Scattering theory for scalar field in de Sitter sp

arXiv:0804.3718v1[hep-th]23Apr2008ScatteringtheoryforscalarfieldindeSitterspacetimeCruceanCosmin∗WestUniversityofTimi¸soara,V.ParvanAvenue4RO-300223Timi¸soara,Romania(Dated:April23,2008)AbstractTheinteractiontheorybetweenchargedscalarfieldandelectromagneticfieldondeSitterback-groundisstudiedinthehypothesisofminimallycoupling.Thenthereductionrulesforscalarfieldareestablishedandfurtherthetheoryofscatteringforscalarfieldisconstructedusingperturba-tiontheory.WiththeseelementsthescatteringofachargedscalarfieldonCoulombpotentialisstudied.OneobtainsthatincontrastwithwhathappensonMinkowskicasethemodulusofmomentumisnotconservedinthescatteringprocessondeSitterspace.Afterthatwestudytheproblemofcalculatingthecrosssection,pointingoutaseriesofdifficultiesthatarises.PACSnumbers:04.62.+v∗Electronicaddress:[e-mail]fieldtheoryondeSitterspace-timewasintensivelystudied.ThereasonforthisspecialattentionisduetothefactthatdeSitterspaceistheuniquemaximallysymmetriccurvedspace-time.ItenjoysthesamedegreeofsymmetryasMinkowskispace(tenKillingvectors).Evenwiththisfacilitiesthepresenceofcurvatureandnon-trivialglobalproperties,introducenewproblemsrelatedtothequantizationoffieldsondeSitterspace.AlsoitturnedoutthatthedeSittermetricplayedacentralroleintheinflationarycosmologicalscenario.ThesereasonsexplaintheenormousamountofliteratureondeSitterquantumfieldtheory.Thequantumscalarfieldisstudiedinproblemsconcerningquantumeffectsinpresenceofgravitationandcosmology.OfaspecialinterestincosmologyisthedeSitterexpandinguniversecarryingscalarfieldsvariouslycoupledwithgravitation.Inthequantumtheoryaspecialroleisplayedbythefreefields(minimallycoupledwithgravitation)becausewiththeirhelpwecancalculatescatteringamplitudesusingperturbationmethods.Inouropinionthecalculationsofscatteringprocessesmustbetheprincipalgoalofthequantumfieldtheoryinthecurvedspace.Thisimpliesatheoryofscatteringoncurvedspacetimewhichmustbeobtainedusinganaloguemethodsasinflatspacecase.ThetheoryofquantumscalarfieldondeSitterspacewasintensivelystudied,buttheexplicitsolutionswaswritteninafewpapers.Inouropinionthesesolutionshaveamainimportance,indevelopingthetheoryofinteractionondeSitterspace-time.Adetailedstudyofinteractionbetweenquantumfieldsminimallycoupledwithgravitationinaexpandinguniversewasnotdonefromourknowledge.WefocusourattentiononthissubjectbecausethelastastronomicalobservationsshowsthattheexpansionoftheUniverseisaccelerating,thusincreasingtheinterestondeSittermodel.Themajorityofinvestigationrelatedtothequantumfieldtheoryoncurvedspace-timesitconcernsproblemsrelatedtothecreationofparticlesoncurvedspace,butneglectthescatteringprocesses.Thistheoryofscatteringcouldbeofinterest,sincethestudyofinteractionsbetweenfermions,bosonsandscalarfieldsonacurvedbackgroundwillhelpustounderstandhowthewellestablishedpicturefromflatspacewillbemodifiedbyagravitationalfield.UntilnowdetailedstudiesrelatedtotheDiracfield[2],[3],[13],[14]andscalarfield[9]ondeSitterspace-timewasdoneobtainingimportantresultssuchasnormalizedanalyticalsolutionsandquantization.Thelessstudiedwastheelectromagneticfield,butherethesituationissimpleisonerecallstheconformal2invariancebetweendeSitterandMinkowskimetrics.ThusatthismomentseemsthatwehaveallelementsforconstructingatheoryofinteractionbetweenquantumfieldsondeSitterbackground.Inthispaperourattentionwillbefocusedontheinteractionbetweenscalarfieldandelectromagneticfield.WestartinthesecondsectionwithabriefreviewofthemainresultsconcerningthequantumscalarfieldtheoryondeSitterspace.InsectionIIIweconstructthetheoryofinteractionbetweenscalarfieldandelectromagneticfieldondeSitterspaceandthenourattentionwillbefocusedondevelopingthereductionformalismforscalarfield.SectionIVisdedicatedtotheperturbationtheorywhichwillhelpustoobtainthedefinitionofscatteringamplitudes.ThescatteringprocessthatimpliesthescatteringofachargedscalarparticleonCoulombpotentialondeSitterspaceisstudiedinsectionVandinsectionVIwestudytheproblemsthatarisewhenwepassfromscatteringamplitudetocrosssection.OurconclusionsissummarizedinsectionVII,andinAppendixeswegivethemainstepsleadingtoourscatteringamplitudeandtheformoftheintegralsthathelpustoestablishthedefinitiveformofthecrosssection.Theseresultsarepresentedinnaturalunits~=c=1.II.PLANEWAVESOLUTIONSFORSCALARFIELDWestartwiththeexactsolutionsofthefreeKlein-GordonequationondeSitterspace-time,whichwasobtainedin[9],[15].LetuswritethedeSitterlineelement[1]:ds2=dt2−e2ωtd~x2,(1)whereωistheexpansionfactor.Inthechart{t,~x}withCartesiancoordinatestheKlein-Gordonequationreads[15]:∂2t−e−2ωtΔ+3ω∂t+m2
φ(x)=0.(2)ItisknownthattheKlein-Gordonequation(2)ondeSitterspacecanbeanalyticallysolvedintermsofBesselfunctions[9],[10].Fromdifferentversionsofsolutionswhicharecurrentlyused,wewilluseherethenormalizedsolutionsofpositivefrequencieswhichcanbewreathedas[15]:f~p(x)=12rπωe−3ωt/2(2π)3/2e−πk/2H(1)ikpωe−ωtei~p·~x,(3)3whereH1μ(z)isaHankelfunctionoffirstkind,p=|~p|andthenotationsare:k=qμ2−94,μ=mω,(4)withm3ω/2andthefundamentalsolutionsofnegativefrequenciesaref∗~p(x).Thesesolutionssatisfiesthe