Renormalised nonequilibrium quantum field theory s

arXiv:0809.0496v1[hep-th]2Sep2008Renormalisednonequilibriumquantumfieldtheory:scalarfieldsSz.Bors´anyi∗DepartmentofPhysicsandAstronomy,UniversityofSussex,Brighton,EastSussexBN19QH,UnitedKingdom†U.Reinosa‡CentredePhysiqueTh´eorique,EcolePolytechnique,CNRS,92198,Palaiseau,France(Dated:September3,2008)Wediscusstherenormalisationoftheinitialvalueprobleminquantumfieldtheoryusingthetwo-particleirreducible(2PI)effectiveactionformalism.Thenonequilibriumdynamicsisrenormalisedbycountertermsdeterminedinequilibrium.WeemphasizetheimportanceoftheappropriatechoiceofinitialconditionsandgobeyondtheGaussianinitialdensityoperatorbydefiningself-consistentinitialconditions.Westudythecorrespondingtimeevolutionandpresentanumericalexamplewhichsupportstheexistenceofacontinuumlimitforthistypeofinitialconditions.I.INTRODUCTIONNonequilibriumfieldtheoryisreceivinganincreasinglevelofattentionfromthesideofcosmologistsaswellasfromtheheavyioncommunity.Thereheatingofthepostinflationaryuniverse[1]thedynamicsofsymmetrybreaking[2]andtheformationanddecayofcosmologicaldefectnetworks[3]aswellasthephasetransitionsintheearlyuniverse[4]withpossiblerelicgravitationalwaves[5]arejustsomeoftheexamplesthatrequirethestudyofout-of-equilibriumfieldsinacosmologicalcontext.Similarly,therapidthermalisationofthehotquark-gluonplasma[6,7]anditsdrivingmechanisms,suchastheWeibelinstability[8]raisequestionsintherealmofnonequilibriumfieldtheory.Oneofthesimplestandmostpopularstrategiestodescribeanout-of-equilibriumfieldtheoryistheclassicalap-proximation.IthasbeenextensivelyusedforreheatingmodelsoftheearlyUniverse[9]andalsoforpredictingthecorrespondingproductionofgravitationalwaves[10].Othercosmologicalapplicationsincludeelectroweakbaryo-genesis[11,12]aswellastheoriesaccomodatingnon-fundamentalstrings[13]ordomainwalls[14,15],wherethenonperturbativetreatmentisessential.Theextremeexcitationofthegluonfieldinaheavyioncollisionhasalsomadetheclassicalstrategyapplicableandverysuccessfulatearlytimesafteracollision[16,17].Thesuccessofclassicalfieldtheoryindicatesthatmanyoftheinterestingphenomenainhighenergyphysicsareactuallyclassical.Indeed,genuinequantumeffectsplaylittleroleiftheclassicalmodesarehighlyexcited,andtheseclassicalfluctuationscanplaytheroleofquantumparticles.InorderfortheclassicalapproachtoworktheUVmodesmustremainunexcitedtoavoidRayleigh-Jeansdivergences.Infact,thisrestrictstheclassicaltreatmenttofar-from-equilibriumsettings,andtheclassicaldynamicsautomaticallydrivesthesystemoutofitsrangeofvalidityinthecourseofequilibration.Itisstillpossibletosplitthemomentumspaceofatheoryintodifferentmomentumregions,whereharddegreesoffreedomfollowaquantum-mechanicallycorrectHardThermalLoop(HTL)dynamicstosomefiniteperturbativeorder,whiletheinfraredpartfollowstheclassicalnon-perturbativedynamics[18,19].Thisallowsakineticdescription[20,21]forthehardmodesintermsofaVlasovequation[22].Theinterplaybetweenclassicalwavesandparticlescangiveaccountfornon-trivialdynamics,suchasthedevelopmentofplasmainstabilities[23,24].Inthiswayonecanavoidtheproblemofultravioletdivergences,butthescaleseparationisnotalwaysnatural,especiallyifthecouplingisnotsmall.AsteptowardstheinclusionofquantumcorrectionsfromfirstprinciplesistheHartreeapproximation[25,26].ItassumesaconstantlyGaussiandensityoperatorandallowstoaccountforquasi-particlespropagatinginarbitraryinhomogeneousbackgrounds,whichcanbeascomplicatedasanetworkoftopologicaldefects[27,28].Althoughitcompletelyneglectsthescatteringofthequasi-particlesoneachother,non-perturbativeparticlecreationmechanisms,liketachyonicinstability[29]orparametricresonance[30]arewithinitsrangeofvalidity.Renormalisationinthisframeworkhasalreadybeendiscussedatlength[31].Despiteofitssimplicityandcleanformulationthefactthat∗Electronicaddress:email:[e-mail]†PartofthisresearchhasbeencarriedoutatKavliInstituteforTheoreticalPhysics,UCSB,SantaBarbara,CA93106,USA.‡Electronicaddress:email:[e-mail]2thermalisationcannotbedescribedinthisway[32,33]explainswhythisapproximationschemecouldnotreachawideacceptance.Thetwo-particleirreducible(2PI)effectiveactionprovidesafirstprinciplesapproachtoquantumfieldtheory[34,35].Thesystematicapproximations,obtainedfromanysmallparameterexpansionofthe2PIeffectiveaction,usuallyresumanentireseriesofladderdiagramswhichinturnplayaparticularlyimportantroleinsolvingthesecularityproblemofout-of-equilibriumperturbationtheory.ThistypeofresummationisalsopresentintheKadanoff-Baymequations[36]aswellasinBoltzmannequations[37].Ithasbeenshownthatthefirstnontrivialtruncationofthe2PIeffectiveactionbeyondtwo-looporderalreadyprovidesasufficientframeworktodescribethermalisationinscalartheories[38,39,40,41,42]aswellasinamodelwithfermions[43].The2PIeffectiveactionhasbecomeastandardframeworkfornonequilibriumquantumfieldtheory[44],atpresent,withmostlyscalarapplicationsofcosmologicalinterest[45,46,47].Thesuccessofthesepracticalapplicationshasalsoencouragedmoreformalinvestigationsontheveryfoundationsofthe2PI(andmoregenerallynPI[48])approach.Tobeconsideredasasensibleapproach,thela