]TJ /F5 11.955 Tf 9.3 0 Td[(V.(2)Thisiscompletelywithoutregardtothegaugesemployedtodenegravitonandphotonpropagators.SupposewenowrestrictattentiontogaugeswhichpreservePoincareinvariance.BecausetheLagrangians( 2 2 )andthebackgroundarealsoPoincareinvariant,thevacuumpolarizationmustinheritthissymmetry.Thendimensionalanalysis,transversalityandthestandard2ofaoneloopquantumgravityresult,togetherimplyaformlikethatof( 2 ), ih3pti(x;x0)=)]TJ /F4 11.955 Tf 9.3 0 Td[(Constant2h@2)]TJ /F3 11.955 Tf 9.96 0 Td[(@@i1 x2D)]TJ /F10 7.97 Tf 6.58 0 Td[(2.(2)However,theconstantprefactormightbegaugedependent,andthatsamegaugedependentconstantwouldmultiplyallofouroneloopcorrections. 38

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Itisusefultobeginatasomewhatearlierpoint.IfdifferentbutPoincareinvariantgravitonandphotonpropagatorshadbeenemployedinexpression( 2 ),thenthecombinationofPoincareinvariance,dimensionalanalysisandthealgebraicsymmetriesofthevertexfunctionandthepropagatorsimplythattheresultcanbeexpressedintermsoftwoconstantsAandB, (i)2@@0(Vihnewi(x;x0)V@@0ihnewi(x;x0))=(i)2)]TJ /F10 7.97 Tf 6.77 4.34 Td[(2(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1) 16D(D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)@@0(A4x[][x] x2D+B[] x2D)]TJ /F10 7.97 Tf 6.59 0 Td[(2), (2) =)]TJ /F3 11.955 Tf 10.49 8.79 Td[(2)]TJ /F10 7.97 Tf 6.78 4.34 Td[(2(D 2)]TJ /F4 11.955 Tf 9.97 0 Td[(1) 16D(D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)hDA)]TJ /F4 11.955 Tf 9.96 0 Td[(2(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)Bi((D+1) x2D)]TJ /F4 11.955 Tf 13.15 8.08 Td[(2Dxx x2D+2), (2) =)]TJ /F3 11.955 Tf 10.49 8.78 Td[(2)]TJ /F10 7.97 Tf 6.78 4.34 Td[(2(D 2)]TJ /F4 11.955 Tf 9.97 0 Td[(1) 16D(D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)[DA)]TJ /F4 11.955 Tf 9.96 0 Td[(2(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)B] 2(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)h@2)]TJ /F3 11.955 Tf 9.96 0 Td[(@@i1 x2D)]TJ /F10 7.97 Tf 6.59 0 Td[(2. (2) Wecanthereforeidentifytheproportionalityconstantin( 2 )as, Constant=)]TJ /F10 7.97 Tf 6.78 4.34 Td[(2(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1) 16D(D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)[DA)]TJ /F4 11.955 Tf 9.97 0 Td[(2(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)B] 2(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1), (2) )]TJ /F10 7.97 Tf 6.78 4.34 Td[(2(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1) 16D1 2D)]TJ /F4 11.955 Tf 9.96 0 Td[(2 D)]TJ /F4 11.955 Tf 9.96 0 Td[(1C. (2) 2.3.2GeneralCovariantGaugesThemostgeneralPoincareinvariantextensionofthegravitongaugexingfunctional( 2 )dependsupontwoparametersaandb, LGRnew=)]TJ /F4 11.955 Tf 13.54 8.09 Td[(1 2aFF,Fh,)]TJ /F5 11.955 Tf 11.16 8.09 Td[(b 2h,.(2)Theassociatedpropagatoris[ 70 ], ihnewi(x;x0)=5XI=1CI(D,a,b)hTIii(x;x0),(2) 39

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wherethecoefcientfunctionsCI(D,a,b)andthetensordifferentialoperators[TI]aregiveninTable 2-1 .Thepropagatorcanbegivenamorerevealingexpressionusingthetransverseprojectionoperator)]TJ /F7 7.97 Tf 13.15 5.86 Td[(@@ @2, ihnewi(x;x0)=(2())]TJ /F4 11.955 Tf 21.38 8.09 Td[(2 D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)]TJ /F4 11.955 Tf 43.83 8.09 Td[(2 (D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)")]TJ /F9 11.955 Tf 9.96 13.27 Td[(Db)]TJ /F4 11.955 Tf 9.96 0 Td[(2 b)]TJ /F4 11.955 Tf 9.97 0 Td[(2@@ @2#")]TJ /F9 11.955 Tf 9.96 13.27 Td[(Db)]TJ /F4 11.955 Tf 9.96 0 Td[(2 b)]TJ /F4 11.955 Tf 9.96 0 Td[(2@@ @2#+4a@()(@) @2+4a (b)]TJ /F4 11.955 Tf 9.96 0 Td[(2)2@@@@ @4). (2) Ofcoursethetranverse-tracelesstermontherstlinerepresentsthecontributionfromdynamical,spintwogravitons.Thistermloomslargeinthequantumgravityliteraturebutitiswelltorecallthatitplaysnoroleinthesolarsystemtestsofgeneralrelativity.Thephenomenologicallymoreimportantpartsofthegravitonpropagatorarethoseonthesecondandthirdlines,whichmediatethegravitationalinteractionbetweensourcesofstress-energy.Notethatthelongitudinaltermsproportionaltothegaugeparameterawouldvanishintheexactgaugeh,=b 2h,.ThemostgeneralPoincareinvariantextensionofthephotongaugexingfunctional( 2 )dependsuponasingleparameterc, LEMnew=)]TJ /F4 11.955 Tf 13.81 8.09 Td[(1 2c(@A)2.(2)Theassociatedpropagatoris, ihnewi(x;x0)="+(c)]TJ /F4 11.955 Tf 9.97 0 Td[(1)@@ @2#i(x;x0).(2)Thelongitudinaltermproportionaltoc)]TJ /F4 11.955 Tf 12.82 0 Td[(1canmakenocontributiontothegeneralgaugevacuumpolarization( 2 )becausethevertexfunction( 2 )isantisymmetricunderinterchangeofitssecondandfourthindices, V=)]TJ /F5 11.955 Tf 9.3 0 Td[(V.(2) 40

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ItremainstoexplainhowtoactthetensordifferentialoperatorsofTable 2-1 onthescalarpropagator( 2 ).Firstnotethatinversed'Alembertiansacton1=xD)]TJ /F10 7.97 Tf 6.59 0 Td[(2togive, 1 @21 xD)]TJ /F10 7.97 Tf 6.59 0 Td[(2=)]TJ /F4 11.955 Tf 28.73 8.09 Td[(1 2(D)]TJ /F4 11.955 Tf 9.96 0 Td[(4)1 xD)]TJ /F10 7.97 Tf 6.59 0 Td[(4, (2) 1 @41 xD)]TJ /F10 7.97 Tf 6.59 0 Td[(2=1 8(D)]TJ /F4 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F4 11.955 Tf 9.96 0 Td[(6)1 xD)]TJ /F10 7.97 Tf 6.59 0 Td[(6. (2) Nowjustactthederivativesinthenumeratortoconclude, @@ @2i(x;x0)=1 2()]TJ /F4 11.955 Tf 11.16 8.09 Td[((D)]TJ /F4 11.955 Tf 9.97 0 Td[(2)xx x2)i(x;x0), (2) @@@@ @4i(x;x0)=1 8(3())]TJ /F4 11.955 Tf 11.15 8.29 Td[(6(D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)(xx) x2 (2) +D(D)]TJ /F4 11.955 Tf 9.97 0 Td[(2)xxxx x4)i(x;x0). (2) 2.3.3GaugeDependentProportionalityConstantWearenowreadytocomputethecrucialproportionalityconstantofrelation( 2 ).Becausethegaugedependenceofthephotonpropagatordropsout,weneedonlyconsiderthegaugedependenceofthegravitonpropagator.Becausethegravitonpropagator( 2 )isasumofgauge-dependentcoefcientsCI(D,a,b)timestensoroperators[TI],actingonthescalarpropagator,wemayaswellworkouttheresultforeachtensoroperatorseparately.Table 2-2 presentsthecoefcientsAI(D)andBI(D)whichweredenedinrelation( 2 ),foreachofthevetensordifferentialoperatorsofTable 2-1 .AlsogivenisthecontributionofeachtensordifferentialoperatortothecoefcientCI(D), CI(D)DAI(D))]TJ /F4 11.955 Tf 11.95 0 Td[(2(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)BI(D).(2)TorecoverthefullresultforC(D,a,b)denedinrelation( 2 )wemultiplyeachCI(D)bytheappropriategaugedependentcoefcientC(D,a,b)fromTable 2-1 C(D,a,b)=5XI=1CI(D,a,b)CI(D)(2) 41

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TheformulaforarbitraryDisnotilluminating,butspecializingtoD=4gives, C(4,a,b)=)]TJ /F4 11.955 Tf 9.3 0 Td[(12a(3b2)]TJ /F4 11.955 Tf 9.96 0 Td[(12b+8) (b)]TJ /F4 11.955 Tf 9.96 0 Td[(2)2)]TJ /F4 11.955 Tf 29.04 8.09 Td[(4 (b)]TJ /F4 11.955 Tf 9.96 0 Td[(2)2.(2)Ouroriginalgaugecorrespondstoa=b=1,whichgivesC(4,a,1)=8.Hencethevariousoneloopcorrectionscomputedinsection3arevalidforgeneralaandbifwemultiplybytheproportionalityconstant, K(a,b)C(4,a,b) C(4,1,1)=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(3 2a(3b2)]TJ /F4 11.955 Tf 9.96 0 Td[(12b+8) (b)]TJ /F4 11.955 Tf 9.96 0 Td[(2)2)]TJ /F4 11.955 Tf 32.18 8.09 Td[(1 2(b)]TJ /F4 11.955 Tf 9.96 0 Td[(2)2.(2)ItisinterestingtonotethatthegaugeindependentcontributionfromdynamicalgravitonsvanishesinD=4dimensions, C1(D))]TJ /F5 11.955 Tf 9.96 0 Td[(C4(D)+2C5(D))]TJ /F4 11.955 Tf 21.37 8.08 Td[(2 D)]TJ /F4 11.955 Tf 9.96 0 Td[(1hC2(D))]TJ /F5 11.955 Tf 9.97 0 Td[(C3(D)+C5(D)i=(D)]TJ /F4 11.955 Tf 9.96 0 Td[(4)(D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)2(D+1)(D+2) 4(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1). (2) Itisapparentthatthegaugedependentproportionalityconstant( 2 )canbemadetohaveeithersignbyvaryingthegaugeparametera.Furthermore,K(a,b)canbemadearbitrarilylargeinmagnitudebytakingthegaugeparameterbcloseto2.Henceitwouldseemthatourresultsarecompletelygaugedependentandunphysical.Gaugedependencehasalsobeennotedintherenormalizationgroupapproach[ 71 72 ].Amoment'sthoughtrevealsthatallisnotlostbecausetheresultofBjerrum-Bohr[ 45 ]forthequantumgravitationalcorrectiontotheCoulombpotentialwasderivedfromthegaugeindependentS-matrixofscalarQED.Roughlyspeaking,thiscorrectionderivesfromthefactthatgravityissourcedbytheelectromagneticeldsofthetwochargedparticlesbeingscattered,andthissourcechangesastheparticlesmovewithrespecttooneanother.Thatisarealeffect,notsomegaugeartifact.AnditiscruciallyimportanttonotethatweagreewithBjerrum-Bohruptoafactorof+9.Weattributedthisfactortoourhavingonlyquantum-correctedthelefthandsideoftheoperator 42

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Maxwellequation, @hp )]TJ /F5 11.955 Tf 9.3 0 Td[(gggF(x)i=J(x).(2)Therighthandsideisalsoanoperatoranditmustalsosufferquantumgravitationalcorrections,oneofwhichisdepictedinFig. 2-2 .ByPoincareinvariance,currentconservationanddimensionalanalysis,thosecorrectionsmusttakeexactlythesameformaswefoundforthelefthandside,uptoanoverallconstant.WeconjecturethatthegaugedependenceK(a,b)wehavejustfoundforcorrectionstothelefthandsideiscanceledbygaugedependenceincorrectionstotherighthandside.Ifthisiscorrect,thentheoverall,gaugeindependentcorrectiontothevariousresultsderivedinsection3canbeinferredbycomparinganyoneofthemwithitsS-matrixanalogue.ForscalarQEDthecorrectionfactorwouldbe9,anditcouldbecomputedforthepointparticlesourceweused.TheresolutionwehavejustproposedtothegaugeissuerecallssomeoldworkbyDeWitt[ 73 ]aboutdependenceuponthegaugexingfunctionalseveninthegaugeinvariantbackgroundeldeffectiveaction)-301(=S+.DeWittstates[ 74 ],Thefunctionalformofisnotindependentofthechoiceoftheseterms.However,thesolutionsoftheeffectiveeldequationcanbeshowntobethesameforallchoices.Attheorderweareworkingthereisnodistinctionbetweenoureffectiveeldequationsandthoseofthegaugeinvariantbackgroundeldeffectiveaction.(Onecanseethisfromthetransversalityofourvacuumpolarization.)However,wehavejustshownthatDeWitt'sstatementcannotbecorrectifthesource(ortheasymptoticeldstrengthsforscatteringsolutions)isnotnormalizedinsomephysicalway.Ourproposalisthatincludingquantumcorrectionstotherighthandsideoftheequationprovidesthisphysicalnormalization.Moreworkisobviouslyrequired,inparticularanexplicitcomputationofthequantumgravitationalcorrectionstothesource,butitwouldbewonderfulifsolutionstotheeffectiveeldequationscouldbephysicallyinterpretedthesamewayasclassicalsolutions. 43

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2.4DiscussionWeuseddimensionalregularizationtocomputetheoneloopquantumgravitationalcontribution( 2 )tothevacuumpolarizationonatspacebackground.Afullyrenormalizedresult( 2 )wasobtainedbyrstpartiallyintegratingtolocalizetheultravioletdivergenceandthenabsorbingitintotheappropriateBPHZcounterterm( 2 )withcoefcient( 2 ).TheSchwinger-Keldyshformalism[ 57 59 ]wasthenemployedtoreachthemanifestlyrealandcausalform( 2 ).ThisisausefulintroductionbecauseIemploythesameprocedureforthefulldeSitterspacecalculationinchapter 5 .Weused( 2 )tosolvethequantumcorrectedMaxwell'sequation( 2 )forvariousspecialcases.Providedtheappropriateperturbativecorrectionstotheinitialstatecancelthesurfacetermsinvolvedinreachingtheform( 2 ),thereisnochangeinthesource-freesolutionsatanyorderintheloopexpansion.However,sourcesinduceavarietyofinterestingeffects.Theeaseofworkinginatspaceallowedustoapplyourresultforthevacuumpolarizationtomanysources;workingindeSitterismuchmorecumbersome,andonlythecaseofdynamicalphotonswillbeconsidered.Gaugedependenceposesamajorobstacletothephysicalinterpretationofsolutionstotheeffectiveeldequations.IfonerestrictstoPoincareinvariantgaugexingfunctionals,theonlypossiblechangetoourvacuumpolarization( 2 )isrescalingbyanoverall,gauge-dependentconstant.Insection 2.3 weconsideredthemostgeneral2-parameterfamilyofgravitongauges( 2 ),andthemostgeneral1-parameterfamilyofphotongauges( 2 ).Weshowedthatthevacuumpolarizationhasnodependenceupontheelectromagneticgaugexingparameterc,butitdependsstronglyonthetwogravitationalgaugexingparametersaandb.Theeffectofbeinginageneralcovariantgaugeistorescale( 2 )bythefunctionK(a,b)giveninequation( 2 ).Byvaryingtheconstantsaandb,onecanmakeK(a,b)assumeanyvaluesfromplusinnitytominusinnity. 44

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Suchmassivegaugedependencewouldseemtoinvalidateanyphysicalinferencefromtheresultsofsection 2.2 ,however,thegaugeindependentresultofBjerrum-Bohrsuggestsasimpleresolution.Thereisnoquestionthatonemustincludequantumgravitationalcorrectionstothecurrentdensityoperator.ThisseemstobewhyBjerrum-Bohr(whoimplicitlydidthis)getsafactorofninedifferentoneloopcorrectiontotheCoulombpotential.Weconjecturethatmakingsuchcorrectionsinageneralgaugewhichseemsquitefeasibleusingthetechniquesofsection 2.3 wouldcompletelycancelthegaugedependenceofourresult.Ifthiscouldbedemonstratedthenitwouldbepossibletorealizetheolddream[ 73 74 ]ofusingsolutionstotheeffectiveeldequationsasfreelyasonedoesclassicalsolutions.Notealsothatitwouldprovideanimportantclassofobservablesincosmology,forwhichtheS-matrixdoesnotexist.Thankfully,dynamicalphotonsdonothaveasourceandtheissueofgaugedependenceisobsolete.Thepointofthisexercisehasbeentoestablishtheatspacecorrespondencelimitforaplannedinvestigationoftheeffectsofinationarygravitonsonelectromagnetism.Inretrospect,wecanrecognizethesimplicityofatspaceastheidealvenueforsortingoutthetroublesomeissuesofdependenceuponthechoiceofeldvariableandthechoiceofgaugewhicharesoimportanttoacorrectinterpretationofthemanysolutionswhichnowexisttolinearizedeffectiveeldequationsondeSitterbackground[ 12 34 51 61 75 78 ]. Figure2-1. Gravitoncontributionstotheoneloopvacuumpolarization. 45

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Figure2-2. Vertexcorrection.Included(alongwithmanyotherdiagrams)intheBjerrum-Bohrresult[ 45 ],butnotineitherourresult( 2 )orthatofRadkowski[ 44 ].Chargedscalarlinesaresolidwithanarrow,photonlinesarewavyandgravitonlinesarewinding. Table2-1. Coefcientfunctionsandtensordifferentialoperators ICI(D,a,b)[TI] 112()2)]TJ /F10 7.97 Tf 17.23 4.71 Td[(2 D)]TJ /F10 7.97 Tf 6.59 0 Td[(234(b)]TJ /F10 7.97 Tf 6.58 0 Td[(1) (D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)(b)]TJ /F10 7.97 Tf 6.58 0 Td[(2)@@ @2+@@ @24a)]TJ /F4 11.955 Tf 9.96 0 Td[(14@()(@) @25)]TJ /F10 7.97 Tf 10.49 5.47 Td[(4a(b)]TJ /F10 7.97 Tf 6.59 0 Td[(1)(b)]TJ /F10 7.97 Tf 6.59 0 Td[(3) (b)]TJ /F10 7.97 Tf 6.59 0 Td[(2)2)]TJ /F10 7.97 Tf 11.16 5.47 Td[(8(b)]TJ /F10 7.97 Tf 6.58 0 Td[(1)(b+D)]TJ /F10 7.97 Tf 6.58 0 Td[(3) (b)]TJ /F10 7.97 Tf 6.59 0 Td[(2)2(D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)@@@@ @4 CoefcientfunctionsCI(D,a,b)andthetensordifferentialoperators[TI]forthegravitonpropagator( 2 )denedwiththegeneralgaugexingfunctional( 2 ). Table2-2. CoefcientsAI,BIandCI IAIBICI 11 2D(3D)]TJ /F4 11.955 Tf 9.97 0 Td[(8)3D)]TJ /F4 11.955 Tf 9.97 0 Td[(81 2(D)]TJ /F4 11.955 Tf 9.97 0 Td[(2)2(3D)]TJ /F4 11.955 Tf 9.96 0 Td[(8)21 4(D)]TJ /F4 11.955 Tf 9.97 0 Td[(4)2D1 2(D)]TJ /F4 11.955 Tf 9.97 0 Td[(4)21 4(D)]TJ /F4 11.955 Tf 9.97 0 Td[(4)2(D)]TJ /F4 11.955 Tf 9.97 0 Td[(2)231 2(D)]TJ /F4 11.955 Tf 9.97 0 Td[(4)2(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)D)]TJ /F4 11.955 Tf 9.96 0 Td[(41 2(D)]TJ /F4 11.955 Tf 9.97 0 Td[(4)(D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)2(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)4(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)(3D)]TJ /F4 11.955 Tf 9.97 0 Td[(8)D2)]TJ /F4 11.955 Tf 9.96 0 Td[(2D)]TJ /F4 11.955 Tf 9.96 0 Td[(2(D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)2(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)51 8(D)]TJ /F4 11.955 Tf 9.97 0 Td[(2)(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)D1 4(D)]TJ /F4 11.955 Tf 9.97 0 Td[(2)(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)1 8(D)]TJ /F4 11.955 Tf 9.97 0 Td[(2)3(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1) CoefcientsAI,BIandCIdenedinrelations( 2 )and( 2 ),underthereplacementi[new](x;x0))166(![TI]i(x;x0)foreachofthevetensordifferentialoperatorsdenedinTable 2-1 46

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CHAPTER3COVARIANTVACUUMPOLARIZATIONSONDESITTERBACKGROUNDHereIreproducetheresultsof[ 79 ],wherewederivedcovariantexpressionsfortheoneloopvacuumpolarizationinducedbyachargedscalarondeSitterbackground.Thisworkbuildsonchapter 2 ,increasingthedifcultyofthecalculationbyusingadeSitterbackgroundandusingthecumbersomedeSitterinvariantformalism.Sinceweonlyconsidertheoneloopvacuumpolarizationinducedbyascalar,itprovidesanicesteppingstonetothefullcalculationinvolvingthemuchmorecomplicatedgravitonpropagtor.Themotivationforthisworkwasndingthemosteffectiverepresentationforthevacuumpolarizationi[](x;x0)fromscalarquantumelectrodynamics(SQED)ondeSitterbackground[ 80 81 ](andindeedbyextrapolationdeterminethebestrepresentationingeneralforperformingquantumeldtheorycalculationsindeSitter).Thisquantityisnotablebecauseitprovidesamechanismbywhichthephotoncangainamassduringination[ 82 ].Ithasbeensuggestedthataresidualeffectfromthismassgenerationmightbetheoriginofprimordialmagneticelds[ 83 ].However,wewillherefocusonlyonhowtorepresentpreviousresultsforthevacuumpolarization[ 80 81 ]asbitensorfunctionsofspaceandtime.ThecontextisthelongcontroversyoverwhichisthemorepowerfulorganizingprincipleforquantumeldtheoryondeSitterbackground:theconformalatnessofdeSitter[ 38 84 85 ],orthemanifold'sisometrygroup[ 86 87 ].ClassicalelectromagnetismisconformallyinvariantinD=4dimensions,whichreducesthesystemtoitswellunderstoodatspacelimit.Thismotivatedthedecisiontoexploitconformalatnessinrepresentingtheoneloopvacuumpolarizationfromamassless,minimallycoupledscalar[ 80 ],andfromalightscalar[ 81 ].However,therehasneverbeenadeSitterinvariantformwithwhichtocompare.Anditisknownthatboththeoneloopeffective 47

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potentialandthetwoloopexpectationvaluesofscalarandeldstrengthbilinears[ 88 ]aresimpliedbyemployingadeSitterinvariantgaugeforthephotonpropagator[ 89 ].Ourgoalistore-expresstheoldresults[ 80 81 ]forthevacuumpolarizationinaformconsistingofcovariantderivativesactingonascalarstructurefunction,muchlikethefamiliaratspacerepresentation, hiat(x;x0)=h@2)]TJ /F3 11.955 Tf 9.97 0 Td[(@@i(x)]TJ /F5 11.955 Tf 9.97 0 Td[(x0)2.(3)TheformweemployisguaranteedtobemanifestlydeSitterinvariantifthevacuumpolarizationisphysicallyinvariant.Themassless,minimallycoupledscalarshowsaphysicalbreakingofdeSitterinvariance[ 90 ],andourcovariantrepresentationrevealsthatthisbreakingafictstheoneloopvacuumpolarization.ThemassivescalarhasadeSitterinvariantpropagatorsoweobtainafullyinvariantresultforitsstructurefunction.Inneithercaseistheresultparticularlyilluminating.Indeed,thedeSitterinvariantformalismseemstoobscuretheessentialphysics,butthiscouldnothavebeenknownbeforehand.ThevacuumpolarizationwewishtocalculatecomesfromthethreediagramsshowninFigs. 3-1 3-3 .WerepresentthedeSitterbackgroundgeometryinopenconformalcoordinates,whichisconducivetoregardingitasalimitingformofprimordialination.Theinvariantelementis ds2=a2)]TJ /F5 11.955 Tf 9.29 0 Td[(d2+d~xd~x,(3)wherea()=)]TJ /F10 7.97 Tf 13.84 4.71 Td[(1 H=eHtisthescalefactorandHistheHubbleparameter.Gravityistreatedasanon-dynamicalbackground,withthevectorpotentialA(x)andthecomplexscalar(x)beingthedynamicalvariables.TheLagrangiandescribingmasslessSQEDis Lm=0=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 4FFggp )]TJ /F5 11.955 Tf 9.3 0 Td[(g)]TJ /F4 11.955 Tf 11.95 0 Td[((@)]TJ /F5 11.955 Tf 11.96 0 Td[(ieA)(@+ieA)gp )]TJ /F5 11.955 Tf 9.3 0 Td[(g,(3) 48

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whereF@A)]TJ /F3 11.955 Tf 11.95 0 Td[(@Aistheeldstrength.TheLagrangianformassiveSQEDis Lm6=0=Lm=0)]TJ /F5 11.955 Tf 11.96 0 Td[(M2p )]TJ /F5 11.955 Tf 9.3 0 Td[(g,(3)whereweareparticularlyinterestedinthecaseMH.Inrepresentingfunctionswhichdependupontwopoints,xandx0,wewillmakeextensiveuseofthedeSitterlengthfunction y(x;x0)a()a(0)H2jj~x)]TJ /F3 11.955 Tf 11.44 .5 Td[(~x0jj2)]TJ /F4 11.955 Tf 11.95 0 Td[((j)]TJ /F3 11.955 Tf 11.95 0 Td[(0j)]TJ /F5 11.955 Tf 17.93 0 Td[(i)2.(3)WealsohaveneedoftwodeSitterbreakingcombinationsofthescalefactoraatxanda0atx0 uln(aa0),andvlna a0.(3)Derivativesofyandufurnishaconvenientbasisforrepresentingbi-vectorfunctionsofxandx0suchasthevacuumpolarization @y,@0y,@@0y,@u,@0u.(3)Wedonotneedderivativesofvbecausetheyarerelatedtothoseofu @v=@u,@0v=)]TJ /F3 11.955 Tf 9.3 0 Td[(@0u.(3)Itturnsoutthateithertakingcovariantderivativesofanyofthevebasistensors( 3 ),orcontractinganytwoofthemintooneanother,producesmetricsandmorebasistensors[ 78 91 ].Section 3.1 developsarepresentation,basedontwocovariantderivatives,forthevacuumpolarizationfromamassless,minimallycoupledscalar[ 80 ].InthiscasethereisdeSitterbreaking,whichrequirestwostructurefunctions.WhenthereisnodeSitterbreakingthevacuumpolarizationcanbeexpressedintermsofjustasinglestructurefunctionusingarepresentationwhichinvolvesfourcovariantderivatives.Section 3.2 derivesthisrepresentationforthecaseofamassivescalar[ 81 ].In 49

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section 3.3 weusetheresultofsection 3.2 tostudytheeffectiveeldequations.Anespeciallycounter-intuitiveandconfusingfeatureofthedeSitterinvariantrepresentationisthatlocalcorrectionstotheeffectiveeldequationsmanifestassurfacetermsfromtheinitialtime.Section 3.4 comprisesourdiscussion. 3.1AMassless,MinimallyCoupledScalarThepurposeofthissectionistoexpresstheoneloopvacuumpolarizationfromamassless,minimallycoupledscalarusinganeutralrepresentationthatwouldbemanifestlydeSitterinvariantifthephysicswas.Intherstsub-sectionwereviewtheprimitiveexpressionforthethreediagramsofFigs. 3-1 3-3 .Thenextsub-sectionderivesthemostgeneraltransverseformthevacuumpolarizationcantakeconsistentwiththesymmetriesofcosmology.Inthenalsub-sectionwegiveasimplesolutionforthestructurefunctionswhichreproducestheprimitiveresult. 3.1.1ThePrimitiveResultIfwecallthescalarpropagatori(x;x0)thenthethreediagramsinFigs. 3-1 3-3 makethefollowingcontributiontothevacuumpolarization ihi(x;x0)=)]TJ /F4 11.955 Tf 9.29 0 Td[(2ie2p )]TJ /F5 11.955 Tf 9.3 0 Td[(ggi(x;x)D(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0)+2e2p )]TJ /F5 11.955 Tf 9.3 0 Td[(ggp )]TJ /F5 11.955 Tf 9.3 0 Td[(g0g0h@i(x;x0)@0i(x;x0))]TJ /F5 11.955 Tf 9.97 0 Td[(i(x;x0)@@0i(x;x0)i+iZ@hp )]TJ /F5 11.955 Tf 9.3 0 Td[(ggg)]TJ /F5 11.955 Tf 9.96 0 Td[(gg@D(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0)i. (3) Equation( 3 )isvalidforanyscalar.Forthespecialcaseofthemassless,minimallycoupledscalarthepropagatorobeys iA(x;x0)=iD(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0) p )]TJ /F5 11.955 Tf 9.3 0 Td[(g,(3)where()]TJ /F5 11.955 Tf 9.3 0 Td[(g))]TJ /F15 5.978 Tf 7.78 3.26 Td[(1 2@(p )]TJ /F5 11.955 Tf 9.3 0 Td[(gg@)isthecovariantscalard'Alembertian.Ithaslongbeenknownthatequation( 3 )hasnodeSitterinvariantsolution[ 90 ],sothescalarpropagatormustbreaksomeofthedeSittersymmetries.WhenusingdeSitterasaparadigmforprimordialinationoneobviouslywishestopreservethesymmetriesof 50

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cosmology:homogeneity,isotropyandspatialatness.Inthatcasetheuniquesolutiontakestheform[ 56 ] iA(x;x0)=Ay(x;x0)+kln(aa0)=A(y)+ku,(3)wheretheconstantkis k=HD)]TJ /F10 7.97 Tf 6.59 0 Td[(2 (4)D 2\(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1) \(D 2).(3)ThedeSitterinvariantpartofthescalarpropagatoris A(y)=HD)]TJ /F10 7.97 Tf 6.59 0 Td[(2 (4)D 2(\(D 2) D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(14 yD 2)]TJ /F10 7.97 Tf 6.59 0 Td[(1+\(D 2+1) D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(24 yD 2)]TJ /F10 7.97 Tf 6.58 0 Td[(2)]TJ /F8 7.97 Tf 15.69 14.94 Td[(1Xn=1"\(n+D 2+1) (n)]TJ /F6 7.97 Tf 11.16 4.71 Td[(D 2+2)(n+1)!y 4n)]TJ /F14 5.978 Tf 7.78 3.26 Td[(D 2+2)]TJ /F4 11.955 Tf 13.15 8.08 Td[(\(n+D)]TJ /F4 11.955 Tf 9.96 0 Td[(1) n\(n+D 2)y 4n#)+A1, (3) whereA1isaD-dependentconstantwhichdivergesforD=4 A1=HD)]TJ /F10 7.97 Tf 6.59 0 Td[(2 (4)D=2\(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1) \(D 2)()]TJ /F3 11.955 Tf 9.3 0 Td[( 1)]TJ /F5 11.955 Tf 11.16 8.09 Td[(D 2+ D)]TJ /F4 11.955 Tf 9.97 0 Td[(1 2+ (D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)+ (1)).(3)Asaconsequenceofthepropagatorequation( 3 )thefunctionA(y)obeys (4y)]TJ /F5 11.955 Tf 9.97 0 Td[(y2)A00(y)+D(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)A0(y)=(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)k.(3)Derivativesofthepropagatorcanbeexpressedusingthetensorbasis( 3 ) @iA(x;x0)=A0(y)@y+k@u, (3) @0iA(x;x0)=A0(y)@0y+k@0u, (3) @@0iA(x;x0)=A0(y)@@0y+A00(y)@y@0y+iD(x)]TJ /F5 11.955 Tf 9.97 0 Td[(x0) H2p )]TJ /F5 11.955 Tf 9.3 0 Td[(g@u@0u. (3) 51

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Byapplyingtheseidentitieswecanwrite( 3 )as ihi(x;x0)=)]TJ /F4 11.955 Tf 9.29 0 Td[(2ie2p )]TJ /F5 11.955 Tf 9.3 0 Td[(ggghg+H)]TJ /F10 7.97 Tf 6.58 0 Td[(2@u@ui(A1+ku)D(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0)+2e2p )]TJ /F5 11.955 Tf 9.3 0 Td[(ggp )]TJ /F5 11.955 Tf 9.3 0 Td[(g0g0(@y@0yA02)]TJ /F5 11.955 Tf 9.96 0 Td[(AA00)]TJ /F5 11.955 Tf 9.96 0 Td[(kuA00)]TJ /F3 11.955 Tf 9.3 0 Td[(@@0yAA0+kuA0+@u@0ykA0+@y@0ukA0+@u@0uk2)+iZ@hp )]TJ /F5 11.955 Tf 9.3 0 Td[(ggg)]TJ /F5 11.955 Tf 9.96 0 Td[(gg@D(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0)i. (3) ThepresenceofdeSitterbreakingin( 3 )isevidentfromthedependenceuponuanditsderivatives. 3.1.2GeneralRepresentationsWewishtoexpress( 3 )usingamanifestlytransverserepresentationanalogoustotheform( 3 )usedforatspace.Animportantpropertyofthisrepresentationisthatitshouldbeneutral.Thatis,itshouldbeintermsofcovariantderivativessothatthestructurefunctionswillbedeSitterinvariantifthephysicalresulthappenstopossessthatsymmetry.Wearethereforeledtotheansatz ihi(x;x0)=p )]TJ /F5 11.955 Tf 9.3 0 Td[(g(x)p )]TJ /F5 11.955 Tf 9.3 0 Td[(g(x0)DD0hTi(x;x0),(3)wherethebi-tensor[T](x;x0)mustbeantisymmetricunder$and$.Itmustalsoobeythereectionidentity hTi(x;x0)=hTi(x0;x).(3)Themostgeneraltensorsatisfyingthesecriteria,consistentwithhomogeneityandisotropy,is hTi=@@0[y@0]@yf1+@[y@]@0[y@0]yf2+@[y@]@0[y@0]uf3+@[u@]@0[y@0]y~f3+@[u@]@0[y@0]uf4+@[y@]u@0[y@0]uf5. (3) 52

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Herethescalarstructurefunctionsfi(y,u,v)dependony,uandv,andwedene~f3(y,u,v)f3(y,u,)]TJ /F5 11.955 Tf 9.3 0 Td[(v).OfcoursemanifestdeSitterinvarianceprecludesf3,f4andf5,andwewillintheendemployonlyf1andf2.Thereisobviouslysomeredundancyinthevestructurefunctionsof( 3 )becauseactingthederivativesin( 3 )resultsinonlyfouralgebraicallyindependenttensors, DD0hTi(x;x0)=@@0yF1+@y@0yF2+@u@0yF3+@y@0ueF3+@u@0uF4. (3) Conservationimpliestwodifferentialrelations[ 91 ], h)]TJ /F4 11.955 Tf 9.3 0 Td[((2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)@y+D+@u+@v+2e)]TJ /F6 7.97 Tf 6.59 0 Td[(v@yiF3=h(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)@y)]TJ /F5 11.955 Tf 9.96 0 Td[(D)]TJ /F3 11.955 Tf 9.96 0 Td[(@u)]TJ /F3 11.955 Tf 9.96 0 Td[(@viF1+h(4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)@y+(D+1)(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)+(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)]TJ /F4 11.955 Tf 9.97 0 Td[(2e)]TJ /F6 7.97 Tf 6.59 0 Td[(v)(@u+@v)iF2, (3) h)]TJ /F4 11.955 Tf 9.3 0 Td[((2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)@y+(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)+@u+@v+2e)]TJ /F6 7.97 Tf 6.58 0 Td[(v@yiF4=)]TJ /F4 11.955 Tf 9.3 0 Td[(2e)]TJ /F6 7.97 Tf 6.59 0 Td[(v(@u+@v)F1)]TJ /F4 11.955 Tf 9.97 0 Td[(2e)]TJ /F6 7.97 Tf 6.59 0 Td[(vF3+h(4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)@y+D(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)+(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)]TJ /F4 11.955 Tf 9.96 0 Td[(2e)]TJ /F6 7.97 Tf 6.59 0 Td[(v)(@u+@v)ieF3. (3) ThissuggeststhatoneshouldbeabletowritethescalarcoefcientfunctionsFi(y,u,v)ofexpression( 3 )intermsofjusttwomasterstructurefunctions,whichwemightcall(y,u,v)and(y,u,v).Afterlongreection,oneseesthatthefollowingsubstitutionsfortheFi(y,u,v)reducetheconservationrelations( 3 3 )totautologies, F1=h)]TJ /F4 11.955 Tf 9.3 0 Td[((4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)@y)]TJ /F4 11.955 Tf 9.96 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y))]TJ /F4 11.955 Tf 9.96 0 Td[(2(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)@u+4cosh(v)@u)]TJ /F4 11.955 Tf 9.96 0 Td[(4sinh(v)@vi+h4cosh(v))]TJ /F4 11.955 Tf 9.97 0 Td[((2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)i)]TJ /F4 11.955 Tf 9.96 0 Td[(, (3) F2=h(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)@y)]TJ /F5 11.955 Tf 9.96 0 Td[(D+1)]TJ /F4 11.955 Tf 9.96 0 Td[(2@ui)]TJ /F4 11.955 Tf 9.96 0 Td[(, (3) F3=)]TJ /F4 11.955 Tf 9.29 0 Td[(2ev(@u)]TJ /F3 11.955 Tf 9.96 0 Td[(@v))]TJ /F4 11.955 Tf 9.96 0 Td[(2ev+, (3) eF3=)]TJ /F4 11.955 Tf 9.29 0 Td[(2e)]TJ /F6 7.97 Tf 6.58 0 Td[(v(@u+@v))]TJ /F4 11.955 Tf 9.96 0 Td[(2e)]TJ /F6 7.97 Tf 6.59 0 Td[(v+, (3) F4=)]TJ /F4 11.955 Tf 9.29 0 Td[(4+(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y). (3) 53

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Heretheauxiliaryfunction(y,u,v)is, (y,u,v)Zdy(@2u)]TJ /F3 11.955 Tf 9.97 0 Td[(@2v)(y,u,v)+(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)(y,u,v))]TJ /F4 11.955 Tf 9.97 0 Td[((D)]TJ /F4 11.955 Tf 9.97 0 Td[(2)Zdy(y,u,v).(3)GiventhecoefcientfunctionsFi(y,u,v),onecanderivearstorderdifferentialequationinonevariableforthemasterstructurefunction(y,u,v)bytakingasuperpositionofrelations( 3 3 ), )]TJ /F4 11.955 Tf 11.96 0 Td[(4(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1))]TJ /F4 11.955 Tf 9.96 0 Td[(8@u=(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)F1+(4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)F2+(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)(F3+eF3))]TJ /F5 11.955 Tf 9.96 0 Td[(F4.(3)Thedifferenceof( 3 )and( 3 )givesanalgebraicrelationfor(y,u,v)once(y,u,v)isknown, )]TJ /F4 11.955 Tf 11.95 0 Td[(4sinh(v)@u+4cosh(v)@v)]TJ /F4 11.955 Tf 9.96 0 Td[(4sinh(v)=F3)]TJ /F9 11.955 Tf 10.99 3.15 Td[(eF3.(3)NotethatdeSitterinvarianceimplies=(y)and=0.Itremainstorelatethemasterstructurefunctions(y,u,v)and(y,u,v)tothestructurefunctionsfi(y,u,v)ofourrepresentation( 3 3 ).Table 3-1 givesthecoefcientfunctionsFi(y,u,v)forthestructurefunctionf1(y,u,v).Applyingrelations( 3 3 )impliesthattheassociatedmasterstructuresare, 1=H4 2n(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)f1)]TJ /F4 11.955 Tf 9.96 0 Td[((2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)@yf1o, (3) 1=H4 2n(@2u)]TJ /F3 11.955 Tf 9.97 0 Td[(@2v)f1o. (3) Table 3-2 givesthecoefcientfunctionsFi(y,u,v)forf2(y,u,v),fromwhichweinfer, 2=H4 4nD(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)f2+(4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)@yf2o, (3) 2=H4 4n(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)(@2u)]TJ /F3 11.955 Tf 9.97 0 Td[(@2v)f2o. (3) 54

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Themasterstructurefunctionsforf3(y,u,v)andef3(y,u,v)=f3(y,u,)]TJ /F5 11.955 Tf 9.3 0 Td[(v)are, 3=H4 4n)]TJ /F4 11.955 Tf 9.3 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)(f3+ef3)+(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)@y(f3+ef3))]TJ /F4 11.955 Tf 9.96 0 Td[(2@y(evf3+e)]TJ /F6 7.97 Tf 6.58 0 Td[(vef3)o, (3) 3=H4 4n2@yh@u(evf3+e)]TJ /F6 7.97 Tf 6.59 0 Td[(vef3))]TJ /F3 11.955 Tf 9.96 0 Td[(@v(evf3)]TJ /F5 11.955 Tf 9.97 0 Td[(e)]TJ /F6 7.97 Tf 6.59 0 Td[(vef3)i)]TJ /F4 11.955 Tf 9.96 0 Td[((@2u)]TJ /F3 11.955 Tf 9.96 0 Td[(@2v)(f3+ef3)o. (3) Theanalogousresultforf4(y,u,v)is, 4=H4 4n@yf4o, (3) 4=H4 4n)]TJ /F4 11.955 Tf 9.29 0 Td[((D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)@yf4+(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)@2yf4)]TJ /F4 11.955 Tf 9.96 0 Td[(2@y@uf4o. (3) Andf5(y,u,v)gives, 5=H4 4n)]TJ /F5 11.955 Tf 9.3 0 Td[(Df5+2(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)@yf5)]TJ /F4 11.955 Tf 9.96 0 Td[(4cosh(v)@yf5o, (3) 5=H4 4n(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)f5)]TJ /F4 11.955 Tf 9.96 0 Td[((D+1)(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)]@yf5)]TJ /F4 11.955 Tf 9.97 0 Td[((4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)@2yf5+2@uf5+[)]TJ /F4 11.955 Tf 7.3 0 Td[(2(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)+4cosh(v)]@u@yf5)]TJ /F4 11.955 Tf 9.96 0 Td[(4@v@y[sinh(v)f5])]TJ /F4 11.955 Tf 11.95 0 Td[((@2u)]TJ /F3 11.955 Tf 9.96 0 Td[(@2v)f5o. (3) Fromtheseexpressionsoneseesthatthevacuumpolarizationcanbedescribedintermsofanytwoofthestructurefunctionsfi(y,u,v).WhentheresultisdeSitterinvariantthenitrequiresonlyasinglestructurefunction,whichcanbeeitherf1(y)orf2(y). 3.1.3RepresentationforthisSystemItremainstoworkoutthestructurefunctionsfortheprimitiveresult( 3 ).SubstitutingthecoefcientfunctionsFi(y,u,v)fromexpression( 3 )intorelation( 3 ),andmakingjudicioususeoftheApropagatorequation( 3 ),givestherstmasterstructurefunction, =)]TJ /F5 11.955 Tf 26.23 8.09 Td[(e2 2(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)[(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)A0)]TJ /F5 11.955 Tf 9.96 0 Td[(k]A+(4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)A02)]TJ /F5 11.955 Tf 10.5 8.09 Td[(ke2 2[(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)A0)]TJ /F5 11.955 Tf 9.96 0 Td[(k]u, (3) 55

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wherewedroppedlocalterms/D(x)]TJ /F5 11.955 Tf 12.61 0 Td[(x0)andalsotheprefactorp )]TJ /F5 11.955 Tf 9.3 0 Td[(gp )]TJ /F5 11.955 Tf 9.3 0 Td[(g0=eDuin( 3 ),asitcontributestotheintegralmeasuretomakeitcovariant.Doingthesamethingforrelation( 3 )impliesthatthesecondmasterstructurefunctionis, =ke2 2[(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)A0)]TJ /F5 11.955 Tf 9.96 0 Td[(k].(3)Atthispointwemustmakeachoicebetweenthetenpossiblepairsofstructurefunctionsfi(y,u,v)thatcouldbeusedtorepresenttheresult.Wechosef1andf2,thetwostructurefunctionsthatwouldbedeSitterinvariantifthesystemwas.Inviewofrelations( 3 ),( 3 )and( 3 ),therequirementthat1+2=impliestheform, f1(y,u,v)=)]TJ /F5 11.955 Tf 10.49 8.08 Td[(k2e2 4H4(u2)]TJ /F5 11.955 Tf 9.96 0 Td[(v2)+f1a(y,u+v)+f1b(y,u)]TJ /F5 11.955 Tf 9.96 0 Td[(v), (3) f2(y,u,v)=ke2 2H4A0(y)(u2)]TJ /F5 11.955 Tf 9.96 0 Td[(v2)+f2a(y,u+v)+f2b(y,u)]TJ /F5 11.955 Tf 9.96 0 Td[(v). (3) Thenrequiring1+2=andrelations( 3 ),( 3 ),and( 3 )implies, f1(y,u,v)=)]TJ /F5 11.955 Tf 10.5 8.09 Td[(k2e2 4H4(u2)]TJ /F5 11.955 Tf 11.96 0 Td[(v2)+1(y)u+2(y), (3) f2(y,u,v)=ke2 2H4A0(y)(u2)]TJ /F5 11.955 Tf 11.96 0 Td[(v2)+1(y)u+2(y), (3) wherethefunctionsi(y)andi(y)satisfy, )]TJ /F4 11.955 Tf 10.49 8.09 Td[(2ke2 H4[(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)A0)]TJ /F5 11.955 Tf 9.96 0 Td[(k]=2(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)1)]TJ /F4 11.955 Tf 9.97 0 Td[(2(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)01+D(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)1+(4y)]TJ /F5 11.955 Tf 11.95 0 Td[(y2)01, (3) )]TJ /F4 11.955 Tf 27.11 8.08 Td[(2e2 (D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)H4n(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)[(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)A0)]TJ /F5 11.955 Tf 9.96 0 Td[(k]A+(4y)]TJ /F5 11.955 Tf 9.97 0 Td[(y2)A02+k2o=2(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)2)]TJ /F4 11.955 Tf 9.97 0 Td[(2(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)02+D(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)2+(4y)]TJ /F5 11.955 Tf 11.95 0 Td[(y2)02. (3) Becausewehaveonlytwoequations( 3 3 )intermsoffourfunctionsi(y)andi(y),therearemanysolutions.Perhapsthenicestandoneofgreatsignicanceforthenextsectionresultsfromtheansatz, i(y)=0i(y),i(y)=)]TJ /F3 11.955 Tf 9.3 0 Td[(00i(y).(3) 56

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Thisansatzeffectsthefollowingsimplicationoftherighthandsidesof( 3 3 ), 2(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)i(y))]TJ /F4 11.955 Tf 9.96 0 Td[(2(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)0i(y)+D(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)i(y)+(4y)]TJ /F5 11.955 Tf 11.95 0 Td[(y2)0i(y)=)]TJ /F3 11.955 Tf 13.92 8.08 Td[(@ @yh(4y)]TJ /F5 11.955 Tf 9.97 0 Td[(y2)00i(y)+D(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)0i(y))]TJ /F4 11.955 Tf 9.97 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)i(y)i, (3) =)]TJ /F5 11.955 Tf 13.35 8.08 Td[(d dyh H2)]TJ /F4 11.955 Tf 9.97 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)ii(y))]TJ /F5 11.955 Tf 25.96 8.08 Td[(d dyDB H2i(y). (3) Onecanseefromrelation( 3 )thatitwouldbehighlydesirabletoexpressthelefthandsidesof( 3 3 )as@=@yofsomething.ThedesiredfunctioninvolvesthepropagatorofascalarwithmassM2=(D)]TJ /F4 11.955 Tf 11.96 0 Td[(2)H2,whichobeystheequation, DBiB(x;x0)=iD(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0) p )]TJ /F5 11.955 Tf 9.3 0 Td[(g.(3)Unlikethemasslessscalar,thepropagatorofthismassivescalarcanbeexpressedintermsofadeSitterinvariantfunctionofy,iB(x;x0)=B(y).Theseriesexpansionofthisfunctionis, B(y)=HD)]TJ /F10 7.97 Tf 6.59 0 Td[(2 (4)D 2()]TJ /F9 11.955 Tf 6.78 13.27 Td[(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(14 yD 2)]TJ /F10 7.97 Tf 6.59 0 Td[(1+1Xn=0"\(n+D 2) \(n+2)y 4n)]TJ /F14 5.978 Tf 7.78 3.26 Td[(D 2+2)]TJ /F4 11.955 Tf 10.49 8.09 Td[(\(n+D)]TJ /F4 11.955 Tf 9.96 0 Td[(2) \(n+D 2)y 4n#). (3) Theseriesexpansions( 3 )and( 3 )implytworelationsbetweenA(y)andB(y)ofgreatutility, (2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)A0(y))]TJ /F5 11.955 Tf 9.96 0 Td[(k=2B0(y), (3) (4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)A0(y)+k(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)=)]TJ /F4 11.955 Tf 9.3 0 Td[(2(D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)B(y). (3) Relation( 3 )impliesthatouransatz( 3 )reducesequation( 3 )totheform, DB1(y)=4ke2 H2B(y).(3)Thesolutionis, 1(y)=4ke2 H2iBB(x;x0),(3) 57

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wherethedeSitterinvariantbiscalariBB(x;x0)wasintroducedinapreviousstudyofthegravitonpropagator[ 92 ].Usingrelation( 3 )similarlyon( 3 )implies, DB2(y)=4e2 (D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)H2Zdyn(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)A(y)B0(y))]TJ /F4 11.955 Tf 11.95 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)A0(y)B(y))]TJ /F5 11.955 Tf 11.96 0 Td[(kB0(y)o.(3)InthenextsectionwewillderiveadeSitterinvariantGreen'sfunctionwhichcanbeusedtosolvefor2(y). 3.2AMassiveScalarThepurposeofthissectionistoderiveadeSitterinvariantrepresentationfortheoneloopvacuumpolarizationfromachargedscalarwithasmallmass.TherstsubsectionisdevotedtopresentingtheprimitiveresultfromthethreediagramsinFigs 3-1 3-3 ,withspecialattentiontotheexpansionofthemassivescalarpropagatorforsmallmass.Inthesecondsubsectionwere-expresstheprimitiveresult.ItisatthispointthatwederivetheGreen'sfunctionforthedifferentialoperatorDBdenedin( 3 ).Thenalsubsectionexplainsrenormalization. 3.2.1ThePrimitiveResultThepropagatorofaminimallycoupledscalarofmassMobeystheequation, ()]TJ /F5 11.955 Tf 9.97 0 Td[(M2)iT(x;x0)=iD(x)]TJ /F5 11.955 Tf 9.97 0 Td[(x0) p )]TJ /F5 11.955 Tf 9.3 0 Td[(g.(3)ForM2>0ithasadeSitterinvariantsolutioniT(x;x0)=T(y), T(y)=HD)]TJ /F10 7.97 Tf 6.58 0 Td[(2 (4)D 2\(D)]TJ /F10 7.97 Tf 6.59 0 Td[(1 2+)\(D)]TJ /F10 7.97 Tf 6.59 0 Td[(1 2)]TJ /F3 11.955 Tf 9.96 0 Td[() \(D 2)2F1D)]TJ /F4 11.955 Tf 11.96 0 Td[(1 2+;D)]TJ /F4 11.955 Tf 11.95 0 Td[(1 2)]TJ /F3 11.955 Tf 9.96 0 Td[(;D 2;1)]TJ /F5 11.955 Tf 11.16 8.08 Td[(y 4,(3)whereq )]TJ /F6 7.97 Tf 6.68 -4.97 Td[(D)]TJ /F10 7.97 Tf 6.59 0 Td[(1 22)]TJ /F6 7.97 Tf 13.15 4.71 Td[(M2 H2[ 93 ].(TheB-typepropagator( 3 3 )representsthespecialcaseofM2=(D)]TJ /F4 11.955 Tf 12.08 0 Td[(2)H2.)ItisusefultoextractthemostultravioletsingulartermfromT(y), T(y)HD)]TJ /F10 7.97 Tf 6.58 0 Td[(2 4D 2)]TJ /F9 11.955 Tf 6.78 13.27 Td[(D 2)]TJ /F4 11.955 Tf 9.97 0 Td[(11 yD 2)]TJ /F10 7.97 Tf 6.58 0 Td[(1+T(y)K1 yD 2)]TJ /F10 7.97 Tf 6.59 0 Td[(1+T(y).(3) 58

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Justasinatspace,nonzeromassgreatlycomplicatesthespacetimedependenceofpropagators.Forexample,theseriesexpansionofT(y)inpowersofydoesnotterminateinD=4dimensionsformostchoicesofM2>0.OnesignalthatthereisnodeSitterinvariantsolutionforthemasslesscaseisthatT(y)divergesasMgoestozero.ThiscanbequantiedbygivingtheLaurentexpansionforT(y)intermsoftheparameters(D)]TJ /F10 7.97 Tf 6.58 0 Td[(1 2))]TJ /F3 11.955 Tf 11.96 0 Td[(.InD=4dimensions,wehave[ 81 ], T(y)=1 2s)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2lny 4)]TJ /F4 11.955 Tf 11.96 0 Td[(1+sT(y)+O(s2),(3)wherethefunctionT(y)is, T(y)=1 2Li2y 4+lny 41 2ln1)]TJ /F5 11.955 Tf 13.15 8.08 Td[(y 4+1)]TJ /F4 11.955 Tf 23.88 8.08 Td[(1 4)]TJ /F5 11.955 Tf 11.96 0 Td[(y,(3)andwhereLi2(z)=P1n=1zn=n2isthepolylogarithmfunction.Forsmallmassthe1=scontributiontothevacuumpolarizationisthedominanteffect.SubstitutingiT(x;x0)forthescalarpropagatorinexpression( 3 )givestheresultforthethreediagramsofFigs 3-1 3-3 .Derivativesofthepropagatorcanbeexpressedusingthetensorbasis( 3 ) @iT(x;x0)=T0(y)@y, (3) @0iT(x;x0)=T0(y)@0y, (3) @@0iT(x;x0)=T0(y)@@0y+T00(y)@y@0y+iD(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0) H2p )]TJ /F5 11.955 Tf 9.29 0 Td[(g@u@0u. (3) Byapplyingtheseidentitieswecanwrite( 3 )as ihi(x;x0)=p )]TJ /F5 11.955 Tf 9.3 0 Td[(g(x)g(x)p )]TJ /F5 11.955 Tf 9.3 0 Td[(g(x0)g(x0)@@0y()]TJ /F4 11.955 Tf 9.3 0 Td[(2e2TT0)+@y@0y2e2T02)]TJ /F5 11.955 Tf 9.96 0 Td[(TT00)]TJ /F4 11.955 Tf 9.96 0 Td[(2e2T(0)hg+H)]TJ /F10 7.97 Tf 6.58 0 Td[(2@u@uiiD(x)]TJ /F5 11.955 Tf 9.97 0 Td[(x0) p )]TJ /F5 11.955 Tf 9.3 0 Td[(g+Zgh+(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)H2i)]TJ /F5 11.955 Tf 9.96 0 Td[(DDiD(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0) p )]TJ /F5 11.955 Tf 9.3 0 Td[(g, (3) 59

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where T(0)=HD)]TJ /F10 7.97 Tf 6.59 0 Td[(2 (4)D 2\(D)]TJ /F10 7.97 Tf 6.58 0 Td[(1 2+)\(D)]TJ /F10 7.97 Tf 6.58 0 Td[(1 2)]TJ /F3 11.955 Tf 9.96 0 Td[() \(1 2+)\(1 2)]TJ /F3 11.955 Tf 11.95 0 Td[(). 3.2.2The(y)RepresentationBecausetheprimitiveresult( 3 )isdeSitterinvariant,itcanbegivenamanifestlydeSitterinvariantexpressionusingthestructurefunctionsf1(y)andf2(y)ofourgeneralrepresentation( 3 3 ).Afurthersimplicationarisesifwerelatethestructurefunctionsasf1(y)=0(y)andf2(y)=)]TJ /F3 11.955 Tf 9.29 0 Td[(00(y), ihi(x;x0)=p )]TJ /F5 11.955 Tf 9.3 0 Td[(g(x)g(x)p )]TJ /F5 11.955 Tf 9.3 0 Td[(g(x0)g(x0)DD0(@@0[y@0]@y0(y))]TJ /F3 11.955 Tf 9.96 0 Td[(@[y@]@0[y@0]y00(y)), (3) =p )]TJ /F5 11.955 Tf 9.3 0 Td[(ggp )]TJ /F5 11.955 Tf 9.3 0 Td[(g0g0DD0D[D0[[n@]@0]]y(y)o. (3) Thedoublebracketsusedhereandhenceforthservetodistinguishwhichindexgroupisanti-symmetrized, D[D0[[n@]@0]]yo1 4DD0n@@0yo)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 4DD0n@@0yo)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 4DD0n@@0yo+1 4DD0n@@0yo. (3) Therelationsf1(y)=0(y)andf2(y)=)]TJ /F3 11.955 Tf 9.3 0 Td[(00(y)arethesameas( 3 )thatwasconsideredintheprevioussection.Asimpleconsequenceofapplyingthatanalysisto( 3 ),withoutworryingaboutthedeltafunctioncontributions,istherelation, (D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)H4@ @yhDB H2(y)i=2e2n(4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)(T02)]TJ /F5 11.955 Tf 9.96 0 Td[(TT00))]TJ /F4 11.955 Tf 9.96 0 Td[((2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)TT0o,(3)whereDB)]TJ /F4 11.955 Tf 10.22 0 Td[((D)]TJ /F4 11.955 Tf 10.22 0 Td[(2)H2.Ifweintroducetheindeniteintegralsymbol,I[f]Rdyf(y),thedifferentialequationfor(y)becomes, DB H2(y)=2e2 (D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)H4Ih(4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)(T02)]TJ /F5 11.955 Tf 9.96 0 Td[(TT00))]TJ /F4 11.955 Tf 9.96 0 Td[((2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)TT0iS(y).(3) 60

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Itturnsoutthatenforcingthisequationautomaticallyrecoverstheundifferentiateddeltafunctionsofexpression( 3 ).Wewillshortlydemonstratethataspecialchoiceforthehomogeneouspartofthesolutionrecoversthecontributionfromtheeldstrengthrenormalization.Ifweignoredeltafunctioncontributions,theactionofDBonafunctionofycanbewrittenas, DB H2(y)=(4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)00(y)+D(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)0(y))]TJ /F4 11.955 Tf 11.95 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)(y).(3)Thisisasecondorder,lineardifferentialoperatorinywhosetwohomogeneoussolutionsareeasilyseentobetheB-typepropagatoranditstranslationtotheantipodalpoint, 1(y)=B(y),2(y)=B(4)]TJ /F5 11.955 Tf 9.96 0 Td[(y).(3)From( 3 )wendthattheirWronskianis, W(y)1(y)02(y))]TJ /F3 11.955 Tf 9.96 0 Td[(01(y)2(y)=H2D)]TJ /F10 7.97 Tf 6.59 0 Td[(4 4D\(D 2)\(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1) (4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)D 2.(3)OfcourseactingDBon1(y)reallyproducesadeltafunctionaty=0,justasactingDBon2(y)producesadeltafunctionattheantipodalpointy=4.TheuniqueGreen'sfunctionwhichavoidsbothpolesis, GB(y;y0)=)]TJ /F3 11.955 Tf 10.49 8.09 Td[((y)]TJ /F5 11.955 Tf 9.96 0 Td[(y0)1(y)2(y0)+(y0)]TJ /F5 11.955 Tf 9.96 0 Td[(y)2(y)1(y0) (4y0)]TJ /F5 11.955 Tf 9.96 0 Td[(y02)W(y0).(3)PossessionofaGreen'sfunctionsuchas( 3 )immediatelydenesthesolutionof( 3 )uptohomogeneouscontributions.Expression( 3 )containsnodeltafunctionswhichbecomesingularattheantipodalpoint,sotherecanbenocontaminationfromB(4)]TJ /F5 11.955 Tf 12.13 0 Td[(y).However,( 3 )doescontaindeltafunctionwhichbecomesingularaty=0,sowemustallowatermproportionaltoB(y).Hencethesolutionfor(y)takestheform, (y)=ConstB(y)+Z40dy0GB(y;y0)S(y0).(3) 61

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Toxthehomogeneoustermweactthefourderivativesof( 3 )onB(y), DD0D[D0[[n@]@0]]yB(y)o=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 4H4(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)2(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)@@0B)]TJ /F5 11.955 Tf 9.29 0 Td[(H2(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)h@yD+@0yD0i@@0B+@@0yDD0@@0B+H2(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)h@y@0B+@0y0@Bi)]TJ /F3 11.955 Tf 11.96 0 Td[(@@0yD0@0B)]TJ /F3 11.955 Tf 9.3 0 Td[(@@0y0D@B+@@0y0B. (3) Ofcoursewealreadyknowthatexpression( 3 )mustvanishexceptfordeltafunctionterms.Wecancollectthevariousdeltafunctiontermsbymakinguseoftheidentities, B=0B=(D)]TJ /F4 11.955 Tf 9.97 0 Td[(2)H2B+iD(x)]TJ /F5 11.955 Tf 11.96 0 Td[(x0) p )]TJ /F5 11.955 Tf 9.3 0 Td[(g, (3) DD0B=@@0yB0+@y@0yB00+@u@0uiD(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0) p )]TJ /F5 11.955 Tf 9.3 0 Td[(g, (3) H2I[B]=(4y)]TJ /F5 11.955 Tf 9.97 0 Td[(y2)B0+D(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)B=2(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)B)]TJ /F4 11.955 Tf 9.96 0 Td[(2k. (3) Aftermuchworkwend, DD0D[D0[[n@]@0]]yB(y)o=)]TJ /F4 11.955 Tf 10.5 8.08 Td[(1 2H2"gh+(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)H2i)]TJ /F5 11.955 Tf 9.96 0 Td[(DD0#iD(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0) p )]TJ /F5 11.955 Tf 9.3 0 Td[(g. (3) Comparisonwith( 3 )revealsthattheconstantinexpression( 3 )mustbe)]TJ /F4 11.955 Tf 9.3 0 Td[(2Z=H2, (y)=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(2Z H2B(y)+Z40dy0GB(y;y0)S(y0).(3) 3.2.3RenormalizationItisbothtediousandunnecessarytomaintainfulldimensionalregularizationwhenintegratingtheGreen'sfunctionupagainstthesourceinexpression( 3 ).Theprimitiveresult( 3 )isonlyquadraticallydivergent,whichmeansitgoeslike1=y3nearcoincidenceinD=4dimensions.Henceextractingfourderivatives,thewaywedoin( 3 ),leavesthemostsingulartermin(y)behavinglike1=ynearcoincidence.This 62

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isintegrable,sowecouldsetD=4directly,exceptforthefactthattheD-dependentcoefcientofthisleadingtermhappenstodivergelike1=(D)]TJ /F4 11.955 Tf 12.45 0 Td[(4).ItisonlynecessarytokeepthisleadingterminDdimensions;theremaindercanbeevaluatedinD=4dimensions.Ofcoursethedivergenceisabsorbedbytheeldstrengthrenormalization.Weworkoutthedetailsbelow.Theanalysisbeginsbyemployingexpressions( 3 3 )toinfer, (4y)]TJ /F5 11.955 Tf 9.97 0 Td[(y2)(T02)]TJ /F5 11.955 Tf 9.97 0 Td[(TT00))]TJ /F4 11.955 Tf 9.96 0 Td[((2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)TT0=K2()]TJ /F4 11.955 Tf 10.49 8.08 Td[((D)]TJ /F4 11.955 Tf 9.97 0 Td[(2) yD)]TJ /F10 7.97 Tf 6.59 0 Td[(1)]TJ /F4 11.955 Tf 16.32 8.08 Td[(3 sy2+3ln(y 4) y2+9 y2+3 4 y+3ln(y 4) (4)]TJ /F5 11.955 Tf 9.97 0 Td[(y)2+3 4 4)]TJ /F5 11.955 Tf 9.96 0 Td[(y+O(s)), (3) wherewerecallthats(D)]TJ /F10 7.97 Tf 6.58 0 Td[(1 2))]TJ /F3 11.955 Tf 12.73 0 Td[(vanishesasthescalarmassgoestozero.Nowsubstitute( 3 )inthedenition( 3 )ofthesourcetoobtain, S(y)2e2 (D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)H4Ih(4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)(T02)]TJ /F5 11.955 Tf 9.96 0 Td[(TT00))]TJ /F4 11.955 Tf 9.96 0 Td[((2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)TT0i, (3) =2e2K2 (D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)H4(1 yD)]TJ /F10 7.97 Tf 6.59 0 Td[(2+3 sy)]TJ /F4 11.955 Tf 11.16 8.79 Td[(3ln(y 4) y)]TJ /F4 11.955 Tf 11.15 8.09 Td[(12 y+3ln(y 4) 4)]TJ /F5 11.955 Tf 9.97 0 Td[(y+O(s)), (3) 2e2K2 (D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)H41 yD)]TJ /F10 7.97 Tf 6.59 0 Td[(2+SR(y). (3) Thisdistinguishesthemostsingularpartofthesourcefromthelesssingularremainder,SR(y).Thenextstepistomakeasimilardistinctionbetweenthemostsingularpartof(y)andthelesssingularremainderR(y), (y))]TJ /F4 11.955 Tf 23.12 8.09 Td[(2Z H2B(y)+ yD)]TJ /F10 7.97 Tf 6.58 0 Td[(3+R(y).(3)Ifwedonotworryaboutdeltafunctions,theresultofactingDB=H2is, DB H2(y)=2(D)]TJ /F4 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F4 11.955 Tf 9.96 0 Td[(4) yD)]TJ /F10 7.97 Tf 6.59 0 Td[(2+(D)]TJ /F4 11.955 Tf 9.96 0 Td[(4) yD)]TJ /F10 7.97 Tf 6.58 0 Td[(3+DB H2R(y).(3) 63

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Equating( 3 )to( 3 )impliesthatthecoefcientis, =e2K2 (D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)(D)]TJ /F4 11.955 Tf 9.97 0 Td[(3)(D)]TJ /F4 11.955 Tf 9.97 0 Td[(4)H4,(3)andalsothat(forD=4)theresidualpartofobeys, DB H2R(y)=SR(y))]TJ /F5 11.955 Tf 19.79 8.08 Td[(e2 4841 y=e2 84()]TJ /F4 11.955 Tf 10.5 8.78 Td[(ln(y 4) y+[1 s)]TJ /F10 7.97 Tf 11.16 4.71 Td[(25 6] y+ln(y 4) 4)]TJ /F5 11.955 Tf 9.96 0 Td[(y+O(s)).(3)Atthisstagewecancombinethepotentiallydivergentpartsof(y), (y))]TJ /F3 11.955 Tf 11.95 0 Td[(R(y)=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(2Z H2B(y)+e2K2 (D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)(D)]TJ /F4 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F4 11.955 Tf 9.96 0 Td[(4)H41 yD)]TJ /F10 7.97 Tf 6.59 0 Td[(3.(3)Nowrecallfrom( 3 )thatB(y)=K=yD 2)]TJ /F10 7.97 Tf 6.59 0 Td[(1+O(D)]TJ /F4 11.955 Tf 12.18 0 Td[(4).Thedivergentpartsof( 3 )willcancelinD=4dimensionsifwechoose, Z=e2K 2(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)(D)]TJ /F4 11.955 Tf 9.96 0 Td[(3)(D)]TJ /F4 11.955 Tf 9.96 0 Td[(4)H2.(3)Thispermitsustonallytaketheunregulatedlimit, limD!4h(y))]TJ /F3 11.955 Tf 11.96 0 Td[(R(y)i=e2 964n)]TJ /F4 11.955 Tf 10.5 8.09 Td[(ln(y) y+ln(y 4) 4)]TJ /F5 11.955 Tf 9.96 0 Td[(yo.(3)Wenoteinpassingthatthedivergentpartoftheeldstrengthrenormalization( 3 )agreeswiththetwopreviousdeSitterresults[ 80 81 ]and,indeed,withtheatspaceresult.ItremainstoevaluateR(y)inD=4dimensions.Fromequations( 3 )and( 3 )wehave, R(y)=)]TJ /F4 11.955 Tf 13.92 8.09 Td[(1 4yZy0dy0y0hSR(y0))]TJ /F5 11.955 Tf 17.8 8.09 Td[(e2 4841 y0i)]TJ /F4 11.955 Tf 27.25 8.09 Td[(1 4(4)]TJ /F5 11.955 Tf 9.96 0 Td[(y)Z4ydy0(4)]TJ /F5 11.955 Tf 9.96 0 Td[(y0)hSR(y0))]TJ /F5 11.955 Tf 17.79 8.09 Td[(e2 4841 y0i, (3) =e2 84()]TJ /F4 11.955 Tf 15.02 8.78 Td[(ln2(y 4) 2(4)]TJ /F5 11.955 Tf 9.96 0 Td[(y)+h1 s)]TJ /F4 11.955 Tf 11.16 8.09 Td[(13 6iln(y 4) 4)]TJ /F5 11.955 Tf 9.96 0 Td[(y+[2 6)]TJ /F4 11.955 Tf 9.96 0 Td[(Li2(1)]TJ /F6 7.97 Tf 11.16 5.04 Td[(y 4)] y+O(s)). (3) 64

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Combining( 3 )and( 3 )givesusthefullyrenormalizedstructurefunction, ren=e2 84()]TJ /F4 11.955 Tf 10.49 8.09 Td[(ln(y) 12y)]TJ /F4 11.955 Tf 17.67 8.79 Td[(ln2(y 4) 2(4)]TJ /F5 11.955 Tf 9.96 0 Td[(y)+h1 s)]TJ /F4 11.955 Tf 13.15 8.09 Td[(25 12iln(y 4) 4)]TJ /F5 11.955 Tf 9.97 0 Td[(y+[2 6)]TJ /F4 11.955 Tf 9.97 0 Td[(Li2(1)]TJ /F6 7.97 Tf 11.16 5.03 Td[(y 4)] y+O(s)).(3)HereLi2(z))]TJ /F9 11.955 Tf 23.91 9.63 Td[(Rz0dtln(1)]TJ /F5 11.955 Tf 11.95 0 Td[(t)=tistheDilogarithmIntegral. 3.3UsingthedeSitterInvariantFormThepurposeofthissectionistodemonstratethatthedeSitterinvariantrepresentationderivedintheprevioussectiondoesnotprovideatransparentexpressionfortheeffectiveeldequations.IntherstsubsectionwepartiallyintegratetheSchwinger-Keldysheffectiveeldequationstoobtainthestartlingresultthatallquantumcorrectionsfordynamicalphotonscanbeexpressedassurfacetermsattheinitialtime.Suchsurfacetermsareusuallysuppressedbyinversepowersofthescalefactor,andignoringthemenormouslysimpliestheanalysis.Toshowthatthisisnotpossiblewiththecumbersome,deSitterinvariantformulation,thesecondsubsectionfocussesonthecontributionwhichdivergeswhenthescalarmassgoestozero.Inthevastlysimpler,noncovariantrepresentationthiscontributiontakestheformofalocalphotonmass[ 81 ].WeatlengthreachthesameformusingGreen's2ndIdentity. 3.3.1AParadoxfromtheEffectiveFieldEquationsThequantum-correctedeffectiveeldequationsare, @hp )]TJ /F5 11.955 Tf 9.3 0 Td[(gggF(x)i+Zd4x0hi(x;x0)A(x0)=J(x).(3)Herethevacuumpolarizationtakestheform( 3 ), hi(x;x0)=)]TJ /F5 11.955 Tf 9.3 0 Td[(ip )]TJ /F5 11.955 Tf 9.29 0 Td[(gp )]TJ /F5 11.955 Tf 9.3 0 Td[(g0DD0D[D0[[nD]D0]]ySKo.(3)However,thestructurefunctionSKisnotquitethein-outstructurefunction( 3 )thatwasderivedintheprevioussection.IftheeffectiveeldA(x)istorepresentthetrueexpectationvalueratherthanthein-outmatrixelementofthevectorpotentialoperatorinthepresenceofsomestate(releasedatsomenitetimei)thenSKmust 65

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betheSchwinger-Keldyshstructurefunction[ 94 ], SK=++++)]TJ /F4 11.955 Tf 9.58 1.79 Td[(.(3)Attheonelooporderweareworking,++isexpression( 3 )withthereplacement[ 36 ], y(x;x0))166(!y++(x;x0)aa0H2h~x)]TJ /F3 11.955 Tf 9.44 .5 Td[(~x02)]TJ /F9 11.955 Tf 11.95 13.27 Td[(j)]TJ /F3 11.955 Tf 9.96 0 Td[(0j)]TJ /F5 11.955 Tf 13.95 0 Td[(i2i.(3)Atthesameorder,+)]TJ /F1 11.955 Tf 9.58 1.79 Td[(isjustminus( 3 )withthereplacement[ 36 ], y(x;x0))166(!y+)]TJ /F4 11.955 Tf 6.25 1.8 Td[((x;x0)aa0H2h~x)]TJ /F3 11.955 Tf 9.44 .49 Td[(~x02)]TJ /F9 11.955 Tf 11.96 13.27 Td[()]TJ /F3 11.955 Tf 9.96 0 Td[(0+i2i.(3)Andthespacetimeintegrationin( 3 )is[ 36 ], Zd4x0=Z0id0Zd3x0(3)For0>weseefromexpressions( 3 )and( 3 )thaty++(x;x0)equalsy+)]TJ /F4 11.955 Tf 6.25 1.79 Td[((x;x0),whichmeans++cancels+)]TJ /F1 11.955 Tf 9.58 1.79 Td[(andSKvanishes.Thereisasimilarcancellationforx0outsidethepastlight-coneofx.Insidethepastlight-conewehave+)]TJ /F4 11.955 Tf 10.27 1.79 Td[(=)]TJ /F3 11.955 Tf 9.3 0 Td[(++,whichcancelsthefactorofiin( 3 ).Therefore,theSchwinger-Keldyshformalismisbothrealandcausal.WecomenowtotheproblemwiththedeSitterinvariantrepresentation( 3 ),whichiswhattodowiththefourexternalderivatives.Itisnaturaltoextractthetwoderivativeswithrespecttoxfrominsidetheintegral, Zd4x0hi(x;x0)A(x0)=)]TJ /F5 11.955 Tf 9.3 0 Td[(ip )]TJ /F5 11.955 Tf 9.3 0 Td[(gDD[Zd4x0p )]TJ /F5 11.955 Tf 9.3 0 Td[(g0D0D0[[nD]D0]]ySKoA(x0). (3) 66

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Infact,thisismandatoryifwearetotaketheunregulatedlimit.Itisjustasnaturalandjustasmandatorytopartiallyintegratethetwoderivativeswithrespecttox0,1 Zd4x0hi(x;x0)A(x0)=i 2p )]TJ /F5 11.955 Tf 9.3 0 Td[(gDD[Zid3x0hD]D00yD0SK)]TJ /F5 11.955 Tf 9.97 0 Td[(D]D0yD00SKiA(x0))]TJ /F5 11.955 Tf 11.69 8.08 Td[(i 2p )]TJ /F5 11.955 Tf 9.3 0 Td[(gDD[Zid3x0D]D0ySKF0(x0)+i 2p )]TJ /F5 11.955 Tf 9.29 0 Td[(gDD[Zd4x0p )]TJ /F5 11.955 Tf 9.3 0 Td[(g0D]D0ySKD0F(x0). (3) NotethattheonlypossiblesurfacetermsintheSchwinger-Keldyshformalismareattheinitialtime;+)]TJ /F1 11.955 Tf 9.58 1.8 Td[(cancels++atthefuturetimesurfaceandatspatialinnity.IntegralsovertheinitialvaluesurfaceareubiquitousincomputationsusingtheSchwinger-Keldyshformalism[ 22 34 36 53 56 61 76 78 80 82 88 95 96 ],andmuchhasbeenlearnedfromallthisexperience.Onetypical(butnotuniversal)featureofsuchsurfacetermsisthattheyfallofflikepowersofthescalefactor,whichrapidlymakesthemirrelevant.Anotherfeatureisthattheyareliabletobeabsorbedbyperturbativecorrectionsoftheinitialstate[ 36 56 ].Therequiredstatecorrectionhasbeenexplicitlyconstructedinonecase[ 58 ].Becausecontributionsfromtheinitialvaluesurfacetendtoredshiftrapidly,andcansometimesbeabsorbedintoperturbativecorrectionsoftheinitialstate,ithasbecomecommontoignorethem[ 22 88 96 ].Thisvastlysimpliescomputations,butitcannotbepossibleforthesurfaceintegralsinexpression( 3 ).Toseewhy,wesubstitute( 3 )inthequantum-correctedMaxwellequation( 3 ),withoutthesurfaceintegrals,andspecializetothecaseofdynamicalphotonswithJ(x)=0.For 1Wenormalizethescalefactortounityat=isothatp )]TJ /F5 11.955 Tf 9.3 0 Td[(g(i,~x)=1. 67

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simplicity,wealsodeletetheoverallfactorofp )]TJ /F5 11.955 Tf 9.3 0 Td[(g, DF(x)+i 2DD[Zd4x0p )]TJ /F5 11.955 Tf 9.3 0 Td[(g0D]D0ySKD0F(x0)=0(3)ItisimmediatelyobviousthatanyeldwhichobeystheclassicalequationDF=0alsosolves( 3 ).Onemightworryaboutthepossibilityofadditionalsolutionsresultingfromcancellationsbetweenthelocalandnonlocaltermsin( 3 ).However,thesesortsofsolutionsiftheyevenexistarenevervalidwithintheperturbativeframeworkimposedbyonlypossessingthevacuumpolarizationtosomeniteorderintheloopexpansion[ 62 ].Wethereforeconcludethatdynamicalphotonswouldreceivenocorrections,toanyorder,ifitisvalidtoignoresurfaceintegrals.Theproblemwiththisconclusionisthatthevastlysimpler,noncovariantformalismallowsonetoprovethatquantumcorrectionsforthismodelchangedynamicalphotonsfrommasslesstomassive[ 81 ]. 3.3.2ResolutionusingGreen's2ndIdentityWhenasingleassumptionleadstofalseresults,thatassumptioncannotbecorrect.Inthissubsectionwedemonstratethatthesurfaceintegralsinexpression( 3 )cannotbedropped.Tosimplifytheanalysiswefocusonthe1=scontributiontoSK,whichshoulddominateforthecaseofsmallscalarmass, SK)166(!se2 84s(ln(y++ 4) 4)]TJ /F5 11.955 Tf 9.96 0 Td[(y++)]TJ /F4 11.955 Tf 13.15 8.79 Td[(ln(y+)]TJ ET q .478 w 302.31 -443.11 m 317.38 -443.11 l S Q BT /F10 7.97 Tf 307.63 -450.22 Td[(4) 4)]TJ /F5 11.955 Tf 9.97 0 Td[(y+)]TJ /F9 11.955 Tf 8.71 30.43 Td[().(3)Foranytransversevectorpotential(DA=0)thenoncovariantformalismimpliesthatthe1=spartofthefullvacuumpolarizationreducestoalocalphotonmassterm[ 81 ], Zd4x0hi(x;x0)A(x0)=)]TJ /F5 11.955 Tf 10.5 8.08 Td[(e2H2 44sp )]TJ /F5 11.955 Tf 9.3 0 Td[(g(x)g(x)A(x)+O(s0).(3)ItmustthereforebethatsgivesthesameresultinthedeSitterinvariantformalism.Ourstartingpointistheobservationthattherighthandsideofexpression( 3 )bearsacloserelationtoGreen'sSecondIdentity.Toseethis,letusdenethevector 68

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pseudo-Green'sfunction, hGi(x;x0)DD[hD]D0yi 2SKi.(3)Foranytransversevectorpotentialexpression( 3 )canbewritten, 1 p )]TJ /F5 11.955 Tf 9.3 0 Td[(gZd4x0hi(x;x0)A(x0)=Zd4x0hGi(x;x0)h0)]TJ /F4 11.955 Tf 9.97 0 Td[((D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)H2iA(x0))]TJ /F9 11.955 Tf 11.29 16.27 Td[(Zid3x0(D00hGi(x;x0)A(x0))]TJ /F9 11.955 Tf 11.96 13.27 Td[(hGi(x;x0)D00A(x0)). (3) Ofcoursetherighthandsideof( 3 )iswhatonegetsbyintegratingGreen'sSecondIdentity, h0)]TJ /F4 11.955 Tf 9.96 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)H2ihGi(x;x0)A(x0)=hGi(x;x0)h0)]TJ /F4 11.955 Tf 9.97 0 Td[((D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)H2iA(x0)+D0(D0hGi(x;x0)A(x0))]TJ /F9 11.955 Tf 9.97 13.27 Td[(hGi(x;x0)D0A(x0)). (3) Relation( 3 )willbeestablishedifwecanshowthatreplacingSKwithjustsin( 3 )givesatrueGreen'sfunction(fortransversevectors)withcoefcient)]TJ /F5 11.955 Tf 9.3 0 Td[(e2H2=(42s).Theinitialpartoftheanalysisdoesnotrequirethepreciseformofthestructurefunction,soweconsiderageneralfunctionf(y), DD[hD]D0yfi=1 2H2(DD0yh)]TJ /F4 11.955 Tf 9.3 0 Td[((4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)f00)]TJ /F4 11.955 Tf 9.97 0 Td[((D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)f0i+DyD0yh(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)f00)]TJ /F4 11.955 Tf 9.96 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)f0i). (3) 69

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Thenextstepistoactthephotonkineticoperator,whichweshalldoignoringpotentialdeltafunctioncontributions, h)]TJ /F4 11.955 Tf 9.96 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)H2iDD[hD]D0yfi=deltafunctions+1 2H4(DD0yh)]TJ /F4 11.955 Tf 9.3 0 Td[((4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)F00)]TJ /F4 11.955 Tf 9.96 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)F0i+DyD0yh(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)F00)]TJ /F4 11.955 Tf 9.96 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)F0i). (3) HerethefunctionF(y)is, F(y)(4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)f00(y)+D(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)f0(y))]TJ /F4 11.955 Tf 9.96 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)f(y)DB H2f(y).(3)Atthisstagewespecializetof(y)=i 2s,andalsosetD=4.BecauseB(y)=H2=42yonehas, DB H2hi 2si=ie2 42sH2hB(y++))]TJ /F5 11.955 Tf 9.96 0 Td[(B(y+)]TJ /F4 11.955 Tf 6.25 1.79 Td[()i.(3)Nowuserelations( 3 3 )and( 3 )toinfer, )]TJ /F4 11.955 Tf 11.95 0 Td[((4y2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)B00(y))]TJ /F4 11.955 Tf 11.96 0 Td[((D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)B0(y)=2A0(y), (3) (2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)B0(y))]TJ /F4 11.955 Tf 11.95 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)B(y)=2A00(y). (3) Combiningtheserelationswith( 3 )implies, h)]TJ /F4 11.955 Tf 9.96 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)H2iDD[hD]D0yi 2si=deltafunctions+ie2H2 42sDD0hA(y++))]TJ /F5 11.955 Tf 9.96 0 Td[(A(y+)]TJ /F4 11.955 Tf 6.26 1.79 Td[()i. (3) Thefactthatthelefthandsideistransversexesthedeltafunctionsontherighthandside, h)]TJ /F4 11.955 Tf 9.96 0 Td[((D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)H2iDD[hD]D0yi 2si=)]TJ /F5 11.955 Tf 10.5 8.09 Td[(e2H2 42s(g4(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0) p )]TJ /F5 11.955 Tf 9.3 0 Td[(g)]TJ /F5 11.955 Tf 9.96 0 Td[(DD0hiA(y++))]TJ /F5 11.955 Tf 9.97 0 Td[(iA(y+)]TJ /F4 11.955 Tf 6.25 1.79 Td[()i). (3) 70

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Wecannowrecognizetherighthandsideasthetransverseprojectionoperator[ 89 ],whichmeansthatrelation( 3 )isindeedcorrect. 3.4DiscussionThefocusofthischapterhasbeenonhowtorepresentthetensorstructureofthevacuumpolarizationondeSitterbackground.TheoriginalstudiesofoneloopcorrectionsfromscalarQEDemployedanoncovariantform[ 80 81 ], ihi(x;x0)=@@0n[]F(x;x0)+ [ ]G(x;x0)o,(3)whereisthespacelikeMinkowskimetricand +00isitspurelyspatialpart.Insection 3.1 wedevelopedacovariantextension( 3 3 )usingcovariantderivatives,ratherthanordinaryones,andantisymmetrizedproductsofthenaturalbasistensors( 3 ).WhenthephysicsbreaksdeSitterinvariance,butpreserveshomogeneityandisotropy,itispossibletorepresentthevacuumpolarizationwithtwostructurefunctions,justliketheoldrepresentation( 3 ).ForthespecialcasewherethevacuumpolarizationisphysicallydeSitterinvariant,onecanemployadeSitterinvariantrepresentation( 3 )basedonjustonestructurefunction.ThevacuumpolarizationfromamassivescalarisdeSitterinvariant[ 81 ]andweworkedoutthefullyrenormalizedstructurefunction( 3 ).However,wesawinsection 3.3 thattheredoesnotseemtobeanyadvantagetousingthisrepresentationovertheoriginalone( 3 ).Withtheoldformalismonecanrathersimplyshowthattheleadingcontribution,forsmallscalarmass,isalocalProcaterm[ 81 ].InthedeSitterinvariantformalismthissameresultappearsinthecumbersomeformofGreen'sSecondIdentity, )]TJ /F5 11.955 Tf 10.5 8.09 Td[(e2H2 42sgA(x)=Zd4x0hGsi(x;x0)h0)]TJ /F4 11.955 Tf 9.96 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)H2iA(x0))]TJ /F9 11.955 Tf 11.29 16.27 Td[(Zid3x0(D00hGsi(x;x0)A(x0))]TJ /F9 11.955 Tf 11.96 13.27 Td[(hGsi(x;x0)D00A(x0)), (3) 71

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where[Gs](x;x0)is, hGsi(x;x0)=ie2 164sDD["D]D0ynln(y++ 4) 4)]TJ /F5 11.955 Tf 9.96 0 Td[(y++)]TJ /F4 11.955 Tf 13.15 8.79 Td[(ln(y++ 4) 4)]TJ /F5 11.955 Tf 9.96 0 Td[(y++o#.(3)Itisdifculttodiscernanyadvantagetherighthandsideof( 3 )mightpossessoverthelefthandside,nomatterhowpassionatelydevotedoneistodeSitterinvariance.Thepresenceofinitialtimesurfaceintegralsinexpression( 3 )isparticularlydisturbing.Longexperiencewiththenoncovariantformalismhasconditionedustoignoresuchsurfaceintegralsbecausetheyarelikelytoredshiftrapidly,andbecausetheycansometimesbeabsorbedintoperturbativecorrectionstotheinitialstate[ 22 36 53 56 58 61 76 78 80 82 88 95 96 117 ].Asweprovedinsection 3.3 ,thesurfaceintegralsof( 3 )cannotbedropped.EmployingthedeSitterinvariantformalismwouldthereforerequireapainstakingstudyofthedistinctionbetweensurfacetermswhichcanandcannotbeabsorbedintostatecorrections.Amajormotivationforthisstudywastodeterminethebestwayofrepresentingthequantumgravitationalcontributiontothevacuumpolarization.Basedonourresults,thereseemslittlepointtoemployingthecovariantrepresentation( 3 3 ).ThegravitonpropagatorbreaksdeSitterinvariance[ 38 84 85 91 92 97 99 ],sothecovariantstructurefunctionswouldnotbedeSitterinvariantinanycase.Butthelargerproblemisthattheywouldnotbesimple,norwouldtheyprovideasimplepictureofthephysics.Wethereforeconcludetheoldrepresentation( 3 )isbest.Shouldthisjudgmentproveincorrect,nothingessentialhasbeenlost;aswewillseeinchapter 4 therearesimpleidentitiesforconvertingbetweenthetworepresentations. Figure3-1. Oneloopcontributiontothevacuumpolarizationfromthe4-point(seagull)interaction. 72

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Figure3-2. Oneloopcontributiontothevacuumpolarizationfromthe3-pointinteraction. Figure3-3. Photoneldstrengthrenormalizationcounterterm. Table3-1. ResultforexpandingDD0(@@0[y)(@0]@y)f1(y,u,v). TensorCoefcient @@0y)]TJ /F4 11.955 Tf 9.3 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)2(2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)f1+(D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)(2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)2)]TJ /F5 11.955 Tf 11.95 0 Td[(D(4y)]TJ /F5 11.955 Tf 11.95 0 Td[(y2)@yf1+(2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)(4y)]TJ /F5 11.955 Tf 11.96 0 Td[(y2)@2yf1)]TJ /F4 11.955 Tf 11.96 0 Td[(2(D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)(2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)@uf1+2)]TJ /F6 7.97 Tf 8.02 -4.98 Td[(a a0+a0 a[(D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)@uf1)]TJ /F4 11.955 Tf 11.95 0 Td[((2)]TJ /F5 11.955 Tf 11.95 0 Td[(y)@u@yf1])]TJ /F4 11.955 Tf 9.3 0 Td[(2)]TJ /F6 7.97 Tf 8.02 -4.97 Td[(a a0)]TJ /F6 7.97 Tf 13.15 4.71 Td[(a0 a[(D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)@vf1)]TJ /F4 11.955 Tf 11.95 0 Td[((2)]TJ /F5 11.955 Tf 11.95 0 Td[(y)@v@yf1]+2(2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)2@u@yf1+2)]TJ /F6 7.97 Tf 8.01 -4.98 Td[(a a0+a0 a)]TJ /F4 11.955 Tf 11.95 0 Td[((2)]TJ /F5 11.955 Tf 11.95 0 Td[(y)(@2u)]TJ /F3 11.955 Tf 11.96 0 Td[(@2v)f1@y@0y)]TJ /F4 11.955 Tf 9.3 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)2f1+(2D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)(2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)@yf1)]TJ /F4 11.955 Tf 11.95 0 Td[((2)]TJ /F5 11.955 Tf 11.95 0 Td[(y)2@2yf1)]TJ /F4 11.955 Tf 9.3 0 Td[(2(D)]TJ /F4 11.955 Tf 11.95 0 Td[(1)@uf1+2(2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)@u@yf1)]TJ /F4 11.955 Tf 11.96 0 Td[((@2u)]TJ /F3 11.955 Tf 11.95 0 Td[(@2v)f1@y@0u2a0 a[)]TJ /F4 11.955 Tf 9.3 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)(@u+@v)f1+(2)]TJ /F5 11.955 Tf 11.95 0 Td[(y)(@u+@v)@yf1)]TJ /F4 11.955 Tf 9.3 0 Td[((@2u)]TJ /F3 11.955 Tf 11.95 0 Td[(@2v)f1@u@0y2a a0[)]TJ /F4 11.955 Tf 9.3 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)(@u)]TJ /F3 11.955 Tf 11.95 0 Td[(@v)f1+(2)]TJ /F5 11.955 Tf 11.95 0 Td[(y)(@u)]TJ /F3 11.955 Tf 11.95 0 Td[(@v)@yf1)]TJ /F4 11.955 Tf 9.3 0 Td[((@2u)]TJ /F3 11.955 Tf 11.95 0 Td[(@2v)f1@u@0u)]TJ /F4 11.955 Tf 9.3 0 Td[(4(@2u)]TJ /F3 11.955 Tf 11.95 0 Td[(@2v)f1 Allcoefcientsaremultipliedbyafactorof1 2H4. 73

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Table3-2. ResultforexpandingDD0(@[y)(@]@0[y)(@0]y)f2(y,u,v). TensorCoefcient @@0y)]TJ /F5 11.955 Tf 9.3 0 Td[(D(D)]TJ /F4 11.955 Tf 11.95 0 Td[(1)(2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)2+D(4y)]TJ /F5 11.955 Tf 11.95 0 Td[(y2)f2)]TJ /F4 11.955 Tf 11.95 0 Td[((4y)]TJ /F5 11.955 Tf 11.96 0 Td[(y2)2@2yf2)]TJ /F4 11.955 Tf 9.3 0 Td[((2D+1)(2)]TJ /F5 11.955 Tf 11.95 0 Td[(y)(4y)]TJ /F5 11.955 Tf 11.96 0 Td[(y2)@yf2)]TJ /F4 11.955 Tf 11.96 0 Td[(2D(2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)2@uf2+2)]TJ /F6 7.97 Tf 8.02 -4.98 Td[(a a0+a0 aD(2)]TJ /F5 11.955 Tf 11.95 0 Td[(y)@uf2+(4y)]TJ /F5 11.955 Tf 11.95 0 Td[(y2)@u@yf2)]TJ /F4 11.955 Tf 9.3 0 Td[(2)]TJ /F6 7.97 Tf 8.02 -4.97 Td[(a a0)]TJ /F6 7.97 Tf 13.15 4.71 Td[(a0 aD(2)]TJ /F5 11.955 Tf 11.95 0 Td[(y)@vf2+(4y)]TJ /F5 11.955 Tf 11.95 0 Td[(y2)@v@yf2)]TJ /F4 11.955 Tf 9.3 0 Td[(2(2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)(4y)]TJ /F5 11.955 Tf 11.96 0 Td[(y2)@u@yf2)]TJ /F4 11.955 Tf 11.95 0 Td[((8)]TJ /F4 11.955 Tf 11.95 0 Td[(4y+y2)(@2u)]TJ /F3 11.955 Tf 11.96 0 Td[(@2v)f2+2(2)]TJ /F5 11.955 Tf 11.96 0 Td[(y))]TJ /F6 7.97 Tf 8.02 -4.97 Td[(a a0+a0 a(@2u)]TJ /F3 11.955 Tf 11.96 0 Td[(@2v)f2@y@0y)]TJ /F5 11.955 Tf 9.3 0 Td[(D2(2)]TJ /F5 11.955 Tf 11.95 0 Td[(y)f2+)]TJ /F4 11.955 Tf 9.3 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)(4y)]TJ /F5 11.955 Tf 11.96 0 Td[(y2)+(D+2)(2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)2@yf2+(2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)(4y)]TJ /F5 11.955 Tf 11.96 0 Td[(y2)@2yf2)]TJ /F4 11.955 Tf 11.96 0 Td[(2D(2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)@uf2)]TJ /F4 11.955 Tf 9.3 0 Td[(2(4y)]TJ /F5 11.955 Tf 11.96 0 Td[(y2)@u@yf2)]TJ /F4 11.955 Tf 11.96 0 Td[((2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)(@2u)]TJ /F3 11.955 Tf 11.96 0 Td[(@2v)f2@y@0u)]TJ /F4 11.955 Tf 9.3 0 Td[(2a0 aD(2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)(@u+@v)f2+(4y)]TJ /F5 11.955 Tf 11.95 0 Td[(y2)(@u+@v)@yf2+4)]TJ /F4 11.955 Tf 11.95 0 Td[(2(2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)a0 a(@2u)]TJ /F3 11.955 Tf 11.96 0 Td[(@2v)f2@u@0y)]TJ /F4 11.955 Tf 9.3 0 Td[(2a a0D(2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)(@u)]TJ /F3 11.955 Tf 11.96 0 Td[(@v)f2+(4y)]TJ /F5 11.955 Tf 11.95 0 Td[(y2)(@u)]TJ /F3 11.955 Tf 11.96 0 Td[(@v)@yf2+4)]TJ /F4 11.955 Tf 11.95 0 Td[(2(2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)a a0(@2u)]TJ /F3 11.955 Tf 11.96 0 Td[(@2v)f2@u@0u0 Allcoefcientsaremultipliedbyafactorof1 4H4. 74

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CHAPTER4REPRESENTINGTHEVACUUMPOLARIZATIONONDESITTERThischapterisanicefollowuptochapter 3 ,cleanlydemonstratinghowtoconvertbetweenthecovariantandnoncovariantrepresentation[ 111 ].Ihavechosentoperformthecalculationinchapter 5 usingthesimplernoncovariantrepresentation,butshouldonepreferthecovariantrepresentationthischapterisanessentialread.ItwillshowhowtoconvertanydeSitterresulttothedesiredformwithrelativeease.Inviewofitsphysicalimportance,considerationshouldclearlybegiventothebestwayofrepresentingthetensorstructureofi[](x;x0).Itistransverseoneachindex,andalsosymmetricunderinterchange, ihi(x;x0)=ihi(x0;x).(4)Thesetwofactsmeani[](x;x0)canhaveatmost1 245)]TJ /F4 11.955 Tf 12.58 0 Td[(4=6independentcomponents.However,onatspacebackgroundthevacuumpolarizationcanberepresentedusingonlyasinglestructurefunction, ihati(x;x0)=)]TJ /F9 11.955 Tf 9.3 13.27 Td[(h@2)]TJ /F3 11.955 Tf 9.96 0 Td[(@@i(x2),(4)whereisthespacelikeMinkowskimetricandx2k~x)]TJ /F3 11.955 Tf 11.43 .5 Td[(~x0k2)]TJ /F4 11.955 Tf 11.96 0 Td[((jx0)]TJ /F5 11.955 Tf 11.96 0 Td[(x00j)]TJ /F5 11.955 Tf 17.93 0 Td[(i)2.Onthehomogeneousandisotropicbackgroundsofcosmologythevacuumpolarizationcanbewrittenasalinearcombinationoffouralgebraicallyindependenttensors[ 91 ].Transversalityprovidestwodifferentialrelationsbetweentheircoefcients,sothereshouldbetwoindependentstructurefunctions[ 91 ]inageneralcosmologicalbackground.TheearliestdeSittercomputations[ 81 106 107 ]weremadeinconformalcoordinates, ds2=a2)]TJ /F5 11.955 Tf 9.3 0 Td[(d2+d~xd~x,a()=)]TJ /F4 11.955 Tf 15.16 8.09 Td[(1 H,(4) 75

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whereHistheHubbleconstant.Theseearlyworksrepresentedthevacuumpolarizationas, ihi(x;x0)=h)]TJ /F3 11.955 Tf 9.96 0 Td[(i@@0F(x;x0)+h )]TJ ET q .478 w 313.18 -64.9 m 319.38 -64.9 l S Q BT /F3 11.955 Tf 313.18 -71.73 Td[( i@@0G(x;x0),(4)where +00isthepurelyspatialpartoftheMinkowskimetric.ThestructurefunctionsF(x;x0)andG(x;x0)haveasimpleinterpretationintermsoftheelectricandmagneticsusceptibilitieseandm[ 112 ], e(x;x0)=)]TJ /F5 11.955 Tf 9.3 0 Td[(iF(x;x0),m 1+m(x;x0)=)]TJ /F5 11.955 Tf 9.3 0 Td[(iG(x;x0).(4)Thesesusceptibilitiescanbeusedtoobtainthepolarizationandthemagnetizationvectors,andthecorrespondingmacroscopicelds,thesameasforatspaceelectrodynamicsinamedium, ~P=e~E=)~D=~E+~P, (4) ~M=m~B=)~H=~B)]TJ /F3 11.955 Tf 13.86 3.16 Td[(~M. (4) Heresigniesconvolution,meaningthatrelations( 4 4 )areactuallynon-localandcaninvolveintegrationswithinthepastlight-cone.Althoughtheoriginalrepresentation( 4 )hasatransparentphysicalinterpretation,andiseasytouseinthequantum-correctedMaxwellequations[ 108 ],itsstructurefunctionsF(x;x0)andG(x;x0)arenotbi-scalardensitiesbecauseand arenotbi-tensors.ThismayappeardisturbingtothosewhobelievethatthedeSittergroupshouldplaythesameroleinorganizingquantumeldtheoryondeSitterbackgroundthatthePoincaregroupdoesonatspace[ 85 87 ].Itwasdifculttoformanopinionastothemeritofthisviewaslongasonlythenoncovariantrepresentation( 4 )hadbeenstudied.Therefore,werecentlyrecasttheoldresultsofSQED(scalarquantum 76

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electrodynamics)[ 81 106 107 ]inacovariantform[ 79 ], ihi(x;x0)=p )]TJ /F5 11.955 Tf 9.3 0 Td[(gp )]TJ /F5 11.955 Tf 9.3 0 Td[(g0DD0(DD0[yD0]Dyf1(y,u,v)+D[yD]D0[yD0]yf2(y,u,v)). (4) HereDandD0standforcovariantderivativeoperators,squarebracketsdenoteanti-symmetrization,andthevariablesy,uandvare, y(x;x0)=aa0H2x2,u(x;x0)=ln(aa0),v(x;x0)=lna a0.(4)ThedeSitterinvariantlengthfunctiony(x;x0)in( 4 )isrelatedtothegeodeticlength`(x;x0)fromxtox0byy=4sin2(1 2H`).WhenthevacuumpolarizationisdeSitterinvariant,thestructurefunctionsareindependentofuandv,andonlyonestructurefunctionisrequired,justasinatspace.Thishappensforthecaseofascalarwithpositivemass-squared[ 81 ].DependenceuponthedeSitterbreakingtermsu(x;x0)andv(x;x0)arisesfromdeSitterbreakinginthemassless,minimallycoupledscalarpropagator[ 90 113 114 ].WewillderiveaclosedformprocedureforconvertingthenoncovariantstructurefunctionsF(x;x0)andG(x;x0)intothestructurefunctionsf1(y,u,v)andf2(y,u,v)ofthecovariantrepresentation( 4 ).Althoughthederivationishighlynontrivial,thenalresultisquitesimple,somathematicalphysicistswhopreferthecovariantrepresentationcaneasilyconverttoit.Section 4.1 reviewssomeimportantresultswhichareemployedintheanalysis.Thederivationismadeinsection 4.2 .Insection 4.3 wecheckthenaltransformationformulaeusingthevacuumpolarizationofSQEDforwhichthestructurefunctionsofbothrepresentationsareknown.Wealsogiveamajorsimplicationintheprocedureforinferringthenoncovariantstructurefunctionsfromprimitivediagrams.Ourconclusionscomprisesection 4.4 .Aftersummarizingresultswediscussthefascinatingissueofhowtorepresentthevacuumpolarizationonageneralmetricbackground. 77

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4.1KnownResultsonTensorStructuresInthissectionwereviewsomerecentlyderivedresultthatarenecessaryforthederivationofsection 4.2 .TherstoftheseresultsconsistsofexpandingtheMinkowskimetricanditsspatialrestrictionintermsofcovariantderivativesofy(x;x0)andu(x;x0)[ 98 100 ], =aa0 2H2n)]TJ /F5 11.955 Tf 9.29 0 Td[(DD0y+DyD0u+DuD0y)]TJ /F5 11.955 Tf 9.96 0 Td[(yDuD0uo, (4) =aa0 2H2n)]TJ /F5 11.955 Tf 9.29 0 Td[(DD0y+DyD0u+DuD0y+(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)DuD0uo. (4) (Notethatcovariantderivativeofv(x;x0)arenotindependentbecauseDv=DuandD0v=)]TJ /F5 11.955 Tf 9.3 0 Td[(D0u.)Thesecondclassofresultsismoreinvolvedandconcernsthemostgeneralformforthevacuumpolarization[ 79 ], ihi(x;x0)=p )]TJ /F5 11.955 Tf 9.3 0 Td[(gp )]TJ /F5 11.955 Tf 9.3 0 Td[(g0DD0hTi(x;x0),(4)where[T](x;x0)=)]TJ /F4 11.955 Tf 9.3 0 Td[([T](x;x0)=)]TJ /F4 11.955 Tf 9.3 0 Td[([T](x;x0)=+[T](x0;x).Thesymmetriesofcosmologyhomogeneityandisotropydonotreduce( 4 )totheform( 4 ).Inadditiontojustf1(y,u,v)andf2(y,u,v),theanti-symmetricbi-tensor[T](x;x0)caninvolvethreeadditionalstructurefunctions, T=@@0[y@0]@yf1+@[y@]@0[y@0]yf2+@[y@]@0[y@0]uf3+@[u@]@0[y@0]y~f3+@[u@]@0[y@0]uf4+@[y@]u@0[y@0]uf5, (4) wherefi=fi(y,u,v)and~f3(y,u,v)=f3(y,u,)]TJ /F5 11.955 Tf 9.3 0 Td[(v),asdictatedbysymmetriesofT.Toseethatonlytwoofthesestructurefunctionsareactuallyindependentwerstnotethatactingthederivativesin( 4 )resultsinonlyfouralgebraicallyindependenttensors, DD0T(x;x0)=@@0yF1+@y@0yF2+@u@0yF3+@y@0ueF3+@u@0uF4, (4) 78

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whereFi=Fi(y,u,v)(i=1,2,3,4)and~F3(y,u,v)=F3(y,u,)]TJ /F5 11.955 Tf 9.3 0 Td[(v).Furthermore,transversalityimpliestwopartialdifferentialrelationsbetweentheFi,oneproportionaltothederivativeofy(x;x0)andtheotherproportionaltothederivativeofu(x;x0)[ 91 ].Allofthissuggeststhatthereareonlytwomasterstructurefunctionswhichwemightcall=(y,u,v)and=(y,u,v).Inref.[ 79 ]thatsuspicionwasconrmedbydemonstratingthetransversalityofthefollowingsubstitutionsforthecoefcientfunctionsFi(y,u,v), F1=h)]TJ /F4 11.955 Tf 9.3 0 Td[((4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)@y)]TJ /F4 11.955 Tf 9.96 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y))]TJ /F4 11.955 Tf 9.96 0 Td[(2(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)@u+4cosh(v)@u)]TJ /F4 11.955 Tf 9.96 0 Td[(4sinh(v)@vi+h4cosh(v))]TJ /F4 11.955 Tf 9.97 0 Td[((2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)i)]TJ /F4 11.955 Tf 9.96 0 Td[(, (4) F2=h(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)@y)]TJ /F5 11.955 Tf 9.96 0 Td[(D+1)]TJ /F4 11.955 Tf 9.96 0 Td[(2@ui)]TJ /F4 11.955 Tf 9.96 0 Td[(, (4) F3=)]TJ /F4 11.955 Tf 9.29 0 Td[(2ev(@u)]TJ /F3 11.955 Tf 9.96 0 Td[(@v))]TJ /F4 11.955 Tf 9.96 0 Td[(2ev+, (4) eF3=)]TJ /F4 11.955 Tf 9.29 0 Td[(2e)]TJ /F6 7.97 Tf 6.58 0 Td[(v(@u+@v))]TJ /F4 11.955 Tf 9.96 0 Td[(2e)]TJ /F6 7.97 Tf 6.59 0 Td[(v+, (4) F4=)]TJ /F4 11.955 Tf 9.29 0 Td[(4+(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y). (4) Heretheauxiliaryfunction(y,u,v)is, (y,u,v)(@2u)]TJ /F3 11.955 Tf 9.96 0 Td[(@2v)Zdy(y,u,v)+(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)(y,u,v))]TJ /F4 11.955 Tf 9.96 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)Zdy(y,u,v).(4)Relations( 4 4 )canbeinvertedtoobtainforthemasterstructurefunctions, [(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)+2@u]=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(1 4[(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)F1+(4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)F2+(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)(F3+eF3))]TJ /F5 11.955 Tf 9.97 0 Td[(F4] (4) sinh(v)=[cosh(v)@v)]TJ /F4 11.955 Tf 9.96 0 Td[(sinh(v)@u])]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 4[F3)]TJ /F9 11.955 Tf 11 3.15 Td[(eF3]. (4) InthedeSitterinvariantcaseonlyonedeSitterinvariantmasterstructurefunctionsufces,thatis=(y)and=0.Ontheotherhand,onecanrelatethemasterstructurefunctionstofiin( 4 ).Belowwequoteseparatelythecontributionsfromeachoffitomasterstructurefunctions 79

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(thisisdenotedbythesubscripti=1,2,3,4,5oniandi): 1=H4 2n(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)f1)]TJ /F4 11.955 Tf 9.96 0 Td[((2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)@yf1o, (4) 1=H4 2n(@2u)]TJ /F3 11.955 Tf 9.96 0 Td[(@2v)f1o, (4) 2=H4 4nD(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)f2+(4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)@yf2o, (4) 2=H4 4n(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)(@2u)]TJ /F3 11.955 Tf 9.96 0 Td[(@2v)f2o, (4) 3=H4 4n)]TJ /F4 11.955 Tf 9.3 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)(f3+ef3)+(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)@y(f3+ef3))]TJ /F4 11.955 Tf 9.96 0 Td[(2@y(evf3+e)]TJ /F6 7.97 Tf 6.59 0 Td[(vef3)o, (4) 3=H4 4n2@yh@u(evf3+e)]TJ /F6 7.97 Tf 6.59 0 Td[(vef3))]TJ /F3 11.955 Tf 9.96 0 Td[(@v(evf3)]TJ /F5 11.955 Tf 9.97 0 Td[(e)]TJ /F6 7.97 Tf 6.59 0 Td[(vef3)i)]TJ /F4 11.955 Tf 9.96 0 Td[((@2u)]TJ /F3 11.955 Tf 9.96 0 Td[(@2v)(f3+ef3)o, (4) 4=H4 4n@yf4o, (4) 4=H4 4n)]TJ /F4 11.955 Tf 9.3 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)@yf4+(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)@2yf4)]TJ /F4 11.955 Tf 9.97 0 Td[(2@y@uf4o. (4) 5=H4 4n)]TJ /F5 11.955 Tf 9.3 0 Td[(Df5+2(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)@yf5)]TJ /F4 11.955 Tf 9.96 0 Td[(4cosh(v)@yf5o, (4) 5=H4 4n(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)f5)]TJ /F4 11.955 Tf 9.96 0 Td[((D+1)(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)@yf5)]TJ /F4 11.955 Tf 9.96 0 Td[((4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)@2yf5+2@uf5+[)]TJ /F4 11.955 Tf 7.31 0 Td[(2(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)+4cosh(v)]@u@yf5)]TJ /F4 11.955 Tf 9.96 0 Td[(4@v@y[sinh(v)f5])]TJ /F4 11.955 Tf 11.96 0 Td[((@2u)]TJ /F3 11.955 Tf 9.96 0 Td[(@2v)f5o. (4) Theseexpressionsalsoimplythatthevacuumpolarizationcanbedescribedintermsofanytwoofthestructurefunctionsfi(y,u,v).WhentheresultisdeSitterinvariantthenitrequiresonlyasinglestructurefunction,whichcanbeeitherf1(y)orf2(y). 4.2TheDerivationTherststepinouranalysisistocovariantizeexpression( 4 )bynotingitcanbewrittenas, ihi(x;x0)=2@0@p )]TJ /F5 11.955 Tf 9.3 0 Td[(gp )]TJ /F5 11.955 Tf 9.3 0 Td[(g0 (aa0)2h[]bF+ [ ]bGi,(4)wherewehaverescaledF(x;x0)andG(x;x0)as F=p )]TJ /F5 11.955 Tf 9.3 0 Td[(gp )]TJ /F5 11.955 Tf 9.3 0 Td[(g0 (aa0)2bF,G=p )]TJ /F5 11.955 Tf 9.3 0 Td[(gp )]TJ /F5 11.955 Tf 9.3 0 Td[(g0 (aa0)2bG.(4) 80

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Now,whenthemeasurefactorsp )]TJ /F5 11.955 Tf 9.3 0 Td[(gandp )]TJ /F5 11.955 Tf 9.3 0 Td[(g0arepulledoutofthecurlybracketsin( 4 ),theexternalderivativesbecomecovariant, ihi(x;x0)=2p )]TJ /F5 11.955 Tf 9.3 0 Td[(gp )]TJ /F5 11.955 Tf 9.29 0 Td[(g0D0Dn(aa0))]TJ /F10 7.97 Tf 6.58 0 Td[(2h[]bF+ [ ]bGio.(4)Thisfollowsfromtheobservationthat,becauseoftheanti-symmetryin$,D[T]=@[T]+)]TJ /F7 7.97 Tf 25.66 5.67 Td[([T],where)]TJ /F7 7.97 Tf 6.77 5.67 Td[(=()]TJ /F5 11.955 Tf 9.3 0 Td[(g))]TJ /F10 7.97 Tf 6.59 0 Td[(1=2@()]TJ /F5 11.955 Tf 9.3 0 Td[(g)1=2.TheanalogousidentityholdsfortheD0derivative.Thenextstepistoputthetermsinvolving=aa0and =aa0in( 4 )intoacovariantformusingrelations( 4 4 ).Uponinsertingtheserelationsinto( 4 ),andthencomparingwithEq.( 4 ),wend, f1=1 2H4(bF+bG),f2=0,f3=1 H4(bF+bG)=~f3,f4=1 H4()]TJ /F5 11.955 Tf 9.29 0 Td[(ybF+(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)bG),f5=)]TJ /F4 11.955 Tf 14.52 8.09 Td[(1 H4(bF+bG). (4) Thethirdstepistoinsertrelations( 4 )intothemasterstructurefunctionsEqs.( 4 4 )foreachofthevestructurefunctionsfi,andthensummingtond, =5Xi=1i=)]TJ /F4 11.955 Tf 10.49 8.08 Td[(1 2@yF, (4) =5Xi=1i=1 2[(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)+y@y+2@u]@yF+@2yG=)]TJ /F4 11.955 Tf 9.3 0 Td[([(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1))]TJ /F4 11.955 Tf 11.95 0 Td[((2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)@y+2@u]+@2y(F+G). (4) Thisimpliesthatwecanexpressthevacuumpolarizationtensorintermsofthetwostructurefunctionsf1andf2asinEq.( 4 ).Indeed,withthehelpofEqs.( 4 4 )weget: 1+2=H4 4)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(@2u)]TJ /F3 11.955 Tf 11.96 0 Td[(@2vg1,g1=2f1+(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)f21+2=H4 4[(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1))]TJ /F4 11.955 Tf 11.96 0 Td[((2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)@y]g1+4@yf2. (4) 81

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Theseequationscanbesolvedforg1andf2,1intermsof1+2and1+2.Usingthe`light-cone'coordinates,x=(uv)=2(x+=ln(a),x)]TJ /F4 11.955 Tf 10.61 1.8 Td[(=ln(a0)),andrequiringinterchangesymmetrygives, g1(y,u,v)=Zx+0dx+Zx)]TJ /F10 7.97 Tf -9.2 -22.84 Td[(0dx)]TJ /F4 11.955 Tf 7.08 1.8 Td[((y,x++x)]TJ /F4 11.955 Tf 7.08 1.8 Td[(,x+)]TJ /F4 11.955 Tf 10.15 0 Td[(x)]TJ /F4 11.955 Tf 7.09 1.8 Td[()+(y,x+)+(y,x)]TJ /F4 11.955 Tf 7.09 1.79 Td[(),f2=2)]TJ /F5 11.955 Tf 9.96 0 Td[(y 4g1+Zdy0h1 H4(y0,u,v))]TJ /F5 11.955 Tf 13.15 8.09 Td[(D)]TJ /F4 11.955 Tf 9.96 0 Td[(2 4g1(y0,u,v)i+(u,v),f1=1 2[g1)]TJ /F4 11.955 Tf 11.96 0 Td[((2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)f2], (4) where(y,x)and(u,v)arearbitraryfunctions,andandaregiveninEqs.( 4 4 ).TogetherwithEqs.( 4 ),theseexpressionsprovidethedesiredprocedurefortransformingthevacuumpolarizationtensorfromtheformgivenin( 4 )into( 4 4 ),andthusconstitutethemainresultofthispaper.InthesubsequentsectionweshowthatthesetransformationsindeedworkforthevacuumpolarizationinducedbytheoneloopvacuumuctuationsofaminimallycoupledmasslessscalarondeSitterbackground,providinganontrivialcheckofEqs.( 4 4 ). 4.3CorrespondenceHereweconsiderthevacuumpolarizationinducedbyoneloopvacuumuctuationsofamasslessminimallycoupledscalarondeSitterbackground.Ifwecallthescalarpropagatori(x;x0)thentheoneloopcontributiontothevacuumpolarizationisoftheform, ihi(x;x0)=)]TJ /F4 11.955 Tf 9.29 0 Td[(2ie2p )]TJ /F5 11.955 Tf 9.3 0 Td[(ggi(x;x)D(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0)+2e2p )]TJ /F5 11.955 Tf 9.3 0 Td[(ggp )]TJ /F5 11.955 Tf 9.3 0 Td[(g0g0h@i(x;x0)@0i(x;x0))]TJ /F5 11.955 Tf 9.97 0 Td[(i(x;x0)@@0i(x;x0)i+iZ@hp )]TJ /F5 11.955 Tf 9.3 0 Td[(ggg)]TJ /F5 11.955 Tf 9.96 0 Td[(gg@D(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0)i, (4) 82

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wereZisacounterterm,andeistheelectromagneticcoupling(e2=(4)e'1=137).Forthespecialcaseofthemassless,minimallycoupledscalarthepropagatorobeys p )]TJ /F5 11.955 Tf 9.3 0 Td[(giA(x;x0)=iD(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0),(4)where()]TJ /F5 11.955 Tf 9.3 0 Td[(g))]TJ /F15 5.978 Tf 7.78 3.26 Td[(1 2@(p )]TJ /F5 11.955 Tf 9.3 0 Td[(gg@)isthecovariantd'Alembertianasitactsona(bi-)scalarfunction.Theuniquesolutionthatpreservesthesymmetriesofthecosmologicalbackground(homogeneity,isotropyandspatialatness)canbewrittenasasumofadeSitterinvariantpartandadeSitterbreakingpart(/u=ln(aa0))as[ 113 114 ]: iA(x;x0)=A)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(y(x;x0)+ku,(4)wheretheconstantkis k=HD)]TJ /F10 7.97 Tf 6.58 0 Td[(2 (4)D 2\(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1) \(D 2)(4)andthedeSitterinvariantpartofthepropagatorA(y)obeysasaconsequenceofthepropagatorequation( 4 )theequation (4y)]TJ /F5 11.955 Tf 9.97 0 Td[(y2)A00(y)+D(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)A0(y)=(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)k.(4)ThesolutionofthisequationcanbewrittenasacombinationoftwoGauss'hypergeometricfunctions.Uponinsertingtheseinto( 4 ),oneobtainsforthevacuumpolarization i(x;x0)=2e2p )]TJ /F5 11.955 Tf 9.3 0 Td[(ggp )]TJ /F5 11.955 Tf 9.3 0 Td[(g0g0n@y@0yA02)]TJ /F5 11.955 Tf 9.96 0 Td[(AA00)]TJ /F5 11.955 Tf 9.97 0 Td[(kuA00 (4) )]TJ /F3 11.955 Tf 9.3 0 Td[(@@0yAA0+kuA0+@u@0ykA0+@y@0ukA0+@u@0uk2opluslocalterms/D(x)]TJ /F5 11.955 Tf 11.42 0 Td[(x0),whicharenotrelevantforthisconsideration(seee.g.[ 79 ]).Inordertoestablishcorrespondence,itsufcestoconsidertwoindependentcomponentsofiin( 4 ).Itturnsoutthattakingthe00andij(i6=j)components 83

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of( 4 )leadstorathersimpleexpressions, i00(x;x0)=2e2p )]TJ /F5 11.955 Tf 9.29 0 Td[(gp )]TJ /F5 11.955 Tf 9.3 0 Td[(g0H2 aa0 (4) n4+(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)2)]TJ /F4 11.955 Tf 11.96 0 Td[(4(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)cosh(v))]TJ /F5 11.955 Tf 10.46 -9.68 Td[(A02)]TJ /F5 11.955 Tf 9.96 0 Td[(AA00)]TJ /F5 11.955 Tf 9.96 0 Td[(kuA00+(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y))]TJ /F4 11.955 Tf 11.95 0 Td[(4cosh(v))]TJ /F5 11.955 Tf 10.46 -9.68 Td[(AA0+kuA0+)]TJ /F4 11.955 Tf 11.95 0 Td[(2(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)+4cosh(v)kA0+k2oiij(x;x0)ji6=j=2e2p )]TJ /F5 11.955 Tf 9.3 0 Td[(gp )]TJ /F5 11.955 Tf 9.3 0 Td[(g0H2 aa0n)]TJ /F4 11.955 Tf 9.96 0 Td[(4axia0xj)]TJ /F5 11.955 Tf 10.46 -9.68 Td[(A02)]TJ /F5 11.955 Tf 9.97 0 Td[(AA00)]TJ /F5 11.955 Tf 9.96 0 Td[(kuA00. (4) Ontheotherhand,fromEq.( 4 )wereadoff, i00=r0rF=2p )]TJ /F5 11.955 Tf 9.3 0 Td[(gp )]TJ /F5 11.955 Tf 9.3 0 Td[(g0H2 aa0h(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1))]TJ /F4 11.955 Tf 11.96 0 Td[(2(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)@y+4cosh(v)@yi@ybF (4) iiji6=j=)]TJ /F3 11.955 Tf 9.3 0 Td[(@0i@j(F+G)=p )]TJ /F5 11.955 Tf 9.3 0 Td[(gp )]TJ /F5 11.955 Tf 9.3 0 Td[(g0H2 aa0n4axia0xj@2y(bF+bG)o. (4) WeseefromEq.( 4 )thati00containsdependenciesintheform:(a)afunctionofy,(b)afunctionofytimesuand(c)afunctionofytimescosh(v).Ontheotherhand,inspecting( 4 )andassumingthatbFdoesnotdependonv(thisinfactfollowsfromthespatialdiagonalcomponentsiii)tellsusthatthefollowingequationsmustbeseparatelysatised, e2n)]TJ /F4 11.955 Tf 9.96 0 Td[(2(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y))]TJ /F5 11.955 Tf 5.48 -9.68 Td[(A02)]TJ /F5 11.955 Tf 9.96 0 Td[(AA00)]TJ /F5 11.955 Tf 9.96 0 Td[(kuA00)]TJ /F4 11.955 Tf 11.96 0 Td[(2)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(A+kuA0+2kA0o=2@2ybFe2n[4+(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)2])]TJ /F5 11.955 Tf 5.48 -9.68 Td[(A02)]TJ /F5 11.955 Tf 9.96 0 Td[(AA00)]TJ /F5 11.955 Tf 9.97 0 Td[(kuA00+2(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y))]TJ /F5 11.955 Tf 5.48 -9.69 Td[(A+kuA0)]TJ /F4 11.955 Tf 11.96 0 Td[(2(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)kA0+k2o=(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1))]TJ /F4 11.955 Tf 11.95 0 Td[(2(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)@y@ybF.Multiplyingtherstequationby(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)andaddingthesecondresultsin, e2n(4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)A02+(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)(A+ku)A0)]TJ /F5 11.955 Tf 9.97 0 Td[(k(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)(A+ku)+k2o=(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)@ybF, 84

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wherewealsomadeuseofEq.( 4 ).Now,multiplyingthisby)]TJ /F4 11.955 Tf 9.3 0 Td[(1=[2(D)]TJ /F4 11.955 Tf 10.89 0 Td[(1)]andseparatingtheu-dependenttermsyields )]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2@ybF=)]TJ /F5 11.955 Tf 26.22 8.09 Td[(e2 2(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)n(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)A0)]TJ /F5 11.955 Tf 11.96 0 Td[(kA+(4y)]TJ /F5 11.955 Tf 9.96 0 Td[(y2)A02+k2o)]TJ /F5 11.955 Tf 12.49 8.08 Td[(e2k 2(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)A0)]TJ /F5 11.955 Tf 9.96 0 Td[(ku=, (4) wherethelastequalityfollowsfrom( 4 ).ThisequationagreeswithEq.(44)inRef.[ 79 ],establishingtherstpartofthecorrespondencefortheminimallycoupledmasslessscalareld.NextweconsiderEqs.( 4 )and( 4 ),whichgiveusinformationaboutthesecondstructurefunctionG, )]TJ /F4 11.955 Tf 11.95 0 Td[(2e2)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(A02)]TJ /F5 11.955 Tf 9.96 0 Td[(AA00)]TJ /F5 11.955 Tf 9.97 0 Td[(kuA00=@2y(bF+bG). (4) Inordertogetthesecondmasterstructurefunction( 4 ),weadd )]TJ /F4 11.955 Tf 11.96 0 Td[([(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1))]TJ /F4 11.955 Tf 11.96 0 Td[((2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)@y+2@u]=2e2)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(A02)]TJ /F5 11.955 Tf 9.96 0 Td[(AA00)]TJ /F5 11.955 Tf 9.97 0 Td[(kuA00+e2k 2(2)]TJ /F5 11.955 Tf 9.97 0 Td[(y)A0)]TJ /F5 11.955 Tf 9.96 0 Td[(k (4) tobothsidesofEq.( 4 ).Thisprocedureresultsin =e2k 2(2)]TJ /F5 11.955 Tf 9.96 0 Td[(y)A0)]TJ /F5 11.955 Tf 9.97 0 Td[(k,(4)andagreeswithEq.(45)ofRef.[ 79 ].Thisestablishesthesecondpartofthecorrespondencefortheminimallycoupledmasslessscalareld,andcompletesour(nontrivial)checkofthemainresults( 4 4 ). 4.4DiscussionWehaveestablishedhowtotransformfromthenon-covariantform( 4 )ofthevacuumpolarizationtensorusedinRefs.[ 81 106 108 ]tothecovariantform( 4 )introducedinourrecentwork[ 79 ].Ourmainresultsarethetransformationformulae( 4 4 ),basedonwhichonecaneffortlesslymovefromonerepresentation 85

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ofthevacuumpolarizationtoanother.Ourresultscanbeusefulforstudiesoftheoneloopvacuumpolarizationinducedbygravitons,scalarsandfermionsonalocallydeSitterbackground[ 22 ].Inparticular,itprovidesawayofcharacterizingtheirreducibleeffectonphotonsofdeSitterbreakinginthegravitationalsector.Animportanttechnicalresultofsomeimportanceistheimprovedprocedure( 4 4 )forextractingthenoncovariantstructurefunctionsF(x;x)andG(x;x0). 86

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CHAPTER5GRAVITONCORRECTIONSTOVACUUMPOLARIZATIONDURINGINFLATIONThischaptercomprisesthemaincalculationofmydissertation:usingdimensionalregularizationtocomputetheoneloopquantumgravitationalcontributiontothevacuumpolarizationondeSitterbackground[ 109 ].AddingtheappropriateBPHZcountertermsgivesafullyrenormalizedresultwhichcanbeusedtoquantumcorrectMaxwell'sequations.Wehaveseenthatinationproducesavastensembleofinfraredscalarsandgravitons.Thisisthoughttobethesourceofprimordialscalarandtensorperturbations[ 115 ].Itisnaturaltowonderwhateffecttheseensembleshaveonotherparticles.Thatsortofquestioncanbeansweredbycomputingthescalarorgravitoncontributiontotheappropriate1PI(one-particle-irreducible)2-pointfunctionandthenusingthatresulttoquantum-correctthelinearizedeldequationfortheparticleinquestion.The1PI2-pointfunctionforascalarisknownasitsself-mass-squared,)]TJ /F5 11.955 Tf 9.3 0 Td[(iM2(x;x0),andthequantum-corrected,linearizedeldequationforamassless,minimallycoupledscalaris, @p )]TJ /F5 11.955 Tf 9.3 0 Td[(gg@'(x))]TJ /F9 11.955 Tf 11.95 16.27 Td[(Zd4x0M2(x;x0)'(x0)=0,(5)whereg(x)isthespacelikemetrictensor.Thefermion's1PI2-pointfunctioniscalleditsself-energy,)]TJ /F5 11.955 Tf 9.3 0 Td[(i[ij](x;x0),andthequantum-corrected,linearizedeldequationforamasslessfermionis, p )]TJ /F5 11.955 Tf 9.3 0 Td[(geaaiji@jk)]TJ /F4 11.955 Tf 11.16 8.09 Td[(1 2AbcJbcjk k(x))]TJ /F9 11.955 Tf 11.96 16.27 Td[(Zd4x0hiji(x;x0) j(x0)=0,(5)whereea(x)isthevierbeineld,aijarethegammamatrices,Abc(x)isthespinconnection,andJbc)]TJ /F6 7.97 Tf 25.08 4.71 Td[(i 4[b,c]aretheLorentzgenerators.The1PI2-pointfunctionforaphotonhastheevocativenamevacuumpolarization,+i[](x;x0),andthequantum-correctedMaxwellequationis, @p )]TJ /F5 11.955 Tf 9.3 0 Td[(gggF(x)+Zd4x0hi(x;x0)A(x0)=J(x),(5) 87

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whereF@A)]TJ /F3 11.955 Tf 13.52 0 Td[(@AistheeldstrengthtensorandJ(x)isthecurrentdensity.Andthe1PI2-pointfunctionforagravitonistermedthegravitonself-energy,)]TJ /F5 11.955 Tf 9.3 0 Td[(i[](x;x0),andthequantum-corrected,linearizedEinsteinequationis, p )]TJ /F5 11.955 Tf 9.3 0 Td[(gDh(x))]TJ /F9 11.955 Tf 11.96 16.27 Td[(Zd4x0hi(x;x0)h(x0)=1 2p )]TJ /F5 11.955 Tf 9.29 0 Td[(gTlin(x),(5)whereDistheLichnerowiczoperator,216GistheloopcountingparameterofquantumgravityandTlinisthelinearizedstresstensor.Manyresultsofthistypehavebeenderivedinrecentyears,withthebackgroundgeometryofprimordialinationmodeledusingthecosmologicalpatchofdeSitterspace.Theeffectsofmassless,minimallycoupled(MMC)scalarsaresimplesttostudy.Aquarticself-interactionleadsMMCscalarstodevelopagrowingmass[ 61 ].ThevacuumpolarizationfromchargedMMCscalarscausesthephotontodevelopamass[ 12 ]andengendersprofoundchangesinelectrodynamicforces[ 95 ].MMCscalarswhichareYukawa-coupledtoafermionmakethefermiondevelopagrowingmass[ 76 ].AndMMCscalarsdonothaveanyeffectongravitonsprovidedonecanabsorbcertainsurfacetermsintoperturbativecorrectionsoftheinitialstate[ 116 ].Theeffectsofinationarygravitonsaremoredifculttoworkout(everythingistougherinquantumgravity!),butstudieshavebeenmadeofwhattheydotofermionsandtoMMCscalars.Theresultsareinterestinglydifferent:whereasinationarygravitonsinduceaslowgrowthofthefermioneldstrength[ 77 ]theyhavenoseculareffectonMMCscalars[ 117 ].Thedifferenceseemstobeduetospin.AMMCscalarcanonlyinteractwithgravitonsthroughitskineticenergybutthiscannotmediateanyseculargrowth,inspiteofthegrowinggravitoneldstrength,becausethescalar'skineticenergyredshiftstozeroexponentiallyfast.Bycontrast,afermioninteractswithgravitonsthroughitsspin,inadditiontoitskineticenergy,andthespin-spininteractionremainseffectiveevenwhenthekineticenergyredshiftstozero[ 35 ].Thesamethingseemstobetrueofasmallmass[ 118 ]. 88

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Theimportanceofspininmediatinginteractionsbetweeninationarygravitonsandmasslessfermionssuggeststhattheremightbecomparablystrongeffectsonotherparticleswithspinsuchasphotonsandgravitons.Thegravitoncontributiontotheoneloopgravitonself-energyhasbeenworkedout[ 103 ]butsofarnotusedtoquantum-correctthelinearizedEinsteinequation.ThepurposeofthispaperistoderivetheoneloopgravitoncontributiontothevacuumpolarizationondeSitterbackground.Wewilluseittosolvethequantum-correctedMaxwellequationinasubsequentpaper.(Theatspaceanalogofthisproblemwascarriedoutasawarmupexercise[ 22 ].)Ourcomputationisdoneindimensionalregularization,fullyrenormalizedwiththenecessaryBPHZ(Bogoliubov-Parasiuk-Hepp-Zimmermann)counterterms[ 119 ],andreportedinthenoncovarianttensorbasiswhoseefcacyhasbeendemonstratedinarecentstudy[ 79 ].WealsousetheHartreeapproximationtoarguethatphotonslikelyexperienceasecularsuppressionoftheirelectriceldstrengths.Thischaptercontainssevensectionsofwhichtherstissection 5.1 givesthoseoftheFeynmanrulesofMaxwell+Einsteinwhichareneededforourcomputation.Thecontributionfromasingle4-pointvertexisderivedinsection 5.2 .Section 5.3 givesthemuchmorecomplicatedcontributionfromtwo3-pointvertices.Renormalizationisaccomplishedinsection 5.4 .Althoughtheuseofourresulttoquantum-correctMaxwell'sequationisdeferredtoalatterwork,theHartreeapproximationisemployedinsection 5.5 toarguethatphotonsexperiencesecularchangesofthesamestrengthbutoppositesignasthoseoffermions[ 77 ].Ourconclusionsaregiveninsection 5.6 5.1FeynmanRulesThepurposeofthissectionistopresenttheformalismusedtocomputetheoneloopquantumgravitycontributiontothevacuumpolarizationdepictedinFig. 5-1 .Webeginbydescribingthebackgroundgeometry.ThenweusetheprimitiveLagrangianstoderiveformalexpressionsforthersttwodiagramsofFig. 5-1 .Thelongestsubsectiondiscussesourconventionsforgaugexingandtheresultingpropagators.Wenext 89

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describehowthevacuumpolarizationcanberepresentedintermsoftwostructurefunctions.Thesectionclosesbygivingthecountertermsneededforthiscomputation. 5.1.1OurdeSitterBackgroundWemodelprimordialinationasthecosmologicalpatchofdeSitterspace.Theinvariantelementis, ds2=a2)]TJ /F5 11.955 Tf 9.29 0 Td[(d2+d~xd~x,(5)wherea()=)]TJ /F10 7.97 Tf 13.84 4.7 Td[(1 H=eHtisthescalefactorandHistheHubbleparameter.Whereasthespatialcoordinates~xtaketheirusualvalues,theconformaltimerunsfrom!(theinnitepast)to!0)]TJ /F1 11.955 Tf 10.4 -4.34 Td[((theinnitefuture).Inrepresentingfunctionssuchaspropagatorswhichdependupontwopoints,xandx0,wewillmakeextensiveuseofthedeSitterlengthfunction, y(x;x0)a()a(0)H2jj~x)]TJ /F3 11.955 Tf 11.44 .5 Td[(~x0jj2)]TJ /F4 11.955 Tf 11.95 0 Td[((j)]TJ /F3 11.955 Tf 11.95 0 Td[(0j)]TJ /F5 11.955 Tf 17.93 0 Td[(i)2.(5)WealsoneedthedeSitterbreakingproductofthescalefactorsaatxanda0atx0, uln(aa0).(5)Derivativesofyandufurnishaconvenientbasisforrepresentingbi-vectorfunctionsofxandx0suchasthevacuumpolarization, @y,@0y,@@0y,@u,@0u.(5)Itturnsoutthateithertakingcovariantderivativesofanyofthevebasistensors( 5 ),orcontractinganytwoofthemintooneanother,producesmetricsandmorebasistensors[ 78 91 ]. 90

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5.1.2OurPrimitiveDiagramsThetotalLagrangianconsistsoftheprimitivecontributionsfromgeneralrelativityandelectromagnetism,plustheBPHZcountertermsnecessaryforthiscomputation, L=LGR+LEM+LBPHZ.(5)TheprimitiveLagrangiansofgeneralrelativityandelectromagnetismare, LGR=1 16GR)]TJ /F4 11.955 Tf 9.97 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)p )]TJ /F5 11.955 Tf 9.3 0 Td[(g,LEM=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(1 4FFggp )]TJ /F5 11.955 Tf 9.3 0 Td[(g.(5)ThesymbolGstandsforNewton'sconstant,while(D)]TJ /F4 11.955 Tf 12.38 0 Td[(1)H2isthecosmologicalconstant.WeemployaD-dimensional,spacelikemetricg,withinverseganddeterminantg=det(g).OurafneconnectionandRiemanntensorare, )]TJ /F7 7.97 Tf 6.77 4.94 Td[(1 2gh@g+@g)]TJ /F3 11.955 Tf 9.96 0 Td[(@gi, (5) R@)]TJ /F7 7.97 Tf 6.77 4.93 Td[()]TJ /F3 11.955 Tf 9.97 0 Td[(@)]TJ /F7 7.97 Tf 6.77 4.93 Td[(+)]TJ /F7 7.97 Tf 16.74 4.93 Td[()]TJ /F7 7.97 Tf 6.78 4.93 Td[()]TJ /F4 11.955 Tf 9.96 0 Td[()]TJ /F7 7.97 Tf 6.78 4.93 Td[()]TJ /F7 7.97 Tf 6.78 4.93 Td[(. (5) OurRiccitensorisRRandtheassociatedRicciscalarisRgR.Theelectromagneticeldstrengthtensoranditsrstcovariantderivativeare, F@A)]TJ /F3 11.955 Tf 9.97 0 Td[(@A,DF@F)]TJ /F4 11.955 Tf 9.97 0 Td[()]TJ /F7 7.97 Tf 6.77 4.94 Td[(F)]TJ /F4 11.955 Tf 9.96 0 Td[()]TJ /F7 7.97 Tf 6.77 4.94 Td[(F.(5)Wedenethegravitoneldh(x)asthedifferencebetweenthefullmetricanditsdeSitterbackgroundvaluea2, g(x)a2()h+h(x)ia2eg(x),(5)where216Gistheloopcountingparameterofquantumgravity.Wefollowtheusualconventionswherebyacommadenotesordinarydifferentiation,thetraceofthegravitoneldishh,andgravitonindicesareraisedandloweredusingtheMinkowskimetric,hhandhh.Uptoasurfacetermthegravitational 91

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Lagrangiancanbewrittenas, LGR)]TJ /F4 11.955 Tf 11.95 0 Td[(Surface=aD)]TJ /F10 7.97 Tf 6.59 0 Td[(2 16Gp )]TJ /F9 11.955 Tf 9.36 .5 Td[(egegegeg(1 2h,h,)]TJ /F4 11.955 Tf 11.16 8.09 Td[(1 2h,h,+1 4h,h,)]TJ /F4 11.955 Tf 11.16 8.09 Td[(1 4h,h,). (5) FromFig. 5-1 onecanseethatweonlyneed( 5 )forthegravitonpropagator.Theonlyinteractionswerequiredescendfromthesecondvariationalderivativeoftheelectromagneticaction, 2SEM A(x)A(x0)=@(p )]TJ /F5 11.955 Tf 9.3 0 Td[(g(x)hg(x)g(x))]TJ /F5 11.955 Tf 9.96 0 Td[(g(x)g(x)i@D(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0)).(5)Thenecessaryvertexfunctionsareobtainedbyexpandingthemetricfactors, p )]TJ /F5 11.955 Tf 9.3 0 Td[(ggg)]TJ /F5 11.955 Tf 9.96 0 Td[(ggaD)]TJ /F10 7.97 Tf 6.58 0 Td[(4)]TJ /F3 11.955 Tf 9.97 0 Td[(+aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4Vh+2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4Uhh+O(3). (5) Thetensorfactorsforthe3-pointand4-pointverticesare, V=[]+4)[][](, (5) U=h1 4)]TJ /F4 11.955 Tf 11.16 8.09 Td[(1 2()i[]+)[][](+)[][](+()[]()+()[]()+()()[]+()()[]+[]()()+[]()(). (5) Parenthesizedindicesaresymmetrizedandindicesenclosedinsquarebracketsareanti-symmetrized.Ifwecallthegravitonpropagatori[](x;x0)andthephotonpropagatori[](x;x0),wecangiveformalexpressionsforthersttwodiagramsofFig. 5-1 .Theoneconstructedfromasingle4-pointvertexis, ih4pti(x;x0)=@(i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4Uihi(x;x)@D(x)]TJ /F5 11.955 Tf 9.97 0 Td[(x0)).(5) 92

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Thediagramconstructedfromtwo3-pointverticesis, ih3pti(x;x0)=@@0(iaD)]TJ /F10 7.97 Tf 6.59 0 Td[(4Vihi(x;x0)ia0D)]TJ /F10 7.97 Tf 6.59 0 Td[(4V@@0ihi(x;x0)). (5) 5.1.3OurPropagatorsThequadraticpartofthegravitationalLagrangian( 5 )is, L(2)GR=aD)]TJ /F10 7.97 Tf 6.59 0 Td[(2(1 2h,h,)]TJ /F4 11.955 Tf 11.16 8.09 Td[(1 2h,h,+1 4h,h,)]TJ /F4 11.955 Tf 11.16 8.09 Td[(1 4h,h,).(5)BeforexingthegaugeandgivingthegravitonpropagatorwemustdigresstosummarizethelongandconfusingdebatebetweencosmologistsandmathematicalphysicistsconcerningthedeSitterinvarianceoffreegravitons[ 85 87 ].AlthoughthepropagatorequationcanbemadedeSitterinvariantbyanappropriatechoiceofgauge,thatdoesnotguaranteethedeSitterinvarianceofthesolution.Theclassiccounter-examplewhichplaysanimportantroleinoursolutionforthegravitonpropagatoristhepropagatoriA(x;x0)ofamassless,minimallycoupledscalar, @p )]TJ /F5 11.955 Tf 9.3 0 Td[(gg@iA(x;x0)p )]TJ /F5 11.955 Tf 9.3 0 Td[(giA(x;x0)=iD(x)]TJ /F5 11.955 Tf 9.97 0 Td[(x0).(5)Equation( 5 )isdeSitterinvariant,butthereisnodeSitterinvariantsolutionforiA(x;x0)[ 90 ].Thiscanbeseenfromthetimedependenceofthecoincidencelimit[ 120 ], iA(x;x)=DivergentConstant+H2 42ln(a).(5) 93

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IfonechoosestheE(3)vacuum[ 121 ]topreservethespatialhomogeneityandisotropyofcosmologythentheuniquesolutionis[ 56 ], iA(x;x0)=icf(x;x0)+HD)]TJ /F10 7.97 Tf 6.59 0 Td[(2 (4)D 2\(D)]TJ /F4 11.955 Tf 9.96 0 Td[(1) \(D 2)(D D)]TJ /F4 11.955 Tf 9.96 0 Td[(4)]TJ /F10 7.97 Tf 6.78 4.34 Td[(2(D 2) \(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)4 yD 2)]TJ /F10 7.97 Tf 6.58 0 Td[(2)]TJ /F3 11.955 Tf 11.95 0 Td[(cot 2D+ln(aa0))+HD)]TJ /F10 7.97 Tf 6.59 0 Td[(2 (4)D 21Xn=1(1 n\(n+D)]TJ /F4 11.955 Tf 9.96 0 Td[(1) \(n+D 2)y 4n)]TJ /F4 11.955 Tf 31.66 8.08 Td[(1 n)]TJ /F6 7.97 Tf 11.16 4.71 Td[(D 2+2\(n+D 2+1) \(n+2)y 4n)]TJ /F14 5.978 Tf 7.78 3.26 Td[(D 2+2), (5) whereicf(x;x0)isthe(deSitterinvariant)propagatorofaconformallycoupledscalar, icf(x;x0)=HD)]TJ /F10 7.97 Tf 6.59 0 Td[(2 (4)D 2)]TJ /F9 11.955 Tf 6.78 13.27 Td[(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(14 yD 2)]TJ /F10 7.97 Tf 6.59 0 Td[(1.(5)Themassless,minimallycoupledscalarisespeciallyrelevanttogravitonsbecauseGrishchukshowedthatthephysicalcomponentsofh(transverse,tracelessandpurelyspatial)obeythesameequation[ 9 ].HencethegravitonpropagatormustbreakdeSitterinvarianceaswell.Acosmologistwouldalsoseethisfromthescaleinvarianceofthetensorpowerspectrum[ 85 ].MathematicalphysicistsforyearsdisputedthatconclusionbecausetheyfoundmanifestlydeSitterinvariantsolutionstothepropagatorequationwhichresultsfromaddingdeSitterinvariantgaugexingfunctionstothequadraticLagrangian( 5 )[ 122 ].Thediscordantviewpointshaverecentlyconvergedsomewhatwiththedemonstrationthatthereisanobstacletoaddinginvariantgaugexingfunctionsonanymanifold,likedeSitter,whichpossessesalinearizationinstability[ 97 ].ThatstillleavesopenthepossibilitiesofeitheraddinganoninvariantgaugexingfunctionorelseenforcingadeSitterinvariantgaugeconditionasastrongoperatorequation.Wheneitherpossibilityispursued,alongwiththerequirementthattheresultingpropagatorcanbeexpressedasasuperpositionofplanewavemodefunctions[ 38 92 99 ],theresultisadeSitterbreakingsolutionofpreciselytheformimpliedbythescaleinvarianceofthetensorpowerspectrum[ 98 123 ].However,whenadeSitterinvariantgaugeconditionis 94

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employed,inconjunctionwithanalyticcontinuationtechniques(eitherfromEuclideandeSitterspace,orinthemass-squaredofcertainscalarpropagators),theresultisdeSitterinvariant[ 124 125 ],exceptforcertaindiscretechoicesofthegaugecondition.Itiswellknownthatanalyticcontinuationfailstorecoverpowerlawinfrareddivergences[ 48 91 126 ].Inthiscontextthediscreteproblems,whichhavelongbeennoted[ 127 128 ],seemsuspiciouslylikethespecialvaluesatwhichapowerlawinfrareddivergencewhichisalwayspresenthappenstobecomelogarithmicandhencevisibletoananalyticcontinuationtechnique.WewillthereforeemployadeSitterbreakinggravitonpropagator,whichseemstobethesaferchoice.Despitethecontinuingdisagreements,itisimportanttonotethatverylittledifferenceremainsbetweencosmologistsandmathematicalphysicists.Inparticular,thevariousdeSitterbreakingsolutionsforthegravitonpropagatorallgivethesameresultforthelinearizedWeyl-Weylcorrelator[ 100 129 ],andthisagreeswiththeresultfromdeSitterinvariantsolutions[ 130 ]oncesomeerrorsarecorrected[ 131 ].Wexthegaugebyadding[ 38 ], LGRx=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(2FF,Fh,)]TJ /F4 11.955 Tf 11.16 8.09 Td[(1 2h,+(D)]TJ /F4 11.955 Tf 9.97 0 Td[(2)Hah0.(5)Theresultinggravitonpropagatorcanbeexpressedasasumofconstanttensorfactorsmultipliedbyscalarpropagators[ 38 ], ihi(x;x0)=XI=A,B,ChTIiiI(x;x0).(5)WehavealreadyseenthedeSitterbreakingA-typepropagator( 5 ).TheB-typeandC-typepropagatorsobeytheequations, h)]TJ /F4 11.955 Tf 9.96 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(2)H2iiB(x;x0)=iD(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0) p )]TJ /F5 11.955 Tf 9.3 0 Td[(g=h)]TJ /F4 11.955 Tf 9.96 0 Td[(2(D)]TJ /F4 11.955 Tf 9.97 0 Td[(3)H2iiC(x;x0).(5) 95

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EachofthesepropagatorsisdeSitterinvariantandconsistsoficf(x;x0)plusaninniteseriesoflesssingulartermswhichvanishinD=4dimensions, iB(x;x0)=B(y)=icf(x;x0))]TJ /F5 11.955 Tf 14.83 8.09 Td[(HD)]TJ /F10 7.97 Tf 6.59 0 Td[(2 (4)D 21Xn=0(\(n+D)]TJ /F4 11.955 Tf 9.96 0 Td[(2) \(n+D 2)y 4n)]TJ /F4 11.955 Tf 10.49 8.79 Td[(\(n+D 2) \(n+2)y 4n)]TJ /F14 5.978 Tf 7.78 3.26 Td[(D 2+2), (5) iC(x;x0)=C(y)=icf(x;x0)+HD)]TJ /F10 7.97 Tf 6.59 0 Td[(2 (4)D 21Xn=0((n+1)\(n+D)]TJ /F4 11.955 Tf 9.96 0 Td[(3) \(n+D 2)y 4n)]TJ /F9 11.955 Tf 9.3 13.27 Td[(n)]TJ /F5 11.955 Tf 11.16 8.09 Td[(D 2+3\(n+D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1) \(n+2)y 4n)]TJ /F14 5.978 Tf 7.78 3.25 Td[(D 2+2). (5) NotethattheB-typeandC-typepropagatorsagreeforD=4dimensions.Thetensorfactorsare, hTAi=2 ( ))]TJ /F4 11.955 Tf 23.36 8.09 Td[(2 D)]TJ /F4 11.955 Tf 9.97 0 Td[(3 (5) hTBi=)]TJ /F4 11.955 Tf 9.3 0 Td[(40( )(0), (5) hTCi=2 (D)]TJ /F4 11.955 Tf 9.97 0 Td[(2)(D)]TJ /F4 11.955 Tf 9.96 0 Td[(3)h(D)]TJ /F4 11.955 Tf 9.97 0 Td[(3)00+ ih(D)]TJ /F4 11.955 Tf 9.96 0 Td[(3)00+ i, (5) where +00isthespatialpartoftheMinkowskimetric.Thequadraticpartoftheelectromagneticactionis, L(2)EM=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4@A@A+1 2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4@A@A.(5)OfcourseitisnomorepossibletoaddadeSitterinvariantgaugexingfunctionforelectromagnetismthanitisforgravity[ 97 ].Unlikegravitons,photonsshownophysicalbreakingofdeSitterinvariance,sothephotonpropagatorismanifestlydeSitterinvariantifoneemploysanexactdeSitterinvariantgaugecondition[ 89 ].However,thedeSitterbreakingofthegravitonpropagatorimpliesthatthereisnopointtokeepingthephotonpropagatorinvariant.Wehavechoseninsteadtoaddthenoncovariantgauge 96

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xingfunctionwhichismostcloselyrelatedtothegravitationalone( 5 )[ 84 ], LGF=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4A,)]TJ /F4 11.955 Tf 11.95 0 Td[((D)]TJ /F4 11.955 Tf 9.97 0 Td[(4)HaA02.(5)Theassociatedphotonpropagatoris[ 84 ], ihi(x;x0)= aa0iB(y))]TJ /F3 11.955 Tf 11.96 0 Td[(00aa0iC(x;x0).(5) 5.1.4OurStructureFunctionsThevacuumpolarizationisabi-vectordensityi[](x;x0)whichistransverseateachpoint, @ @xhi(x;x0)=0=@ @x0hi(x;x0).(5)Althoughitpossesses16componentsinD=4dimensions,thecombinationoftransversality( 5 ),reectioninvariance[](x;x0)=[](x0;x)andthecoordinatesymmetriesofthevacuumrelatethesecomponentssothattheycanbeexpressedintermsofaveryfewstructurefunctions.Forexample,Poincareinvarianceimpliesthefollowingsimpleformintheatspacelimit, hiat=)]TJ /F3 11.955 Tf 9.96 0 Td[(@@0(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0).(5)Theoneloopfermionandscalarcontributionsto(x)]TJ /F5 11.955 Tf 9.54 0 Td[(x0)havebeenknownfordecades,andanexplicitresultfortheoneloopgravitoncontributionhasrecentlybeenderived[ 22 ].BecausethegravitonvacuumondeSitterbackgrounddoesnotrespectfulldeSitterinvariance,butonlyspatialhomogeneityandisotropy,itturnsoutthattwostructurefunctionsareneeded[ 79 ].Wecouldstillchoosetorepresentthetransverseprojectionoperatorsusingcovariantderivatives.However,adetailedexaminationofthisformforthealreadyderivedvacuumpolarizationfromSQED[ 12 ]revealsthatitiscumbersomeandthatitobscuresratherthansimpliestheessentialphysics[ 79 ].ThisseemstobebecausetheconformalinvarianceofclassicalelectromagnetisminD=4dimensions 97

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isamorepowerfulorganizingprinciplethatthebackground'sisometries.WehavethereforechosentoemploythenoncovariantformoriginallyusedtorepresenttheSQEDresult, ihi(x;x0)=)]TJ /F3 11.955 Tf 9.96 0 Td[(@@0F(x;x0)+ )]TJ ET q .478 w 311.35 -88.81 m 317.55 -88.81 l S Q BT /F3 11.955 Tf 311.35 -95.63 Td[( @@0G(x;x0).(5)Weremindthereaderthat +00isthepurelyspatialpartoftheMinkowskimetric.Bycomparing( 5 )with( 5 )oneseesthatourstructurefunctionF(x;x0)mustagreewiththeatspaceresulti(x)]TJ /F5 11.955 Tf 12.84 0 Td[(x0)inthelimitthatHvanisheswiththeco-movingtimet=ln(a)=Hheldxed.HencetheleadingdivergencesarecontainedinF(x;x0).AlltermsinG(x;x0)mustcontainatleastonefactorofH2,andtheyarecorrespondinglylessdivergent.Althoughourrepresentation( 5 )isnotdeSittercovariant,thetwostructurefunctionshaveverysimplephysicalinterpretationsintermsofchangesintheelectricpermittivityandthemagneticpermeability[ 112 ].However,thereisastraightforwardprocedureforconvertingourresultsforF(x;x0)andG(x;x0)tothephysicallyopaque,deSittercovariantrepresentation[ 111 ]ifthatisdesired. 5.1.5OurCountertermsDeserandvanNieuwenhuizenshowedthatEinstein+Maxwellisnotrenormalizableatonelooporder[ 47 ].However,itisstraightforwardtoabsorbthedivergences,orderbyorder,withBPHZcounterterms[ 119 ].Wecanthensolvethequantum-correctedMaxwellequation( 5 )inthestandardsenseofeffectiveeldtheory[ 19 ].Thishasalreadybeendoneforquantumgravitationalcorrectionstoelectrodynamicsonatbackground[ 22 45 ].Thevacuumpolarizationhastwoexternalphotonlinessoourcomputationrequirescountertermswithtwovectorpotentials.Thesupercialdegreeofdivergenceisfouratonelooporder,whichmeanstheremustbefourderivativesactingeitheruponvectorpotentialsormetrics.However,U(1)gaugeinvarianceimpliesthatatleasttwoofthe 98

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derivativesmustactonvectorpotentials.Ifbothoftheremainingtwoderivativesactuponvectorpotentialswehavethesinglecountertermwhichsurvivesinatspace[ 22 ], C4DFDFgggp )]TJ /F5 11.955 Tf 9.3 0 Td[(g.(5)Becauseourgaugexingfunctions( 5 )and( 5 )reduce,forH!0atxedco-movingtime,tothoseemployedinourpreviousatspacecomputation[ 22 ],thedivergentpartofC4mustagreeaswell, C4=2 128D 2D\(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1) (D)]TJ /F4 11.955 Tf 9.96 0 Td[(1)(D)]TJ /F4 11.955 Tf 9.96 0 Td[(4).(5)Therearethreeinvariantcountertermswithtwoderivativesactingontwovectorpotentialsandtwoactingonmetrics, C1FFggRp )]TJ /F5 11.955 Tf 9.3 0 Td[(g+C2FFgRp )]TJ /F5 11.955 Tf 9.3 0 Td[(g+C3FFRp )]TJ /F5 11.955 Tf 9.3 0 Td[(g.(5)However,allthecurvaturesarerelatedindeSitterbackground, R)166(!H2a)]TJ /F10 7.97 Tf 6.58 0 Td[(4h)]TJ /F3 11.955 Tf 9.96 0 Td[(i, (5) R)166(!(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)H2a)]TJ /F10 7.97 Tf 6.58 0 Td[(2, (5) R)166(!(D)]TJ /F4 11.955 Tf 9.97 0 Td[(1)DH2. (5) Ourcomputationthereforedeterminesonlythecombination C(D)]TJ /F4 11.955 Tf 9.68 0 Td[(1)DC1+(D)]TJ /F4 11.955 Tf 9.69 0 Td[(1)C2+2C3,andwecanwritetheresultingcountertermasjust CH2FFggp )]TJ /F5 11.955 Tf 9.3 0 Td[(g.Becauseourgaugexingfunctions( 5 )and( 5 )breakdeSitterinvariance,itisnecessarytoconsidercountertermswhichareU(1)invariantbutnotgenerallycoordinateinvariant.Twopropertiesofourgaugeconditionsrestrictthenumberofpossibilities: TheybecomePoincareinvariantintheatspacelimitofH!0atxedco-movingtime;and 99

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Theypreservespatialhomogeneityandisotropy,aswellasthedilatationsymmetryx!kx.Therstpropertymeansthat( 5 )istheonlycountertermwithoutexplicitfactorsofH.Thesecondpropertyimpliesthattheonlyextracountertermwerequirehastheform,CH2FijFk`gikgj`p )]TJ /F5 11.955 Tf 9.3 0 Td[(g.Ourcomputationthereforerequiresonlythreecounterterms, LBPHZ=CH2FijFk`gikgj`p )]TJ /F5 11.955 Tf 9.3 0 Td[(g+ CH2FFggp )]TJ /F5 11.955 Tf 9.3 0 Td[(g+C4DFDFgggp )]TJ /F5 11.955 Tf 9.29 0 Td[(g. (5) Itremainstoworkouthowthecounterterms( 5 )affectthestructurefunctionsF(x;x0)andG(x;x0)inourrepresentation( 5 )ofthevacuumpolarization.TherststepistakingthevariationwithrespecttoA(x)andA(x0), SBPHZ A(x)A(x0)=)]TJ /F4 11.955 Tf 9.29 0 Td[(4CH2@(p )]TJ /F5 11.955 Tf 9.3 0 Td[(g(x) g(x) g(x)F(x) A(x0))+@(p )]TJ /F5 11.955 Tf 9.3 0 Td[(g(x)g(x)g(x)h4 CH2)]TJ /F4 11.955 Tf 9.97 0 Td[(4C4g(x)DDiF(x) A(x0)). (5) ThenextstepisspecializingtodeSitter,withg=a2and)]TJ /F7 7.97 Tf 6.78 4.33 Td[(=aH(0+0)]TJ /F3 11.955 Tf -447.44 -23.91 Td[(0).ThisisverysimplefortheCand Ccounterterms, 4CH2@(p )]TJ /F5 11.955 Tf 9.29 0 Td[(g g gF(x) A(x0))= )]TJ ET q .478 w 238.89 -413.4 m 245.1 -413.4 l S Q BT /F3 11.955 Tf 238.89 -420.22 Td[( @@0(4CH2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4D(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0)), (5) 4 CH2@(p )]TJ /F5 11.955 Tf 9.29 0 Td[(gggF(x) A(x0))=)]TJ /F3 11.955 Tf 9.96 0 Td[(@@0(aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4h4 CH2)]TJ /F4 11.955 Tf 9.96 0 Td[(4C4iD(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0)). (5) Ofcourse( 5 )contributesdirectlytothestructurefunctionF(x;x0),and( 5 )contributestoG(x;x0).TheC4countertermiscomplicatedbecauseofthewaythetensord'AlembertianactsonF, F=1 a2(h@2)]TJ /F4 11.955 Tf 9.96 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(4)Ha@0+2(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)H2a2iF)]TJ /F4 11.955 Tf 9.29 0 Td[(2Hah0@F)]TJ /F3 11.955 Tf 9.97 0 Td[(0@Fi+(D)]TJ /F4 11.955 Tf 9.97 0 Td[(4)H2a2h0F0)]TJ /F3 11.955 Tf 9.96 0 Td[(0F0i). (5) 100

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Thersttermof( 5 )contributesonlytothestructurefunctionF(x;x0),whereasthesecondandthirdtermscontributetobothstructurefunctions.Aftersometedioustensoralgebraandreectionofderivativesonthedeltafunction(@0D(x)]TJ /F5 11.955 Tf 11.4 0 Td[(x0)=)]TJ /F3 11.955 Tf 9.3 0 Td[(@D(x)]TJ /F5 11.955 Tf 11.4 0 Td[(x0)),wendthatthecountertermsmakethefollowingcontributionstothetwostructurefunctions, F(x;x0)=4h C)]TJ /F4 11.955 Tf 9.96 0 Td[((3D)]TJ /F4 11.955 Tf 9.96 0 Td[(8)C4iH2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4iD(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0))]TJ /F4 11.955 Tf 7.31 0 Td[(4C4aD)]TJ /F10 7.97 Tf 6.58 0 Td[(6h@2)]TJ /F4 11.955 Tf 9.96 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(6)Ha@0iiD(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0), (5) G(x;x0)=4hC)]TJ /F4 11.955 Tf 9.96 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(6)C4iH2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4iD(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0). (5) Thesimplicityofthisformisonemoreindication,addedtothemanyalreadyfound[ 79 ],ofthesuperiorityofthenoncovariantrepresentationoverthecovariantone. 5.2The4-PointContributionThissectionsummarizesthederivationofthecontributiontothevacuumpolarizationfromthe4-pointdiagraminFig. 5-1 ,byderivingthestructurefunctionsrequiredforthedesiredrepresentationofthevacuumpolarization( 5 ).Thisisachievedbyrstcompletingallnaiveindexcontractionsof( 5 ),followedbyasubstitutionofthegravitonpropagator,whichallowsallnalindexcontractionstobetaken.Lastly,severalcontractionidentitiesareintroducedfromwhichthe4-pointstructurefunctionscanbededuced.Theprocedureoutlinedinthissectionwillbesimilartothatusedforderivingthe3-pointcontributionandservesasasimpleguidetoextractingthedesiredstructurefunctions. 5.2.1NaiveContractionsInordertoderivethetwoscalarstructurefunctionsfromthe4-pointcontributiontothevacuumpolarization( 5 )therststepwillbetosubstitute( 5 )forthe4-pointvertexfunctionandapplythenaivecontractions.Thiswillnaturallyengendermanyterms,andtomaketheprocesssimpleforthereadertofollowwewillbreak( 5 )into 101

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piecesandthendenethe4-pointcontributionasasumofthosepieces ih4pti(x;x0)=@(i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(46XI=1UIihi(x;x)@D(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0)),(5)whereTable 5-1 liststhevariousUI.Oncethevertexfunctiontermshavebeeninsertedintothevacuumpolarizationthenaiveindexcontractionscanbecarriedouttermbyterm,theresultsofwhicharelistedinTable 5-2 5.2.2SubstitutionofGravitonPropagatorTocompletetheindexcontractionsthefullgravitonpropagatormustbeinserted.Thiswillagaincreatemanymoretermssoitisusefultobreakthegravitonpropagatorintothreepiecesandconsidereachpartseparately.Uponconsiderationofthegravitonpropagator( 5 )weseethatifeachofthethreetypesofscalarpropagatorsaresetequal,tosayB(y),thenthetensorcomponentscombinetogivetheconformalgravitonpropagatortensorcomponent hTAi+hTBi+hTCi=hTcfi=2())]TJ /F4 11.955 Tf 30.24 8.09 Td[(2 (D)]TJ /F4 11.955 Tf 11.95 0 Td[(2).(5)ByaddingandsubtractingB(y)fromeachofthescalarpropagatorsin( 5 )thegravitonpropagatorcanberewrittenas ihi(x;x0)=hTcfiB+hTAi(iA)]TJ /F5 11.955 Tf 11.95 0 Td[(B)+hTCi(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B). (5) Writingthegravitonpropagatorinthiswaycancelstheconformalpartsofthescalarpropagatorsinthesecondtwoterms.Thispropertyisnotusefulforthe4-pointcontribution,butitwillbehelpfulforrenormalizingthe3-pointcontributionandweuseitagainhereforconsistency.Wewillrefertothethreecomponentsof( 5 )separatelyastheconformalpart,theA-typepartandandC-typepartasassociatedwiththetensorfactorofeachpiece.BeforemakingsubstitutionsforthegravitonpropagatorweneedtoknowhoweachofthetensorcomponentscontractinthevariouswaysappearinginTable 5-2 .The 102

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relevantcontractionsarelistedinTable 5-3 ,whereeachelementrepresentsthetensorfactorinthetoprowcontractedwiththecombinationofatspacemetricslistedinthelefthandcolumn.Asstatedabovewedissectthegravitonpropagatorsubstitutionintothreeparts.ThesubstitutionandfollowingindexcontractionsfortheconformalpartofthegravitonpropagatorarelistedinTable 5-4 ,thetermsresultingfromtheA-typepartarelistedinTable 5-5 ,andthetermsfromtheC-typepartinTable 5-6 .Notethatinthesetableswesuppressthe(x)]TJ /F5 11.955 Tf 12.51 0 Td[(x0)factoronthedeltafunctionsforbrevity.UponfurtherinspectionoftheconformalpartofthegravitonpropagatorinTable 5-4 weseethatallsixsetsoftermshavethesametensorstructure.Combiningallofthetermswendthesimpliedexpression ih4pticf(x;x0)=)]TJ /F4 11.955 Tf 10.5 8.09 Td[((D3)]TJ /F4 11.955 Tf 11.95 0 Td[(9D2+10D+16) 4(D)]TJ /F4 11.955 Tf 11.96 0 Td[(2)@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4B(0)@D(x)]TJ /F5 11.955 Tf 11.96 0 Td[(x0))]TJ /F3 11.955 Tf 9.3 0 Td[(@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4B(0)@D(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0). (5) Equation( 5 ),andTables 5-5 and 5-6 makeupthe4-pointcontributiontothevacuumpolarization.Nextwewilltransformtheseresultsintothedesiredmanifestlytransverseform. 5.2.3Findingthe4-PointStructureFunctionsRecallthatitisourgoaltowritetheresultforthevacuumpolarizationintheformof( 5 ),wheretwotransverseprojectionoperatorsareactingontwoscalarstructurefunctionsF(x;x0)andG(x;x0).UsingtheTablesofsection 5.2.2 wecannowderivethesestructurefunctions.Theeasiestwaywefoundtoextractthestructurefunctionswastoexploittheknowntransversalityofthevacuumpolarizationandisolatethestructurefunctionsviatwocontractions,resultingintwoequationsforthetwostructurefunctions.Therst 103

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contractionismadewith00,appliedtoequation( 5 )thisgivestheidentity ih00i(x;x0)=r2F(x;x0),(5)whichprovidesasimpleequationforF.TondGthesecondcontractionistakenwithij(j6=i),appliedto( 5 )wendthesecondidentity ihiji(x;x0)=)]TJ /F3 11.955 Tf 9.3 0 Td[(@0i@j[F(x;x0)+G(x;x0)].(5)KnowingFitisnowtrivialtondG.Itistruethatthereareotherpairsofcontractionsthatwouldworkequallyaswell.Forexample,contractingwith0iandleadstotheidentities ih0ii(x;x0)=)]TJ /F3 11.955 Tf 9.3 0 Td[(@00@iF(x;x0), (5) ihi(x;x0)=)]TJ /F4 11.955 Tf 9.3 0 Td[((D)]TJ /F4 11.955 Tf 11.95 0 Td[(1)@0@00F(x;x0))]TJ /F4 11.955 Tf 11.95 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(2)r2[F(x;x0)+G(x;x0)]. (5) Theseidentitiesprovideausefulcheckonthestructurefunctionsderivedfromidentities( 5 )and( 5 ).Therearelikelymorecontractionsthatwouldprovidesimilarsetsofequations,butthesetwosetsweresufcientforourpurposes.Itisquiteatedioustasktoshowhowapplyingthesecontractionsplaysoutfortheentire4-pointcontribution,andtheprocedureisextremelyrepetitive.Forthereader'ssanityandourownwewillworkoutoneexampleandassumethattheprocedurefortherestofthe4-pointcontributioncanbeeasilydeduced.WewilldemonstratehowtondFandGfromtheconformalpartofthegravitonpropagatorasgivenin( 5 ). 104

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Contracting00with( 5 )wehave ih004pticf(x;x0)=)]TJ /F4 11.955 Tf 10.5 8.09 Td[((D3)]TJ /F4 11.955 Tf 11.95 0 Td[(9D2+10D+16) 4(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)n)]TJ /F3 11.955 Tf 9.3 0 Td[(@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4B@D(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0))]TJ /F3 11.955 Tf 9.3 0 Td[(@0i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4B@0D(x)]TJ /F5 11.955 Tf 11.96 0 Td[(x0)o=)]TJ /F4 11.955 Tf 10.5 8.09 Td[((D3)]TJ /F4 11.955 Tf 11.95 0 Td[(9D2+10D+16) 4(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)n)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(@@0+@0@00i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4BD(x)]TJ /F5 11.955 Tf 11.96 0 Td[(x0)o=(D3)]TJ /F4 11.955 Tf 11.96 0 Td[(9D2+10D+16) 4(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)r2i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4B(0)D(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0), (5) whereingoingfromtherstlinetothesecondweusedthedeltafunctiontomakethechange@!)]TJ /F3 11.955 Tf 25.79 0 Td[(@0.Itisthentrivialtopulltheinnerderivativeoutsidethecurlybracketssinceallprefactorsareonlyfunctionsofx.AlsotheB-typepropagatorcanbeevaluatedaty=0sinceitisbeingevaluatedatcoincidence(x=x0).Uponcomparisonwithidentity( 5 )wendtherststructurefunction F4,cf(x;x0)=(D3)]TJ /F4 11.955 Tf 11.95 0 Td[(9D2+10D+16) 4(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4B(0)D(x)]TJ /F5 11.955 Tf 11.96 0 Td[(x0).(5)Tondthesecondstructurefunctionwecontractij(j6=i)with( 5 ) ihijicf(x;x0)=)]TJ /F4 11.955 Tf 10.49 8.08 Td[((D3)]TJ /F4 11.955 Tf 11.95 0 Td[(9D2+10D+16) 4(D)]TJ /F4 11.955 Tf 11.96 0 Td[(2)n0)]TJ /F3 11.955 Tf 11.96 0 Td[(@ji2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4B@iD(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0)o=)]TJ /F4 11.955 Tf 10.49 8.08 Td[((D3)]TJ /F4 11.955 Tf 11.95 0 Td[(9D2+10D+16) 4(D)]TJ /F4 11.955 Tf 11.96 0 Td[(2)@0i@ji2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4BD(x)]TJ /F5 11.955 Tf 11.96 0 Td[(x0). (5) Invokingidentity( 5 )andmakingthepropersubstitutionforF4,cfitiseasytosee G4,cf(x;x0)=0.(5)ThisconcludestheexamplecaseforndingFandG.ThesameprocedurecanbeappliedtoallofthetermsinTables 5-5 and 5-6 tondtherestofthestructurefunctions.CombiningallFandGcontributionsfromthethreepartsofthegravitonpropagator 105

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producesthefullresultforthe4-pointstructurefunctions F4(x;x0)=2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4nD(D)]TJ /F4 11.955 Tf 9.96 0 Td[(5) 4iA(0))]TJ /F4 11.955 Tf 13.15 8.09 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(1) 2B(0))]TJ /F4 11.955 Tf 10.49 8.09 Td[((3D)]TJ /F4 11.955 Tf 9.96 0 Td[(10) 2(D)]TJ /F4 11.955 Tf 9.97 0 Td[(2)C(0)oiD(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0) (5) G4(x;x0)=)]TJ /F3 11.955 Tf 9.3 0 Td[(2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4n(D2)]TJ /F4 11.955 Tf 9.96 0 Td[(4D+1) (D)]TJ /F4 11.955 Tf 9.97 0 Td[(3)iA(0))]TJ /F4 11.955 Tf 11.96 0 Td[((D)]TJ /F4 11.955 Tf 9.96 0 Td[(3)B(0))]TJ /F4 11.955 Tf 9.29 0 Td[(2(D)]TJ /F4 11.955 Tf 9.96 0 Td[(4) (D)]TJ /F4 11.955 Tf 9.96 0 Td[(3)C(0)oiD(x)]TJ /F5 11.955 Tf 9.97 0 Td[(x0) (5) Wecannowmakesubstitutionsforthepropagators.TheB-andC-typepropagatorsareaniteconstantatcoincidenceinD=4,buttheA-typepropagatorisdivergent.Therefore,itisusefultobreakthestructurefunctionsintotheirniteanddivergentpieces F4,div(x;x0)=)]TJ /F5 11.955 Tf 10.49 8.09 Td[(D(D)]TJ /F4 11.955 Tf 11.95 0 Td[(5) 42aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4HD)]TJ /F10 7.97 Tf 6.59 0 Td[(2\(D)]TJ /F4 11.955 Tf 11.96 0 Td[(1) (4)D=2\(D 2)cot 2DiD(x)]TJ /F5 11.955 Tf 11.96 0 Td[(x0), (5) F4,nite(x;x0)=2H2 421 4)]TJ /F4 11.955 Tf 11.95 0 Td[(ln(a)i4(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0), (5) G4,div(x;x0)=(D2)]TJ /F4 11.955 Tf 11.95 0 Td[(4D+1) (D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4HD)]TJ /F10 7.97 Tf 6.59 0 Td[(2\(D)]TJ /F4 11.955 Tf 11.95 0 Td[(1) (4)D=2\(D 2)cot 2DiD(x)]TJ /F5 11.955 Tf 11.96 0 Td[(x0), (5) G4,nite(x;x0)=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(2H2 421 4+ln(a)i4(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0). (5) Thisconcludesourderivationofthe4-pointcontributionstothestructurefunctions,wherewenotethatonlythetermsproportionaltoln(a)in( 5 5 )cannotbeabsorbedintothecounterterms.Wewillnowderivethe3-pointcontribution. 106

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5.3The3-PointContributionThissectionwillcoverthederivationofthecontributiontothevacuumpolarizationattributedtothe3-pointdiagraminFig. 5-1 .Theprocedureforderivingthescalarstructurefunctionsfromthediagramisverysimilartotheoneusedinsection 5.2 .Firstthenaiveindexcontractionsarecompletedinpiecesbydividingthe3-pointvertexfunctionappropriately.Thenoncethepropersubstitutionshavebeenmadeforthegravitonandphotonpropagatorsallremainingindexcontractionscanbecompleted.Finallythe3-pointcontributionstoFandGwillbepresentedinseveralparts,brokenupaccordingtowhichpiecesofthephotonandgravitonpropagatorsthestructurefunctionoriginatedfrom. 5.3.1NaivecontractionsFollowingthesameorganizingprocedureasintheprevioussection,wecompletethenaiveindexcontractionsrstbybreakingthe3-pointvertexfunction( 5 )intopieces,showninTable 5-7 ,andrewritingthe3-pointcontributionasasumoftheseterms ih3pti(x;x0)=@@0niaD)]TJ /F10 7.97 Tf 6.59 0 Td[(42XI=1VIihi(x;x0)ia0D)]TJ /F10 7.97 Tf 6.59 0 Td[(42XJ=1VJ@@0ihi(x;x0)o. (5) ( 5 )containsproductsofthepartsofthetwo3-pointvertexfunctions;theresultforcompletingthenaiveindexcontractionsforallsaidproductsareshowninTable 5-8 .Tosimplifytheresultsinthistableashorthandnotationforantisymmetrizationhasbeenadopted,wherebothsquarebracketsanddoublesquarebracketsimplyantisymmetrization.Theindexstructureforthesetermscanbeconfusinganditshouldbenotedthattheantisymmetrizationonlyappliestotheimmediateindiceswithinthe 107

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brackets.Hereisaworkedoutexample ih[[[i@]@0]]=1 4nihi@@0)]TJ /F5 11.955 Tf 11.95 0 Td[(ihi@@0+ihi@@0)]TJ /F5 11.955 Tf 9.3 0 Td[(ihi@@0o. (5) Tocompletetheindexcontractions,substitutionsforthegravitonandphotonpropagatormustbemade. 5.3.2GravitonandPhotonPropagatorSubstitutionsTomakesubstitutionsforthegravitonandphotonpropagatorsandcompletingtheindexcontractionsasclearaspossiblewewillbeusingthenewformofthegravitonpropagator( 5 )andalsobreakthephotonpropagatorupinasimilarmanner.Tomodifytheorginalphotonpropagator( 5 )wecanagainaddandsubtractB(x;x0)fromeachterm.Rearrangingwend ihi(x;x0)=aa0B(y)+00[B(y))]TJ /F5 11.955 Tf 11.96 0 Td[(C(y)].(5)Sincewewillbeconsideringdifferentpartsofthepropagatorsindividuallyfortherestofthissectionitisnecessarytotakeamomenttoexplainournotation.Wewillrefertotherst,second,andthirdtermof( 5 )withthesubscriptsB,A,andCrespectively.Likewise,for( 5 )wewillrefertotherstandsecondtermswiththesubscriptsBandCrespectively.Inthe3-pointcontributionitisalwaysaproductofthegravitonandphotonpropagatorthatappear,sothereareingeneralsixcombinationsoftermsthatcanariseBB,AB,CB,BC,AC,andCC,wheretherstletterstandsforthepartofthegravitonpropagatorbeingconsideredandthesecondletterstandsforthepartofthephotonpropagator.However,itcanbeshownthatthelasttwocombinationsdonotneedtobecalculatedsincetheyaremadeoftheproductoftwodifferencesofscalarpropagators.Forthesecasestheconformalpartsofthescalarpropagatorswillcancelandwhatremainsareonlytheinnitesumsineachpropagator,butthesetermsgolike 108

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y0.TheirdegreeofdivergenceissuchthatwecantakeD=4immediately,andinthiscasealloftheinniteseriesvanish.ItistruethattheA-typescalarpropagatorhasanextratermgoinglikey2)]TJ /F6 7.97 Tf 6.59 0 Td[(D=2;however,intheD=4limittheinniteseriesmultiplyingthisextratermwillcauseittovanishtoo.So,intheendtherearereallyonlyfourcombinationsthatneedtobecalculated.Oneothernotationalcommentisneededbeforethecalculationcancontinue.Allfourpossiblecombinationsofthepartsofthepropagatorswillresultintermsofthesameform,namely #@@0(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4ix@@0(aa0iy),(5)whereix,yisactuallyadifferenceofpropagatorsforthecasesx=A,Cory=C.Sinceallofthetermswilllookalmostidentical,withonlythepropagatorsandindiceschanging,wecanvastlysimplifyreportingourresultsbyadoptinganotationofonlywritingoutthederivativesandtheirassociatedindices.Thesearetheonlypartsneededbecauseitistheindicesthatwillmakeuptheprimarydifferenceineachterm,andthespecicpropagatorcombinationcanbedenotedinthesubscriptofthevacuumpolarization.Inthisnotation( 5 )wouldtakethesimpleform@@0(@@0).Wearenowreadytodiveintothecalculation.FirstwewillperformthesubstitutionsfortheBpartofthephotonpropagatorwiththeBpartofthegravitonpropagator.Allofthetermsforthisportionwilltaketheform #@@0(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4B@@0(aa0B).(5)Thefullresultforthissetofpropagatorpiecesis ih3ptiB,B(x;x0)=(D2)]TJ /F4 11.955 Tf 11.95 0 Td[(4D+2) (D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)2n)]TJ /F3 11.955 Tf 9.3 0 Td[(@@0(@@0)+@@0(@@0))]TJ /F3 11.955 Tf 11.96 0 Td[(@@0(@@0)+@@0(@@0)o+2n)]TJ /F4 11.955 Tf 9.3 0 Td[(2@@0(@@0))]TJ /F3 11.955 Tf 11.95 0 Td[(@@0(@@0)+2@@0(@@0)+@@0(@@0))]TJ /F3 11.955 Tf 9.3 0 Td[(@@0(@@0)+@@0(@@0)o. (5) 109

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NextwecomputethevacuumpolarizationfromtheBpartofthegravitonpropagatorandtheCpartofthephotonpropagator.Thesetermswillalltakethegenericform #@@0(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(4B@@0(aa0(B)]TJ /F5 11.955 Tf 11.95 0 Td[(C)).(5)Thefullresultis ih3ptiB,C(x;x0)=2 (D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)2n00@@0(@@0))]TJ /F3 11.955 Tf 11.95 0 Td[(0@0@0(@@0)+@0@00(@@0))]TJ /F3 11.955 Tf 9.3 0 Td[(0@@00(@@0)o+2n)]TJ /F3 11.955 Tf 9.3 0 Td[(@0@00(@@0))]TJ /F3 11.955 Tf 11.96 0 Td[(@0@00(@@0)+@@00(@@00)+@@0(@@0)+@0@0(@0@0)+0@@00(@@0)+0@@00(@@0))]TJ /F3 11.955 Tf 11.96 0 Td[(@@00(@@00))]TJ /F3 11.955 Tf 9.29 0 Td[(@@0(@@0))]TJ /F3 11.955 Tf 11.96 0 Td[(0@@0(@0@0))]TJ /F3 11.955 Tf 11.95 0 Td[(00@@0(@@0))]TJ /F3 11.955 Tf 9.29 0 Td[(00@@0(@@0)+@@0(@@0)+0@0@0(@@0))]TJ /F3 11.955 Tf 9.29 0 Td[(0@@0(@@00)+0@@0(@@00)+0@@0(@0@0))]TJ /F3 11.955 Tf 11.96 0 Td[(@@0(@@0))]TJ /F3 11.955 Tf 9.29 0 Td[(@0@0(@0@0)+0@0@0(@@0)o. (5) WhenweconsidertheApartofthegravitonpropagatorandtheBpartofthephotonpropagatorthetermsallhavetheform #@@0(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(4(iA)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@@0(aa0B),(5) 110

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andthefullresultis ih3ptiA,B(x;x0)=(D)]TJ /F4 11.955 Tf 11.95 0 Td[(1)(D)]TJ /F4 11.955 Tf 11.96 0 Td[(2) (D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)2n)]TJ /F4 11.955 Tf 9.91 0 Td[(@@0(@@0)+@@0(@@0))]TJ /F4 11.955 Tf 10.55 2.66 Td[(@@0(@@0)+@@0(@@0)o+2n)]TJ /F3 11.955 Tf 9.3 0 Td[(@@0(@@0))]TJ /F3 11.955 Tf 11.95 0 Td[(@@0(@@0)+@@0(@@0)+@@0(@@0))]TJ /F3 11.955 Tf 9.3 0 Td[(@@0(@@0))]TJ /F3 11.955 Tf 11.95 0 Td[(@@0(@@0)+@@0(@@0)+@@0(@@0)o+2 (D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)2n@@0(@@0))]TJ /F3 11.955 Tf 11.95 0 Td[(@@0(@@0))]TJ /F3 11.955 Tf 11.96 0 Td[(@@0(@@0)+@@0(@@0)+@@0(@@0))]TJ /F3 11.955 Tf 11.95 0 Td[(@@0(@@0))]TJ /F3 11.955 Tf 11.96 0 Td[(@@0(@@0)+@@0(@@0)+@@0(@@0))]TJ /F4 11.955 Tf 13.21 2.65 Td[(@@0(@@0)+@@0(@@0))]TJ /F3 11.955 Tf 11.96 0 Td[(@@0(@@0)o+(D)]TJ /F4 11.955 Tf 11.95 0 Td[(1) (D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)2n@@0(@@0))]TJ /F3 11.955 Tf 11.96 0 Td[(@@0(@@0)+@@0(@@0))]TJ /F3 11.955 Tf 11.96 0 Td[(@@0(@@0)+@@0(@@0)+@@0(@@0))]TJ /F4 11.955 Tf 13.21 2.66 Td[(@@0(@@0))]TJ /F3 11.955 Tf 11.95 0 Td[(@@0(@@0))]TJ /F4 11.955 Tf 10.55 2.66 Td[(@@0(@@0)+@@0(@@0)]TJ /F4 11.955 Tf 13.21 2.66 Td[(@@0(@@0)+@@0(@@0)o. (5) ThelastcaseistheCpartofthegravitonpropagatorandtheBpartofthephotonpropagator.Thesetermstaketheform #@@0(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(4(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@@0(aa0B),(5) 111

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andthefullresultis ih3ptiC,B(x;x0)=2(D)]TJ /F4 11.955 Tf 11.96 0 Td[(1) (D)]TJ /F4 11.955 Tf 11.96 0 Td[(2)2n)]TJ /F3 11.955 Tf 9.3 0 Td[(0@@00(@@0)+00@@0(@@0)+@0@00(@@0))]TJ /F3 11.955 Tf 9.3 0 Td[(0@0@0(@@0)o+2(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3) (D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)2n)]TJ /F3 11.955 Tf 9.29 0 Td[(00@@0(@0@00)+0@0@0(@0@00))]TJ /F3 11.955 Tf 11.95 0 Td[(@0@00(@0@00)+0@@00(@0@00)o+2 (D)]TJ /F4 11.955 Tf 11.96 0 Td[(2)(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)2n)]TJ /F3 11.955 Tf 9.3 0 Td[(@@0(@@0)+@@0(@@0))]TJ /F3 11.955 Tf 11.95 0 Td[(@@0(@@0)+@@0(@@0))]TJ /F3 11.955 Tf 9.3 0 Td[(@@0(@@0)+@@0(@@0)+@@0(@@0))]TJ /F4 11.955 Tf 13.2 2.65 Td[(@@0(@@0))]TJ /F3 11.955 Tf 9.3 0 Td[(@@0(@@0)+@@0(@@0)+@@0(@@0))]TJ /F3 11.955 Tf 11.95 0 Td[(@@0(@@0))]TJ /F3 11.955 Tf 9.29 0 Td[(@@0(@@0)+@@0(@@0)+@@0(@@0))]TJ /F4 11.955 Tf 13.21 2.66 Td[(@@0(@@0))]TJ /F3 11.955 Tf 9.29 0 Td[(@@0(@@0)+@@0(@@0)+@@0(@@0)+@@0(@@0))]TJ /F4 11.955 Tf 10.55 2.66 Td[(@@0(@@0))]TJ /F4 11.955 Tf 12.57 0 Td[(@@0(@@0))]TJ /F3 11.955 Tf 11.95 0 Td[(@@0(@@0)+@@0(@@0)+@@0(@@0)+@@0(@@0))]TJ /F4 11.955 Tf 12.57 0 Td[(@@0(@@0))]TJ /F4 11.955 Tf 13.21 2.66 Td[(@@0(@@0)o+2 (D)]TJ /F4 11.955 Tf 11.96 0 Td[(2)2n)]TJ /F3 11.955 Tf 9.3 0 Td[(0@@0(@@00))]TJ /F3 11.955 Tf 11.95 0 Td[(0@@0(@0@0)+@@00(@@00)+@0@0(@0@0)+0@@0(@@00)+0@@0(@0@0))]TJ /F3 11.955 Tf 11.95 0 Td[(@@00(@@00))]TJ /F3 11.955 Tf 11.96 0 Td[(@0@0(@0@0))]TJ /F3 11.955 Tf 9.29 0 Td[(0@@0(@0@0))]TJ /F3 11.955 Tf 11.96 0 Td[(0@@0(@@00)+0@@00(@@0)+0@@0(@@00)+0@@0(@0@0)+0@0@0(@@0)+@0@0(@0@0)+0@@0(@@00))]TJ /F3 11.955 Tf 9.29 0 Td[(@0@00(@@0))]TJ /F3 11.955 Tf 11.96 0 Td[(0@@0(@@00))]TJ /F4 11.955 Tf 12.57 0 Td[(@0@0(@0@0))]TJ /F3 11.955 Tf 11.96 0 Td[(00@@0(@@0))]TJ /F3 11.955 Tf 9.29 0 Td[(@0@0(@0@0))]TJ /F3 11.955 Tf 11.96 0 Td[(@@00(@@00)+0@0@0(@@0)+@@00(@@00)+@0@0(@0@0)+0@@00(@@0)+0@@0(@0@0)+@@00(@@00))]TJ /F3 11.955 Tf 9.29 0 Td[(00@@0(@@0))]TJ /F4 11.955 Tf 12.57 0 Td[(@@00(@@00))]TJ /F3 11.955 Tf 11.95 0 Td[(0@@0(@0@0))]TJ /F3 11.955 Tf 9.29 0 Td[(@0@00(@@0)o. (5) 112

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5.3.3Findingthe3-PointStructureFunctionsRecallingtheprocedureoutlinedinsection 5.2.3 wewillnowndthe3-pointcontributionstoFandGinmuchthesameway.Theprocesswillbealittlemorelaborintensivesincewenolongerhaveadeltafunctiononeachtermtoassistinpullingoutinternalderivatives;thereisalsotheaddedcomplicationofhavingtwointernalderivativesinsteadofone.Thesechangeswillbeaccountedforasfollows:Firstwecanstillusethesameidentities( 5 )and( 5 )asoursetofequationsforndingF3andG4.Nextwenoticethatintheabsenceofadeltafunctionwecannolongersimplychange@!)]TJ /F3 11.955 Tf 24.6 0 Td[(@0,insteadwehavetocarefullyaccountforfeeddowntermsthatwillarisefromextractingderivatives.Allofthe3-pointtermstaketheformof( 5 ),thusthereareonlyfourpossiblecombinationsofinternalderivativeswewillencounter,theyare@i@0j,@0@0i,@i@00,and@0@00.Extractingthesesetsofderivativeswillalwaysresultinthesamefeeddowntermsregardlessofthepropagatorsinvolved.AslightmodicationisneededfortheA-typepropagator,butingeneralextractingthesederivativesresultsinthefollowingidentities (aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(4ix@0@0i(aa0iy)=@0@0i(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3I2[ixiy]+@0inH(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3a)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(I[ixi_y])]TJ /F4 11.955 Tf 9.3 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)I2[ixiy]+I2[i_xi_y]o, (5) (aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4ix@i@00(aa0iy)=@i@00(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3I2[ixiy]+@inH(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3a0)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(I[ixi_y])]TJ /F4 11.955 Tf 9.3 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)I2[ixi00y]+I2[i_xi_y]o, (5) (aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4ix@i@0j(aa0iy)=@i@0j(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3I2[ixiy])]TJ /F4 11.955 Tf 11.96 0 Td[(2ijH2(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(2I[i_xi_y], (5) 113

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(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4ix@0@00(aa0iy)=)]TJ /F4 11.955 Tf 9.3 0 Td[(2(D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)H2(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(2I[ixi_y]+@0@00(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3I2[ixiy]+(@0a0+@00a)nH(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3I[ixi_y])]TJ /F4 11.955 Tf 11.95 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)I2[ixiy]o+1 2(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3r2I3[i_xi_y]+H2(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(2nixiy+(2)]TJ /F5 11.955 Tf 11.95 0 Td[(y)I[i_xi_y])]TJ /F4 11.955 Tf 9.3 0 Td[((D)]TJ /F4 11.955 Tf 11.95 0 Td[(1)I2[i_xi_y]+(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)2I2[ixiy]o, (5) whereIrepresentsandanindeniteintegralwithrespecttoy,andadotoverthepropagatorsrepresentsaderivativewithrespecttoy1.TheA-typepropagatorisuniqueinthatitisnotjustafunctionofy,butalsocontainsadeSitterbreakingpiece.Itcanberewrittenintheform iA(x;x0)=A(y)+ku,(5)wherek=HD)]TJ /F15 5.978 Tf 5.76 0 Td[(2 (4)D=2\(D)]TJ /F10 7.97 Tf 6.59 0 Td[(1) \(D=2).Inthisformwecanseethatidentities( 5 5 )willmissthefeeddowntermsthatarisewhenoneoftheinternalderivativesactonthesecondtermin( 5 ).Toaccountforthisthefollowingtermsneedtobeinsertedineachoftheaboveidentitieswhenix=(iA)]TJ /F5 11.955 Tf 11.96 0 Td[(B) (aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(4ku@0@0i(aa0B)!)]TJ /F3 11.955 Tf 24.57 0 Td[(@0iHa(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3kB (5) (aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4ku@i@00(aa0B)!)]TJ /F3 11.955 Tf 24.58 0 Td[(@iHa0(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3kB (5) (aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4ku@i@0j(aa0B)!0 (5) 1Onemightworryaboutlocaldeltafunctioncontributionsfromtermsoftheform@0@00B,butactuallythesecontributionsarefullyaccountedforindeltafunctionsarisingfromfeeddowntermsoftheform@0@00(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3I2[ixiy]. 114

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(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4ku@0@00(aa0B)!)]TJ /F4 11.955 Tf 24.58 0 Td[((@0a0+@00a)H(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3kB+2H2(D)]TJ /F4 11.955 Tf 11.96 0 Td[(4)(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(2kB (5) Wearenowreadytoapplythecontractionsinidentities( 5 )and( 5 )andthenusethesubstitutions( 5 5 ),makinguseof( 5 5 )whereappropriate,tondF3andG3.Again,wewillworkthroughanexampleforthereaderandthenstatethenalresultforalloftheterms.Weconsideralltermswiththecoefcient(D2)]TJ /F10 7.97 Tf 6.58 0 Td[(4D+2) (D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)2in( 5 ),sincetheseformthesmallestsetoftransverseterms.Performingtherstcontractionwith00resultsin ih003pt,exiB,B=(D2)]TJ /F4 11.955 Tf 11.95 0 Td[(4D+2) (D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)2n@@0(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(4B@@0(aa0B)+@0@0(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4B@0@0(aa0B))]TJ /F3 11.955 Tf 11.95 0 Td[(@@0(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4B@0@00(aa0B)+@@00(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(4B@@00(aa0B)o=(D2)]TJ /F4 11.955 Tf 11.95 0 Td[(4D+2) (D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)2n@i@0j(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4B@i@0j(aa0B)+r2(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4B@0@00(aa0B)o (5) wherewehavetakenthe3+1decompositioningoingfromtherstlinetothesecond.Nowwecanmaketheappropriatesubstitutionsusingidentities( 5 5 )andcomparewith( 5 )tond F3,ex(x;x0)=(D2)]TJ /F4 11.955 Tf 11.96 0 Td[(4D+2) (D)]TJ /F4 11.955 Tf 11.96 0 Td[(2)2n)]TJ /F4 11.955 Tf 9.3 0 Td[(2(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)H2(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(2I[B_B]+(@0@00+r2)(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3I[BB]+(@0a0+@00a)H(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(I[B_B])]TJ /F4 11.955 Tf 11.96 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)I2[BB]+H2(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(2B2+(4)]TJ /F5 11.955 Tf 11.95 0 Td[(y)I[_B2])]TJ /F4 11.955 Tf 11.95 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)I2[_B2]+(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)2I2[BB]+1 2(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3r2I3[_B2]o. (5) 115

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TondthecompanionG3,exwecontractij(i6=j)withthesamesetoftermsin( 5 ) ihij3pt,exiB,B=(D2)]TJ /F4 11.955 Tf 11.96 0 Td[(4D+2) (D)]TJ /F4 11.955 Tf 11.96 0 Td[(2)2n0+@j@0(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(4B@i@0(aa0B))]TJ /F3 11.955 Tf 9.3 0 Td[(@@0(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4B@i@0j(aa0B)+@@0i(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(4B@@0j(aa0B)o=(D2)]TJ /F4 11.955 Tf 11.95 0 Td[(4D+2) (D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)2n)]TJ /F3 11.955 Tf 9.3 0 Td[(@j@00(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4B@i@00(aa0B)+@j@0k(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4B@i@0k(aa0B)+@0@00(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4B@i@0j(aa0B)+r2(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4B@i@0j(aa0B))]TJ /F3 11.955 Tf 9.3 0 Td[(@0@0i(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(4B@0@0j(aa0B)+@k@0i(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(4B@k@0j(aa0B), (5) whereagainwehavetakenthe3+1decompositioningoingfromtherstlinetothesecond.Applyingidentities( 5 5 ),( 5 )canberewrittenintheform ihij3pt,exiB,B=)]TJ /F4 11.955 Tf 10.49 8.09 Td[((D2)]TJ /F4 11.955 Tf 11.95 0 Td[(4D+2) (D)]TJ /F4 11.955 Tf 11.96 0 Td[(2)2@0i@jn(r2)]TJ /F3 11.955 Tf 11.96 0 Td[(@0@00)]TJ /F3 11.955 Tf 11.95 0 Td[(@002)]TJ /F3 11.955 Tf 11.96 0 Td[(@02)(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3I2[BB])]TJ /F3 11.955 Tf 9.3 0 Td[(@00H(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3a0)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(I[B_B])]TJ /F4 11.955 Tf 11.96 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)I2[BB]+I2[_B])]TJ /F3 11.955 Tf 9.3 0 Td[(@0H(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3a)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(I[B_B])]TJ /F4 11.955 Tf 11.95 0 Td[((D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)I2[BB]+I2[_B]+4H2(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(2I[_B2]o. (5) SubstitutingF3,exinto( 5 )wend G3,ex(x;x0)=)]TJ /F4 11.955 Tf 10.5 8.09 Td[((D2)]TJ /F4 11.955 Tf 11.95 0 Td[(4D+2) (D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)2H2(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(2)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(B2+(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)2I2[BB])]TJ /F4 11.955 Tf 9.3 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)I2[_B2])]TJ /F4 11.955 Tf 11.95 0 Td[(2(D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)I[B_B])]TJ /F5 11.955 Tf 11.95 0 Td[(yI[_B2]+1 2(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3r2I3[_B2]+(@0a0+@00a)H(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(I[B_B])]TJ /F4 11.955 Tf 11.95 0 Td[((D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)I2[BB]+@0H(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3a)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(I[B_B])]TJ /F4 11.955 Tf 11.96 0 Td[((D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)I2[BB]+I2[_B2]+@00H(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3a0)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(I[B_B])]TJ /F4 11.955 Tf 11.95 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)I2[BB]+I2[_B2]+(@0+@00)2(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3I2[BB]. (5) 116

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Thisconcludesthe3-pointexample;itshouldbeclearhowtoproceedwiththerestofthe3-pointcontribution.Wenowpresenttheresultsforthescalarstructurefunctionsorganizedaccordingtopropagatorcombination.From( 5 )wend FBB(x;x0)=(D2)]TJ /F5 11.955 Tf 11.96 0 Td[(D)]TJ /F4 11.955 Tf 11.95 0 Td[(4) (D)]TJ /F4 11.955 Tf 11.96 0 Td[(2)2nH2(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(2B2+(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)2I2[BB])]TJ /F4 11.955 Tf 11.95 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)I2[_B2])]TJ /F4 11.955 Tf 9.3 0 Td[(2(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)I[B_B]+1 2(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3r2I3[_B2]+)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(r2+@0@00(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3I2[BB]+H(@0a0+@00a)(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(I[B_B])]TJ /F4 11.955 Tf 11.95 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)I2[BB]o+2H2 (D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(24D(2D)]TJ /F4 11.955 Tf 11.95 0 Td[(5))]TJ /F4 11.955 Tf 11.95 0 Td[((D2)]TJ /F5 11.955 Tf 11.96 0 Td[(D)]TJ /F4 11.955 Tf 11.95 0 Td[(4)yI[B2], (5) GBB(x;x0)=)]TJ /F3 11.955 Tf 24.89 8.09 Td[(2 (D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)nD(D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)H2(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(2B2+(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)2I2[BB])]TJ /F4 11.955 Tf 11.95 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)I2[_B2])]TJ /F4 11.955 Tf 9.3 0 Td[(2(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)I[B_B])]TJ /F5 11.955 Tf 11.95 0 Td[(yI[_B2]+1 2D(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3r2I3[_B2])]TJ /F5 11.955 Tf 9.3 0 Td[(H(@0a0+@00a)(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(I2[_B2])]TJ /F4 11.955 Tf 11.96 0 Td[((D2)]TJ /F4 11.955 Tf 11.96 0 Td[(4D+2)(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)I2[BB]+(D2)]TJ /F4 11.955 Tf 11.96 0 Td[(4D+2)I[B_B])]TJ /F4 11.955 Tf 9.3 0 Td[((D2)]TJ /F4 11.955 Tf 11.95 0 Td[(4D+2)H@0a(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(31 (D)]TJ /F4 11.955 Tf 11.96 0 Td[(2)I2[_B2]+(D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)I2[BB])]TJ /F5 11.955 Tf 11.95 0 Td[(I[B_B])]TJ /F4 11.955 Tf 9.3 0 Td[((D2)]TJ /F4 11.955 Tf 11.95 0 Td[(4D+2)H@00a0(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(31 (D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)I2[_B2]+(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)I2[BB])]TJ /F5 11.955 Tf 11.96 0 Td[(I[B_B]+(D2)]TJ /F4 11.955 Tf 11.95 0 Td[(4D+2)(@0+@00)2(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3I2[BB]o. (5) From( 5 )wend FBC(x;x0)=)]TJ /F3 11.955 Tf 24.89 8.09 Td[(2 (D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)n2(D2)]TJ /F5 11.955 Tf 11.95 0 Td[(D)]TJ /F4 11.955 Tf 11.96 0 Td[(4)H2(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(2I[_B(_B)]TJ /F4 11.955 Tf 14.93 2.66 Td[(_C)]+(3D)]TJ /F4 11.955 Tf 11.95 0 Td[(8)(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3r2I2[B(B)]TJ /F4 11.955 Tf 13.53 2.65 Td[(C)]o, (5) GBC(x;x0)=22(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3) (D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)nDH2(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(2I[_B(_B)]TJ /F4 11.955 Tf 14.93 2.65 Td[(_C)]+)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(r2+@0@00(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3I2[B(B)]TJ /F4 11.955 Tf 13.53 2.65 Td[(C)]o. (5) 117

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From( 5 )wehave FAB(x;x0)=2nH2(D)]TJ /F4 11.955 Tf 11.95 0 Td[(1)(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(2B(iA)]TJ /F5 11.955 Tf 11.96 0 Td[(B)+(D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)2I2[(iA)]TJ /F5 11.955 Tf 11.96 0 Td[(B)B])]TJ /F4 11.955 Tf 9.3 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)I2[(i_A)]TJ /F4 11.955 Tf 14.81 2.65 Td[(_B)_B])]TJ /F4 11.955 Tf 11.96 0 Td[(2(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)I[(iA)]TJ /F5 11.955 Tf 11.96 0 Td[(B)_B]+(4)]TJ /F5 11.955 Tf 11.96 0 Td[(y)I[(i_A)]TJ /F4 11.955 Tf 14.81 2.65 Td[(_B)_B]+1 2(D)]TJ /F4 11.955 Tf 11.95 0 Td[(1)(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3r2I3[(i_A)]TJ /F4 11.955 Tf 14.81 2.66 Td[(_B)_B]+)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(r2+(D)]TJ /F4 11.955 Tf 11.95 0 Td[(1)@0@00(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3I2[(iA)]TJ /F5 11.955 Tf 11.95 0 Td[(B)B]+(D)]TJ /F4 11.955 Tf 11.95 0 Td[(1)H(@0a0+@00a)(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(I[(iA)]TJ /F5 11.955 Tf 11.95 0 Td[(B)_B])]TJ /F4 11.955 Tf 11.95 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)I2[(iA)]TJ /F5 11.955 Tf 11.96 0 Td[(B)B]o+(D)]TJ /F4 11.955 Tf 11.95 0 Td[(1)k2n2(D)]TJ /F4 11.955 Tf 11.95 0 Td[(4)H2(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(2B)]TJ /F5 11.955 Tf 11.96 0 Td[(H(@0a0+@00a)(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3Bo, (5) GAB(x;x0)=)]TJ /F3 11.955 Tf 9.3 0 Td[(2H2(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(2(D)]TJ /F4 11.955 Tf 11.95 0 Td[(1)B(iA)]TJ /F5 11.955 Tf 11.95 0 Td[(B)+(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)2I2[(iA)]TJ /F5 11.955 Tf 11.95 0 Td[(B)B])]TJ /F4 11.955 Tf 9.29 0 Td[((D)]TJ /F4 11.955 Tf 11.95 0 Td[(1)I2[(i_A)]TJ /F4 11.955 Tf 14.81 2.66 Td[(_B)_B])]TJ /F4 11.955 Tf 11.96 0 Td[(2(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)I[(iA)]TJ /F5 11.955 Tf 11.96 0 Td[(B)_B])]TJ /F9 11.955 Tf 11.29 16.85 Td[(4(D)]TJ /F4 11.955 Tf 11.95 0 Td[(4) (D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)+yI[(i_A)]TJ /F4 11.955 Tf 14.8 2.65 Td[(_B)_B]+)]TJ /F4 11.955 Tf 10.49 8.09 Td[((D2)]TJ /F4 11.955 Tf 11.96 0 Td[(4D+1) (D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)r2+(D)]TJ /F4 11.955 Tf 11.96 0 Td[(2)(@20+@002)+(D)]TJ /F4 11.955 Tf 11.95 0 Td[(1)@0@00(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3I2[(iA)]TJ /F5 11.955 Tf 11.95 0 Td[(B)B]+1 2(D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3r2I3[(i_A)]TJ /F4 11.955 Tf 14.81 2.66 Td[(_B)_B]+H(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)@00a0(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(I2[(i_A)]TJ /F4 11.955 Tf 14.81 2.66 Td[(_B)_B]+I[(iA)]TJ /F5 11.955 Tf 11.96 0 Td[(B)_B])]TJ /F4 11.955 Tf 9.29 0 Td[((D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)I2[(iA)]TJ /F5 11.955 Tf 11.96 0 Td[(B)B]+H(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)@0a(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(I2[(i_A)]TJ /F4 11.955 Tf 14.81 2.66 Td[(_B)_B]+I[(iA)]TJ /F5 11.955 Tf 11.95 0 Td[(B)_B])]TJ /F4 11.955 Tf 9.29 0 Td[((D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)I2[(iA)]TJ /F5 11.955 Tf 11.96 0 Td[(B)B]+H(D)]TJ /F4 11.955 Tf 11.95 0 Td[(1)(@0a0+@00a)(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(I[(iA)]TJ /F5 11.955 Tf 11.96 0 Td[(B)_B])]TJ /F4 11.955 Tf 11.96 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)I2[(iA)]TJ /F5 11.955 Tf 11.95 0 Td[(B)B]o)]TJ /F5 11.955 Tf 9.29 0 Td[(k2n2(D)]TJ /F4 11.955 Tf 11.96 0 Td[(4)(D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)H2(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(2B)]TJ /F4 11.955 Tf 11.95 0 Td[((D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)(@0a0+@00a)(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3B)]TJ /F5 11.955 Tf 9.29 0 Td[(H(D)]TJ /F4 11.955 Tf 11.95 0 Td[(1)@00a0(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3B)]TJ /F5 11.955 Tf 11.95 0 Td[(H(D)]TJ /F4 11.955 Tf 11.95 0 Td[(1)@0a(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3Bo. (5) 118

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Andlastly,from( 5 )wehave FCB(x;x0)=2(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3) (D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)2nH2(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(2B(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)+(D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)2I2[(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B)B])]TJ /F4 11.955 Tf 9.29 0 Td[((D)]TJ /F4 11.955 Tf 11.95 0 Td[(1)I2[(_C)]TJ /F4 11.955 Tf 14.81 2.65 Td[(_B)_B])]TJ /F4 11.955 Tf 11.96 0 Td[(2(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)I[(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B)_B]+(4)]TJ /F5 11.955 Tf 11.96 0 Td[(y)I[(_C)]TJ /F4 11.955 Tf 14.81 2.66 Td[(_B)_B]+1 2(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3r2I3[(_C)]TJ /F4 11.955 Tf 14.81 2.66 Td[(_B)_B]+H(@0a0+@00a)(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)]TJ /F5 11.955 Tf 5.47 -9.69 Td[(I[(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)_B])]TJ /F4 11.955 Tf 11.95 0 Td[((D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)I2[(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B)B]+(r2+@0@00)(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3I2[(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)B]o, (5) GCB(x;x0)=22 (D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)nH2(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)]TJ /F4 11.955 Tf 9.3 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)B(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B))]TJ /F4 11.955 Tf 11.96 0 Td[((D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)2I2[(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)B]+(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)(D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)I2[(_C)]TJ /F4 11.955 Tf 14.81 2.66 Td[(_B)_B]+2(D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)2I[(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B)_B])]TJ /F9 11.955 Tf 12.48 16.85 Td[(4(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)(D)]TJ /F4 11.955 Tf 11.96 0 Td[(4) (D)]TJ /F4 11.955 Tf 11.95 0 Td[(3))]TJ /F4 11.955 Tf 11.95 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)yI[(_C)]TJ /F4 11.955 Tf 14.81 2.65 Td[(_B)_B])]TJ /F9 11.955 Tf 11.29 16.86 Td[((D)]TJ /F4 11.955 Tf 11.95 0 Td[(4)(D)]TJ /F4 11.955 Tf 11.96 0 Td[(2) (D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)r2+2(D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)@0@00)]TJ /F3 11.955 Tf 11.96 0 Td[(@20)]TJ /F3 11.955 Tf 11.95 0 Td[(@002(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3I2[(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)B])]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 2(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(3r2I3[(_C)]TJ /F4 11.955 Tf 14.81 2.66 Td[(_B)_B]+H(D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)(@0a0+@00a)(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)]TJ /F4 11.955 Tf 5.48 -9.68 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)I2[(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B)B])]TJ /F5 11.955 Tf 11.96 0 Td[(I[(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)_B])]TJ /F5 11.955 Tf 9.3 0 Td[(H@0a(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)]TJ /F4 11.955 Tf 5.48 -9.68 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)I2[(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)B])]TJ /F5 11.955 Tf 11.95 0 Td[(I2[(_C)]TJ /F4 11.955 Tf 14.81 2.66 Td[(_B)_B])]TJ /F5 11.955 Tf 9.3 0 Td[(I[(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B)_B])]TJ /F5 11.955 Tf 9.3 0 Td[(H@00a0(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)]TJ /F4 11.955 Tf 5.47 -9.68 Td[((D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)I2[(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B)B])]TJ /F5 11.955 Tf 11.96 0 Td[(I2[(_C)]TJ /F4 11.955 Tf 14.81 2.65 Td[(_B)_B])]TJ /F5 11.955 Tf 9.3 0 Td[(I[(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B)_B]o. (5) BecauseofthedifferentcombinationsofpropagatorsappearingineachsetofFandGitwasnotusefulorenlighteningtocombinethemhere.Instead,oncetheyarerenormalizedinthenextsectiontheyarecombinedeasily. 5.4RenormalizationThissectionisdevotedtorenormalizingthe3and4-pointcontributionstoFandG.First,allofthe3-pointcontributionsmustbeputinthesameformsoastobeeasilycombined.Wecanthenlocalizetheultravioletdivergentpieces,andcombinethemwith 119

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the4-pointdivergences.Byreadingoffthecorrectcountertermcoefcentsweremovealldivergences,andareleftwiththefullyrenormalizedstructurefunctionsofthevacuumpolarization. 5.4.1ConvertingtoFunctionsofxTheexpressionswederivedinsection 5.3 containmanyindeniteintegralsofproductsofderivativesofthethreepropagatorfunctionsiA(y),B(y)andC(y).Eachoftheseproductsconsistsofafewpowersofy(x;x0)whicharesingularatcoincidence(x0=x)andwhosecoefcientsarenonzeroforD=4,plusaninniteseriesoflessandlesssingularpowersofywhosecoefcientsvanishforD=4.Becausethevacuumpolarizationisusedinsidethe4-dimensionalintegralofthequantum-correctedMaxwellequation( 5 ),theonlytermswhichrequiredimensionalregularizationarethosewhichareatleastassingularas1=y2.AnylesssingulartermcanbeevaluatedforD=4,atwhichpointmostofthetediousinniteseriescontributionsvanish.Recallingaswellthaty(x;x0)=H2aa0x2,wecanmakethefollowingsimplications:ForFBBandGBBweusetheidentities B2!)]TJ /F10 7.97 Tf 6.77 4.34 Td[(2)]TJ /F6 7.97 Tf 6.67 -4.98 Td[(D 2)]TJ /F4 11.955 Tf 11.95 0 Td[(1 16D(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(2x2D)]TJ /F10 7.97 Tf 6.59 0 Td[(4 (5) I[B_B]!)]TJ /F10 7.97 Tf 6.77 4.34 Td[(2)]TJ /F6 7.97 Tf 6.67 -4.98 Td[(D 2)]TJ /F4 11.955 Tf 11.95 0 Td[(1 32D(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(2x2D)]TJ /F10 7.97 Tf 6.59 0 Td[(4 (5) I[_B2]!)]TJ /F4 11.955 Tf 32.41 8.09 Td[((D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)2 (D)]TJ /F4 11.955 Tf 11.96 0 Td[(1))]TJ /F10 7.97 Tf 6.78 4.34 Td[(2)]TJ /F6 7.97 Tf 6.67 -4.98 Td[(D 2)]TJ /F4 11.955 Tf 11.96 0 Td[(1 64D(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(1H2x2D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)]TJ /F4 11.955 Tf 10.49 8.09 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(2) 512D)]TJ /F10 7.97 Tf 6.77 4.34 Td[(2)]TJ /F6 7.97 Tf 6.67 -4.97 Td[(D 2)]TJ /F4 11.955 Tf 11.95 0 Td[(1(D)]TJ /F4 11.955 Tf 11.96 0 Td[(4) (aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(2x2D)]TJ /F10 7.97 Tf 6.58 0 Td[(4 (5) I2[_B2]!(D)]TJ /F4 11.955 Tf 11.96 0 Td[(2) (D)]TJ /F4 11.955 Tf 11.96 0 Td[(1))]TJ /F10 7.97 Tf 6.78 4.34 Td[(2)]TJ /F6 7.97 Tf 6.68 -4.97 Td[(D 2)]TJ /F4 11.955 Tf 11.96 0 Td[(1 64D(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(2x2D)]TJ /F10 7.97 Tf 6.58 0 Td[(4 (5) I3[_B2]!H2 9641 aa0x2 (5) I2[BB]!D (D)]TJ /F4 11.955 Tf 11.96 0 Td[(1))]TJ /F10 7.97 Tf 6.78 4.33 Td[(2)]TJ /F6 7.97 Tf 6.68 -4.98 Td[(D 2)]TJ /F4 11.955 Tf 11.96 0 Td[(1 64D(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(2x2D)]TJ /F10 7.97 Tf 6.58 0 Td[(4 (5) 120

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wherewenotethatforthissetofidentitiesonly( 5 )couldbeputintheD=4limit.ForFBCandGBCmostofthetermswillgotozeroandthereisonlyoneidentityneeded I[_B(_B)]TJ /F4 11.955 Tf 14.93 2.66 Td[(_C))]!)]TJ /F4 11.955 Tf 25.77 8.09 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(4) 128)]TJ /F10 7.97 Tf 6.77 4.34 Td[(2)]TJ /F6 7.97 Tf 6.67 -4.97 Td[(D 2)]TJ /F4 11.955 Tf 11.96 0 Td[(1 D(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(2x2D)]TJ /F10 7.97 Tf 6.58 0 Td[(4.(5)ForFABandGABwendthatwecantakeD=4inmostoftheterms B(iA)]TJ /F5 11.955 Tf 11.95 0 Td[(B)!)]TJ /F5 11.955 Tf 37.52 8.09 Td[(H2 3241 2+ln1 4H2x21 aa0x2 (5) I[(iA)]TJ /F5 11.955 Tf 11.96 0 Td[(B)_B]!)]TJ /F5 11.955 Tf 37.52 8.09 Td[(H2 3243 2+ln1 4H2x21 aa0x2 (5) I[(i_A)]TJ /F4 11.955 Tf 14.81 2.65 Td[(_B)_B]!)]TJ /F4 11.955 Tf 32.41 8.09 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(2) 2)]TJ /F10 7.97 Tf 6.77 4.34 Td[(2)]TJ /F6 7.97 Tf 6.67 -4.98 Td[(D 2)]TJ /F4 11.955 Tf 11.95 0 Td[(1 64D1 (aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(2x2D)]TJ /F10 7.97 Tf 6.59 0 Td[(4 (5) I2[(i_A)]TJ /F4 11.955 Tf 14.81 2.66 Td[(_B)_B]!H2 6241 aa0x2 (5) I3[(i_A)]TJ /F4 11.955 Tf 14.81 2.65 Td[(_B)_B]!H4 644ln1 4H2x2+u (5) I2[(iA)]TJ /F5 11.955 Tf 11.96 0 Td[(B)B]!)]TJ /F5 11.955 Tf 37.52 8.08 Td[(H2 3242+ln1 4H2x21 aa0x2 (5) kB!H2 3241 aa0x2 (5) Noticethat( 5 )istheonlyidentitythatneedstobekeptinDdimensions.LastlyforFCBandGCBthereisagainonlyonerelevantidentitysincemostofthetermswillbezero I[(_C)]TJ /F4 11.955 Tf 14.81 2.66 Td[(_B)_B]!(D)]TJ /F4 11.955 Tf 11.96 0 Td[(4) 128)]TJ /F10 7.97 Tf 6.77 4.34 Td[(2)]TJ /F6 7.97 Tf 6.67 -4.97 Td[(D 2)]TJ /F4 11.955 Tf 11.96 0 Td[(1 D(aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(2x2D)]TJ /F10 7.97 Tf 6.58 0 Td[(4.(5) 5.4.2FindingtheFiniteandDivergentPartsofFandGItiseasiesttorenormalizethestructurefunctionstermbyterm;thuswewilldemonstratetheprocedureforonetermandthereadercanextrapolatefromtheexampletoderivetherestoftheterms.First,itisusefultonotethenecessaryidentitiesforrenormalization.Thetermsproportionalto1=x2arealreadyintegrableanddonotneedtoberenormalized.Therearealsotermsproportional1=x2D)]TJ /F10 7.97 Tf 6.59 0 Td[(2and1=x2D)]TJ /F10 7.97 Tf 6.58 0 Td[(4;thesewillneedtoberenormalized.Weusedimensionalregulationtopartiallyintegrate 121

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thesetermsuntiltheyareintegrable 1 x2D)]TJ /F10 7.97 Tf 6.59 0 Td[(2=@4 4(D)]TJ /F4 11.955 Tf 11.96 0 Td[(2)2(D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)(D)]TJ /F4 11.955 Tf 11.96 0 Td[(4)1 x2D)]TJ /F10 7.97 Tf 6.59 0 Td[(6, (5) 1 x2D)]TJ /F10 7.97 Tf 6.59 0 Td[(4=@2 2(D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)(D)]TJ /F4 11.955 Tf 11.95 0 Td[(4)1 x2D)]TJ /F10 7.97 Tf 6.59 0 Td[(6, (5) butitisclearthattheseidentitiescontainadivergenceinthefactorsof1 (D)]TJ /F10 7.97 Tf 6.59 0 Td[(4).Wecanlocalizethedivergencebyaddingzerointheform @21 x2=i4D=2 )]TJ /F9 11.955 Tf 8.77 9.69 Td[()]TJ /F6 7.97 Tf 6.68 -4.98 Td[(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1D(x)]TJ /F5 11.955 Tf 11.96 0 Td[(x0).(5)Thenbyadding( 5 )tothedivergentpartsof( 5 )and( 5 )wend @2 (D)]TJ /F4 11.955 Tf 11.95 0 Td[(4)1 x2D)]TJ /F10 7.97 Tf 6.58 0 Td[(6=i4D=2 )]TJ /F9 11.955 Tf 8.77 9.68 Td[()]TJ /F6 7.97 Tf 6.67 -4.98 Td[(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1D)]TJ /F10 7.97 Tf 6.59 0 Td[(4D(x)]TJ /F5 11.955 Tf 11.96 0 Td[(x0) (D)]TJ /F4 11.955 Tf 11.96 0 Td[(4))]TJ /F3 11.955 Tf 27.5 8.09 Td[(@2 (D)]TJ /F4 11.955 Tf 11.96 0 Td[(4)1 x2D)]TJ /F10 7.97 Tf 6.58 0 Td[(6)]TJ /F3 11.955 Tf 17.13 8.09 Td[(D)]TJ /F10 7.97 Tf 6.59 0 Td[(4 xD)]TJ /F10 7.97 Tf 6.59 0 Td[(2=i4D=2 )]TJ /F9 11.955 Tf 8.77 9.68 Td[()]TJ /F6 7.97 Tf 6.67 -4.98 Td[(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1D)]TJ /F10 7.97 Tf 6.59 0 Td[(4D(x)]TJ /F5 11.955 Tf 11.96 0 Td[(x0) (D)]TJ /F4 11.955 Tf 11.96 0 Td[(4))]TJ /F3 11.955 Tf 13.15 8.09 Td[(@2 2ln(2x2) x2+O(D)]TJ /F4 11.955 Tf 11.95 0 Td[(4), (5) wherethefactorofisaddedfordimensionalconsistency.Wemaynowbeginourexample:WeconsidertheeighthterminFBB Fex(x;x0)=(D2)]TJ /F5 11.955 Tf 11.96 0 Td[(D)]TJ /F4 11.955 Tf 11.95 0 Td[(4) (D)]TJ /F4 11.955 Tf 11.96 0 Td[(2)2(r2+@0@00)(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(3I2[BB].(5)Applyingidentity( 5 )wend Fex(x;x0)=(D2)]TJ /F5 11.955 Tf 11.96 0 Td[(D)]TJ /F4 11.955 Tf 11.95 0 Td[(4) (D)]TJ /F4 11.955 Tf 11.96 0 Td[(2)2(r2+@0@00)"D 64D(D)]TJ /F4 11.955 Tf 11.95 0 Td[(1)\(D 2)]TJ /F4 11.955 Tf 11.96 0 Td[(1) aa0x2D)]TJ /F10 7.97 Tf 6.59 0 Td[(4#.(5) 122

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Wecannowuseidentities( 5 )and( 5 )tobreakthisintoitsdivergentandnitepieces.Wend Fex,div(x;x0)=D(D2)]TJ /F5 11.955 Tf 11.95 0 Td[(D)]TJ /F4 11.955 Tf 11.96 0 Td[(4) 8(D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)(D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)2 16D)]TJ /F9 11.955 Tf 8.76 16.86 Td[(D 2)]TJ /F4 11.955 Tf 9.97 0 Td[(11 a2@2+2H a@0i4D=2 (D)]TJ /F4 11.955 Tf 11.95 0 Td[(4)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4D(x)]TJ /F5 11.955 Tf 11.96 0 Td[(x0), (5) Fex,nite(x;x0)=)]TJ /F3 11.955 Tf 16.94 8.09 Td[(2 484)]TJ /F2 11.955 Tf 5.47 -9.69 Td[(r2+@0@001 aa0@2ln(2x2) x2, (5) whereinderivingthedivergentpartwehaveusedthefactthat,inconjunctionwiththedeltafunction,@0@001 aa0!2H a@0)]TJ /F10 7.97 Tf 15.46 4.71 Td[(1 a2@20.Renormalizingallothertermswillfollowaverysimilarprocedure.Onceallofthetermshavebeenrenormalizedandcombinedwenallyarriveatthefullresultforthestructurefunctionscomingfromthe3-pointdiagram,forconveniencewesplittheresultsintotheirniteanddivergentpieces F3,div(x;x0)=2)]TJ /F9 11.955 Tf 8.77 9.68 Td[()]TJ /F6 7.97 Tf 6.67 -4.98 Td[(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1D)]TJ /F10 7.97 Tf 6.59 0 Td[(4 32D=2(D)]TJ /F4 11.955 Tf 11.95 0 Td[(4)(D3)]TJ /F4 11.955 Tf 11.96 0 Td[(9D2+8D+24) (D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)H2+D (D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)1 a2@2+2H a@0iD(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0), (5) F3,nite(x;x0)=)]TJ /F3 11.955 Tf 27.45 8.09 Td[(2 1924aa0@4ln(2x2) x2+2H2 164(3 4@2ln(2x2) x2)]TJ /F4 11.955 Tf 13.16 8.09 Td[(1 2(@2)]TJ /F4 11.955 Tf 11.95 0 Td[(2@20)ln(1 4H2x2) x2+2@201 x2)]TJ /F4 11.955 Tf 9.3 0 Td[(42i4(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0)), (5) G3,div(x;x0)=)]TJ /F3 11.955 Tf 10.5 8.85 Td[(2H2)]TJ /F9 11.955 Tf 8.77 9.68 Td[()]TJ /F6 7.97 Tf 6.67 -4.98 Td[(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1D)]TJ /F10 7.97 Tf 6.58 0 Td[(4 32D=2(D5)]TJ /F4 11.955 Tf 11.95 0 Td[(12D4+50D3)]TJ /F4 11.955 Tf 11.95 0 Td[(77D2+20D+16) (D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)(D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)(D)]TJ /F4 11.955 Tf 11.96 0 Td[(4)iD(x)]TJ /F5 11.955 Tf 11.96 0 Td[(x0), (5) 123

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G3,nite(x;x0)=2H2 164(1 6@2ln(2x2) x2)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2@2ln(1 4H2x2) x2)]TJ /F4 11.955 Tf 9.3 0 Td[(112i4(x)]TJ /F5 11.955 Tf 11.96 0 Td[(x0)). (5) IntheatspacelimitH!0,a=a0=1thesestructurefunctionsrecovertheoldresult[ 22 ].Wearenowreadytocombinethe3-pointand4-pointcontributionstondthefullvacuumpolarization. 5.4.3FullResultWewillnowndtheappropriatecountertermcoefcientssuchastocancelalldivergences,andbeleftwiththefullrenormalizedvacuumpolarization.Recallthatwecanactuallyabsorballofthe4-pointcontribution,minusthetermsproportionaltoln(a),intothecounterterms.Thustondthefullcountertermcoefcientswesimplyhavetoaddthedivergentcoefcientsofthe3-pointcontributiontothethe4-pointcontribution.From( 5 ),( 5 ),and( 5 )wesee C=)]TJ /F3 11.955 Tf 10.93 8.09 Td[(2 16((D4)]TJ /F4 11.955 Tf 11.95 0 Td[(13D3+31D2)]TJ /F4 11.955 Tf 11.96 0 Td[(24) 8(D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)(D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)(D)]TJ /F4 11.955 Tf 11.95 0 Td[(4))]TJ /F9 11.955 Tf 8.77 9.68 Td[()]TJ /F6 7.97 Tf 6.68 -4.98 Td[(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1D)]TJ /F10 7.97 Tf 6.59 0 Td[(4 D=2)]TJ /F5 11.955 Tf 9.3 0 Td[(D(D)]TJ /F4 11.955 Tf 11.96 0 Td[(5)HD)]TJ /F10 7.97 Tf 6.59 0 Td[(4\(D)]TJ /F4 11.955 Tf 11.95 0 Td[(1) (4)D=2\(D 2)cot 2D+1 42), (5) from( 5 )wend C4=D (D)]TJ /F4 11.955 Tf 11.96 0 Td[(4)(D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)2)]TJ /F9 11.955 Tf 8.77 9.68 Td[()]TJ /F6 7.97 Tf 6.67 -4.97 Td[(D 2)]TJ /F4 11.955 Tf 11.95 0 Td[(1 128D=2D)]TJ /F10 7.97 Tf 6.59 0 Td[(4,(5)andfrom( 5 ),( 5 ),and( 5 )wend C=2 16((D4)]TJ /F4 11.955 Tf 11.96 0 Td[(7D3+11D2+3D)]TJ /F4 11.955 Tf 11.95 0 Td[(4) 8(D)]TJ /F4 11.955 Tf 11.96 0 Td[(1)(D)]TJ /F4 11.955 Tf 11.95 0 Td[(2)(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)(D)]TJ /F4 11.955 Tf 11.96 0 Td[(4))]TJ /F9 11.955 Tf 8.77 9.69 Td[()]TJ /F6 7.97 Tf 6.67 -4.98 Td[(D 2)]TJ /F4 11.955 Tf 9.96 0 Td[(1D)]TJ /F10 7.97 Tf 6.58 0 Td[(4 D=2)]TJ /F4 11.955 Tf 10.49 8.08 Td[((D2)]TJ /F4 11.955 Tf 11.96 0 Td[(4D+1) (D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)4HD)]TJ /F10 7.97 Tf 6.58 0 Td[(4\(D)]TJ /F4 11.955 Tf 11.95 0 Td[(1) (4)D=2\(D 2)cot 2D+1 42). (5) 124

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Usingthesecoefcientsinthecounterterms( 5 )and( 5 )wehavesufcientlyremovedalldivergencesandarriveatthenalrenormalizedformforthestructurefunctions F(x;x0)=2H2 82ln(a)i4(x)]TJ /F5 11.955 Tf 11.96 0 Td[(x0))]TJ /F3 11.955 Tf 30.11 8.08 Td[(2 1924aa0@4ln(2x2) x2+2 242(H a@0)]TJ /F4 11.955 Tf 11.95 0 Td[(ln(a)1 a2@2+2H a@0)i4(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0)+2H2 164(3 4@2ln(2x2) x2)]TJ /F4 11.955 Tf 13.16 8.09 Td[(1 2(@2)]TJ /F4 11.955 Tf 11.95 0 Td[(2@20)ln(1 4H2x2) x2+2@201 x2)]TJ /F4 11.955 Tf 9.3 0 Td[(42i4(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0)), (5) G(x;x0)=)]TJ /F3 11.955 Tf 10.49 8.08 Td[(2H2 62ln(a)i4(x)]TJ /F5 11.955 Tf 11.96 0 Td[(x0)+2H2 164(1 6@2ln(2x2) x2)]TJ /F4 11.955 Tf 13.16 8.09 Td[(1 2@2ln(1 4H2x2) x2)]TJ /F4 11.955 Tf 11.96 0 Td[(112i4(x)]TJ /F5 11.955 Tf 11.95 0 Td[(x0)). (5) Thesestructurefunctionscompriseourmainresult. 5.5HartreeApproximationWecangainaqualitativeunderstandingofwhattheresultwillshowbyapplyingtheHartreeapproximation[ 132 ].Thishasbeenusedpreviouslytostudytheeffectofchargedinationaryscalarsonphotons[ 83 ]and(especiallyrelevanttothecurrentproblem)theeffectofinationarygravitonsonmasslessfermions[ 77 ].IneachofthepreviouscasestheHartreeapproximationgavethecorrectspacetimedependenceoftheoneloopcorrectiontothemodefunctionsandthecorrectsignrelativetothetreeorderresult.TheHartreeapproximationto( 5 )consistsofreplacingtheHeisenbergoperatoreldequationbyitsexpectationvalueinfreegravitonvacuumandthenexpandingin 125

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termsofcoincidentgravitonpropagators, 0=@(p )]TJ /F5 11.955 Tf 9.3 0 Td[(g(x)g(x)g(x)F(x)), (5) )167(!0=@(aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4Dhp )]TJ /F9 11.955 Tf 9.36 .5 Td[(eg(x)eg(x)eg(x)hEF(x)), (5) =@naD)]TJ /F10 7.97 Tf 6.59 0 Td[(4Fo+2 2@(UaD)]TJ /F10 7.97 Tf 6.59 0 Td[(4ihi(x;x)F(x))+... (5) HereUdenotesthetensorfactorofthe2-graviton-2-photonvertex,giveninexpression( 5 ).Mostofthecoincidentgravitonpropagatorisadivergentconstant;theseculareffectsforwhichwearesearchingderivefromonlythelogarithmpartoftheA-typepropagator( 5 ).AtthispointwecanalsotakeD=4,2soourHartreeapproximationtotheeffectiveeldequation( 5 )is, 0=@F+2H2 82@(UhTAiln(a)F)+O(4).(5)RecallthattheA-typetensorfactorwasdenedinexpression( 5 ).Substituting( 5 )and( 5 )in( 5 )givesasimpleresult, 0=@F+2H2 42@(ln(a)h)]TJ /F4 11.955 Tf 9.3 0 Td[(3F+4F +4F )]TJ /F4 11.955 Tf 9.96 0 Td[(3F i)+O(4).(5)Hereabarredindexonanytensorindicatesthatits0-componenthasbeensuppressed,forexample,V V=V)]TJ /F3 11.955 Tf 12.82 0 Td[(0V0.Wecandistinguishinexpression( 5 )betweenthecasesof=0and=i.Theconstraintequationiseffectivelymultipliedby 2Onemightworryaboutfactorsofln(a)whichcouldarisewhenadivergentconstantmultipliesaD)]TJ /F10 7.97 Tf 6.59 0 Td[(4.However,wecanseefromexpressions( 5 )and( 5 )thattheverysamefactorofaD)]TJ /F10 7.97 Tf 6.59 0 Td[(4multipliesthecountertermswhichabsorbdivergencesfromthe4-pointdiagram.SotherecanbenosecularcontributionsfromthissourceandwemayaswelldropthedivergentconstantsandtakeD=4. 126

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asecularfactor, 0=@jFj0+2H2 42@j(ln(a)Fj0)+O(4), (5) =(1+2H2 42ln(a))@jFj0+O(4). (5) ThishasnoeffectondynamicalphotonsalthoughitwouldleadtoasecularscreeningofapointchargeofthesortreportedbyKitamotoandKitazawa[ 133 ]providedthereisnocompensatingsecularfactoronthechargedensity.Theequationofrelevancefordynamicalphotonsis=i, @Fi+2H2 42(@0hln(a)F0ii+@jh2ln(a)Fjii)+O(4).(5)Theoneloopcorrectiontotheeffectiveeldequationofcoursexesonlytheoneloopcorrectionstotheeldstrength.Wethereforeexpandinpowersof2, F=F(0)+2F(1)+4F(2)+...(5)Equations( 5 )and( 5 )implythefollowingrelationsfortheoneloopeldstrengths, @j2Fj0(1)=0, (5) @2Fi(1)=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(2H2 42(@0hln(a)F0i(0)i+@jh2ln(a)Fji(0)i). (5) WiththeU(1)Bianchiidentity,theleadingsecularbehavioris, 2F0i(1))166(!)]TJ /F3 11.955 Tf 39.72 8.09 Td[(2H2 42ln(a)F0i(0), (5) 2Fij(1))166(!)]TJ /F3 11.955 Tf 39.72 8.08 Td[(2H2 42ln(a) Hah@iF0j(0))]TJ /F3 11.955 Tf 9.96 0 Td[(@jF0i(0)i. (5) Weseethattheoneloopcorrectiontotheelectriceldstrengthofaphotontendstocancelitsclassicalvaluewhereastheoneloopcorrectiontothemagneticeldstrengthdiesoff. 127

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5.6DiscussionWehaveuseddimensionalregularizationtocomputetheoneloopquantumgravitationalcontributiontothevacuumpolarizationondeSitterbackground.Werstcalculatedthe4-pointcontributioninsection 5.2 ,andthenderivedthemuchmorecumbersome3-pointcontributioninsection 5.3 .Eachresultwasexpressedintheform( 5 )asthesumoftwotransverseprojectionoperatorsactingonstructurefunctions.Insub-section 5.1.5 therelevantBPHZcounterterms( 5 )werealsoreducedtothisform,resultinginexpressions( 5 5 ).Renormalizationwasimplementedinsection 5.4 togiveournalresults( 5 )and( 5 )forthestructurefunctionsF(x;x0)andG(x;x0).Ourultimategoalistostudyhowinationarygravitonsaffectelectrodynamicsusingthequantum-correctedMaxwellequation( 5 ).SpecializingtodeSitterinconformalcoordinates,substitutingourform( 5 )forrepresentingthevacuumpolarization,andpartiallyintegratingtheprimedderivatives,allowsustoexpressthequantum-correctedMaxwellequationsas, @F(x)+@Zd4x0(iF(x;x0)F(x0)+iG(x;x0)F(x0))=J.(5)(Recallthatabarredindexispurelyspatial.)Equation( 5 )canbeemployedthesamewayoneusestheclassicalMaxwellequationtostudydynamicalphotons(J=0solutions)andtheelectricandmagneticeldsinducedbystandardsources.Wehavealreadydonethisfortheone-loopvacuumpolarizationfromgravitonsonatspacebackground[ 22 ].Closelyrelatedstudieshavealsobeenmadeoftheeffectsthatinationaryscalarshaveondynamicalphotons[ 12 ]andonelectrodynamicforces[ 95 ].AsimpleestimateofwhatthequantumcorrectedMaxwell'sequationmightgivewasderivedinsection 5.5 bymakingtheHartreeapproximation[ 132 ]tolocalizetheeffectiveeldequation.Wendthattheoneloopelectriceldstrength( 5 )ofdynamicalphotonsexperiencesaseculargrowthwhichtendstocancelitsfreeeld 128

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value,whereastheoneloopcorrectiontothemagneticeldstrength( 5 )diesawaycomparedtoitsfreeeldvalue.Itisinterestingtonotethatthemagneticresponsetoinationaryscalarsalsoseemstobesubdominanttotheelectricrepsonse[ 95 ].Workingoutwhatthefullequation( 5 )givesfordynamicalphotonsisimportanttochecktheobservationin[ 35 ]thatthespin-spininteractionbetweengravitonsandfermionsseemstoexplainwhyinationarygravitonscausethefermionmodefunctiontogrow[ 77 ]whereastheyhavenoseculareffectonthemodefunctionofamassless,minimallycoupledscalar[ 117 ].Anotherimportantexerciseistoworkouttheeffectofinationarygravitonsontheelectriceldofapointcharge.ThisisthenaturalwaytocheckthesurprisingclaimofKitamotoandKitazawathatinfraredgravitonsscreensub-horizoninteractionsduringination[ 133 ].Beforeclosing,weshouldcommentonthegaugeissue.ThevacuumpolarizationrequiresxingboththeU(1)anddiffeomorphismsymmetries,andthemannerinwhichthisisaccomplishedcanaffecttheresult.Ourpreviousstudyofgravitonsonatbackgroundrevealednodependenceuponthechoiceofelectromagneticgauge,butahugevariationwiththegravitationalgauge[ 22 ].WebelievethereisnotlikelytobeanygaugedependenceintheleadingsecularinfraredeffectsonendsfromdeSittergravitonsbecausethespintwopartofthegravitonpropagatorhasthesameinfraredlogarithmterminanygauge[ 98 99 ].Notethatitisperfectlypossiblefora1PIfunctionsuchasthevacuumpolarizationtochangewiththegauge,whileaparticularfeatureofitsdependenceonspaceandtimeisthesameinallgauges[ 134 ].Thatispreciselywhathappenswiththepoletermsof1PIfunctionsinatspacequantumeldtheory,andwesuspectthatthesameappliesfortheleadingseculardependenceondeSitter. 129

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Figure5-1. Gravitoncontributionstotheoneloopvacuumpolarization.Photonpropagatorsarewavyandgravitonpropagatorsarecurly. Table5-1. Partsof4-pointvertexfunction IUI 11 4[]2)]TJ /F10 7.97 Tf 10.49 4.71 Td[(1 2[]()3)[][](+)[][](4()[]()+()[]()5()()[]+()()[]6[]()()+[]()() Table5-2. Termsof4-pointcontributionafternaiveindexcontractions. I@(i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4UIihi(x;x)@D(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0)) 11 8@ni2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4ihi@Do)]TJ /F10 7.97 Tf 13.15 4.71 Td[(1 8@ni2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4ihi@Do2)]TJ /F10 7.97 Tf 10.49 4.71 Td[(1 4@ni2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4ihi@Do+1 4@ni2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4ihi@Do3)]TJ /F10 7.97 Tf 10.49 4.71 Td[(1 2@ni2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4ihi@Do+1 2@ni2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4ihi@Do)]TJ /F10 7.97 Tf 10.49 4.71 Td[(1 2@ni2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4ihi@Do+1 2@ni2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4ihi@Do4@ni2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4ihi@Do)]TJ /F3 11.955 Tf 11.95 0 Td[(@ni2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4ihi@Do5@ni2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4ihi@Do)]TJ /F3 11.955 Tf 11.95 0 Td[(@ni2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4ihi@Do6@ni2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4ihi@Do)]TJ /F3 11.955 Tf 11.95 0 Td[(@ni2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4ihi@Do Table5-3. Variousgravitontensorfactorcontractions. hTcfihTAihTCi )]TJ /F10 7.97 Tf 17.23 4.7 Td[(4D (D)]TJ /F10 7.97 Tf 6.58 0 Td[(2))]TJ /F10 7.97 Tf 10.5 5.47 Td[(4(D)]TJ /F10 7.97 Tf 6.59 0 Td[(1) (D)]TJ /F10 7.97 Tf 6.58 0 Td[(3)8 (D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)(D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)D(D2)]TJ /F6 7.97 Tf 6.58 0 Td[(D)]TJ /F10 7.97 Tf 6.58 0 Td[(4) (D)]TJ /F10 7.97 Tf 6.58 0 Td[(2)(D3)]TJ /F10 7.97 Tf 6.58 0 Td[(4D2+D+2) (D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)2(D2)]TJ /F10 7.97 Tf 6.59 0 Td[(5D+8) (D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)(D)]TJ /F10 7.97 Tf 6.59 0 Td[(3))]TJ /F10 7.97 Tf 20.68 4.71 Td[(4 (D)]TJ /F10 7.97 Tf 6.58 0 Td[(2))]TJ /F10 7.97 Tf 20.69 4.71 Td[(4 (D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)4 (D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)00+4 (D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)(D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)(D2)]TJ /F6 7.97 Tf 6.59 0 Td[(D)]TJ /F10 7.97 Tf 6.59 0 Td[(4) (D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)(D2)]TJ /F10 7.97 Tf 6.58 0 Td[(3D)]TJ /F10 7.97 Tf 6.59 0 Td[(2) (D)]TJ /F10 7.97 Tf 6.59 0 Td[(3))]TJ /F4 11.955 Tf 9.3 0 Td[(2(D)]TJ /F10 7.97 Tf 6.58 0 Td[(3) (D)]TJ /F10 7.97 Tf 6.58 0 Td[(2)00+2 (D)]TJ /F10 7.97 Tf 6.58 0 Td[(2)(D)]TJ /F10 7.97 Tf 6.58 0 Td[(3) 130

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Table5-4. 4-pointcontributionscomingfromtheconformalpartofthegravitonpropagator. I@(i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4UIihTcfiB(0)@D(x)]TJ /F5 11.955 Tf 9.97 0 Td[(x0)) 1)]TJ /F6 7.97 Tf 21.67 4.7 Td[(D 2(D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4B@D)]TJ /F3 11.955 Tf 11.95 0 Td[(@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4B@D2)]TJ /F6 7.97 Tf 10.49 5.48 Td[(D(D2)]TJ /F6 7.97 Tf 6.58 0 Td[(D)]TJ /F10 7.97 Tf 6.58 0 Td[(4) 4(D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4B@D)]TJ /F3 11.955 Tf 11.95 0 Td[(@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4B@D34 (D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4B@D)]TJ /F3 11.955 Tf 11.95 0 Td[(@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4B@D4)]TJ /F6 7.97 Tf 19.45 4.7 Td[(D (D)]TJ /F10 7.97 Tf 6.58 0 Td[(2)@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4B@D)]TJ /F3 11.955 Tf 11.95 0 Td[(@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4B@D5(D2)]TJ /F6 7.97 Tf 6.59 0 Td[(D)]TJ /F10 7.97 Tf 6.59 0 Td[(4) (D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4B@D)]TJ /F3 11.955 Tf 11.96 0 Td[(@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4B@D6(D2)]TJ /F6 7.97 Tf 6.59 0 Td[(D)]TJ /F10 7.97 Tf 6.59 0 Td[(4) (D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4B@D)]TJ /F3 11.955 Tf 11.96 0 Td[(@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4B@D Table5-5. 4-pointcontributionscomingfromtheA-typepartofthegravitonpropagator. I@(i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4UIihTAi(iA(x;x))]TJ /F5 11.955 Tf 11.96 0 Td[(B(0))@D(x)]TJ /F5 11.955 Tf 9.96 0 Td[(x0)) 1)]TJ /F10 7.97 Tf 12.71 5.48 Td[((D)]TJ /F10 7.97 Tf 6.59 0 Td[(1) 2(D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4(iA)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D)]TJ /F3 11.955 Tf 11.96 0 Td[(@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(iA)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D2)]TJ /F10 7.97 Tf 10.49 5.47 Td[((D3)]TJ /F10 7.97 Tf 6.59 0 Td[(4D2+D+2) 4(D)]TJ /F10 7.97 Tf 6.58 0 Td[(3)@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4(iA)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D)]TJ /F3 11.955 Tf 11.96 0 Td[(@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(iA)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D32 (D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(iA)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D)]TJ /F3 11.955 Tf 11.95 0 Td[(@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(iA)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D+@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(iA)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D)]TJ /F4 11.955 Tf 13.21 2.66 Td[(@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(iA)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D4)]TJ /F10 7.97 Tf 10.49 5.47 Td[((D)]TJ /F10 7.97 Tf 6.58 0 Td[(1) (D)]TJ /F10 7.97 Tf 6.58 0 Td[(3)@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(iA)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D)]TJ /F4 11.955 Tf 13.21 2.65 Td[(@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(iA)]TJ /F5 11.955 Tf 11.96 0 Td[(B)@D5(D2)]TJ /F10 7.97 Tf 6.59 0 Td[(3D)]TJ /F10 7.97 Tf 6.58 0 Td[(2) (D)]TJ /F10 7.97 Tf 6.58 0 Td[(3)@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(iA)]TJ /F5 11.955 Tf 11.96 0 Td[(B)@D)]TJ /F4 11.955 Tf 13.21 2.66 Td[(@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(iA)]TJ /F5 11.955 Tf 11.96 0 Td[(B)@D6(D2)]TJ /F10 7.97 Tf 6.59 0 Td[(3D)]TJ /F10 7.97 Tf 6.58 0 Td[(2) (D)]TJ /F10 7.97 Tf 6.58 0 Td[(3)@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(iA)]TJ /F5 11.955 Tf 11.96 0 Td[(B)@D)]TJ /F3 11.955 Tf 11.95 0 Td[(@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(iA)]TJ /F5 11.955 Tf 11.96 0 Td[(B)@D Table5-6. 4-pointcontributionscomingfromtheC-typepartofthegravitonpropagator. I@(i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4UIihTCi(C(0))]TJ /F5 11.955 Tf 11.95 0 Td[(B(0))@D(x)]TJ /F5 11.955 Tf 9.97 0 Td[(x0)) 11 (D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)(D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D)]TJ /F3 11.955 Tf 11.96 0 Td[(@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D2)]TJ /F10 7.97 Tf 14.25 5.48 Td[((D2)]TJ /F10 7.97 Tf 6.59 0 Td[(5D+8) 2(D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)(D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D)]TJ /F3 11.955 Tf 11.96 0 Td[(@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D32 (D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)n0@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@0D+1 (D)]TJ /F10 7.97 Tf 6.58 0 Td[(3)@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B)@D)]TJ /F3 11.955 Tf 9.3 0 Td[(@0i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@0D)]TJ /F10 7.97 Tf 23.34 4.71 Td[(1 (D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B)@D+0@0i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D+1 (D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B)@D)]TJ /F3 11.955 Tf 9.29 0 Td[(00@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B)@D)]TJ /F10 7.97 Tf 23.34 4.71 Td[(1 (D)]TJ /F10 7.97 Tf 6.58 0 Td[(3)@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@Do42 (D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)(d)]TJ /F10 7.97 Tf 6.58 0 Td[(3)(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)@0i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B)@0D+@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B)@D+(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)00@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D)]TJ /F4 11.955 Tf 11.96 0 Td[((D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)0@0i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B)@D)]TJ /F4 11.955 Tf 9.29 0 Td[((D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)0@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@0D)]TJ /F4 11.955 Tf 13.2 2.66 Td[(@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B)@D52 (D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)(D)]TJ /F10 7.97 Tf 6.59 0 Td[(3))]TJ /F4 11.955 Tf 9.3 0 Td[((D)]TJ /F4 11.955 Tf 11.96 0 Td[(3)2@0i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@0D)]TJ /F4 11.955 Tf 13.21 2.66 Td[(@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D+(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)20@0i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B)@D+@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D62 (D)]TJ /F10 7.97 Tf 6.59 0 Td[(2)(D)]TJ /F10 7.97 Tf 6.59 0 Td[(3)@i2aD)]TJ /F10 7.97 Tf 6.58 0 Td[(4(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D)]TJ /F3 11.955 Tf 11.96 0 Td[(@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D)]TJ /F4 11.955 Tf 9.29 0 Td[((D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)200@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(C)]TJ /F5 11.955 Tf 11.95 0 Td[(B)@D+(D)]TJ /F4 11.955 Tf 11.95 0 Td[(3)20@i2aD)]TJ /F10 7.97 Tf 6.59 0 Td[(4(C)]TJ /F5 11.955 Tf 11.96 0 Td[(B)@0D 131

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Table5-7. Piecesof3-pointvertexfunction. IV 1[]24)[][]( Table5-8. 3-pointresultsafternaiveindexcontractions. IJ)]TJ /F3 11.955 Tf 9.3 0 Td[(@@0n2(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4VIihi(x;x0)VJ@@0ihi(x;x0)o 11)]TJ /F3 11.955 Tf 9.3 0 Td[(2@@0n(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4ihi@[@0[[ih]]]io122@@0n(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4ih[i@[[@0ih]]]i)]TJ /F4 11.955 Tf 9.3 0 Td[((aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4ih[i@[[@0]ih]]io212@@0n(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4ih[i@@0[[ih]]]i)]TJ /F4 11.955 Tf 9.29 0 Td[((aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4ih[i@]@0[ih]io22)]TJ /F3 11.955 Tf 9.3 0 Td[(2@@0n(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4ih[[[i@@0ih]]]i)]TJ /F4 11.955 Tf 9.3 0 Td[((aa0)D)]TJ /F10 7.97 Tf 6.58 0 Td[(4ih[[[i@]@0ih]]i+(aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4ih[[[i@]@0]]ihi)]TJ /F4 11.955 Tf 9.3 0 Td[((aa0)D)]TJ /F10 7.97 Tf 6.59 0 Td[(4ih[[[i@@0]]ih]io 132

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CHAPTER6CONCLUSIONWehaveuseddimensionalregularizationtocomputetheoneloopquantumgravitationalcontributiontothevacuumpolarizationondeSitterbackground.( 5 5 )and( 5 5 )giveourdimensionallyregulatedresult.ThefullBPHZrenormalizedresultisgivenin( 5 )and( 5 ).Chapter 3 discussesourdecisiontousethenoncovariantrepresentationforournalresult.Shouldacovariantrepresentationbedesired,chapter 4 providesthenecessaryidentities.Thethreecountertermsthatwerenecessarytorenormalizeourresultweredeterminedbythesupercialdegreeofdivergenceatone-looporderandU(1)gaugeinvariance.Twoofthecountertermspreservegeneralcoordinateinvariance,butonedoesnot,whichmustbeallowedforgaugexingfunctionsthatbreakdeSitterinvariance.Thesecountertermsareguaranteedtoexistsbydimensionalanalysis,U(1)symmetry,andthesupercialdegreeofdivergence.Therefore,ifonedayacompletequantumgravitytheoryisfound,itcanonlyalterourresultsbymodifyingthenitepartsoftheBPHZcounterterms.However,ourresultswillstillbecorrectinthefarinfraredregimebecausetheln(a)termsinourrenormalizedresultswillalwaysdominateanymodicationtothenitepartsofthecounterterms.Thisisthestandardargumentforthevalidityofeffectiveeldtheories,andthefactthatourcalculationinvolvesgravitydoesnotmaketheargumentanylessappealing.Giventhatarealphysicalresultmayinprinciplebeobtainedforthiscalculationitisimportanttodiscusswhywebelieveourcalculationtobecorrect.Therearethreemainreasons,therstofwhichisthatourresultshouldreducetotheatspaceresultofchapter 2 ,whichindeeditdoeswhenH!0.Secondly,allofthedivergenttermswereabletobeabsorbedusingthecountertermspredictedusingthemeansdescribedabove.Thisisnotatallatrivialresultgiventhenumberoftermsthatneededtobereduced.Thelastfactinsupportofourresultisthatthegeneralpropertiesofthe 133

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vacuumpolarizationarepreserved;namely,symmetryunderinterchangeofcoordinatesandtransversality.Onemightbeconcernedthatourresultismerelyagaugeartifact.ThevacuumpolarizationrequiresxingboththeU(1)anddiffeomorphismsymmetries,andthemannerinwhichthisisaccomplishedcanaffecttheresult.Ourpreviousstudyofgravitonsonatbackgroundrevealednodependenceuponthechoiceofelectromagneticgauge,butahugevariationwiththegravitationalgauge[ 22 ].WebelievethereisnotlikelytobeanygaugedependenceintheleadingsecularinfraredeffectsonendsfromdeSittergravitonsbecausethespintwopartofthegravitonpropagatorhasthesameinfraredlogarithmterminanygauge[ 98 99 ]. 134

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BIOGRAPHICALSKETCH KatieLeonardisfromHartlandWisconsin.ShereceivedherundergraduatedegreesinphysicsandastrophysicsattheUniversityofMinnesota(UMN)in2009.WhileatUMNshereceivedanundergraduateresearchgrantforherworkondoubleradiogalaxiesunderProfessorLawrenceRudnick.Forherphysicsthesisshestudiedtheenergydistributionofcosmicmuons,andcompletedherastrophysicsthesisonmodelingtheeffectsofsubstructureinlensinggalaxiessupervisedbyProfessorLiliyaWilliams.LeonardcametotheUniversityofFloridain2009.WhilecompletingherPh.D.researchshereceivedanAPS/SBFawardtoconductresearchattheUniversidadedeS~aoPaulounderthedirectionofProfessorL.RaulAbramo.ShealsoreceivedtheRaymondE.AndrewsawardforseniorgraduateresearchandseveraltravelgrantstosupportworkdoneattheUniversityofUtrechtunderthedirectionofProfessorTomislavProkopec.LeonardreceivedherPh.D.inthesummerof2013. 143