Divergences in formal variational calculus and bou

arXiv:hep-th/9511130v117Nov1995***********************************************BANACHCENTERPUBLICATIONS,VOLUME**INSTITUTEOFMATHEMATICSPOLISHACADEMYOFSCIENCESWARSZAWA19**DIVERGENCESINFORMALVARIATIONALCALCULUSANDBOUNDARYTERMSINHAMILTONIANFORMALISMVLADIMIRO.SOLOVIEVInstituteforHighEnergyPhysics142284,Protvino,Moscowregion,RussiaE-mail:[e-mail]ormalvariationalcalculusinordertoincorporateintegralsofdivergencesintoit.Suchageneralizationpermitstostudynontrivialboundaryproblemsinfieldtheoryonthebaseofcanonicalformalism.1.Introduction.TheHamiltonianformulationofclassicalmechanics[Arn]isbasedongeometricalconstructionswhichusesuchnotionsasdifferentialforms,vectorfieldsandmultivectors,andsuchoperationsasdifferential,interiorproduct,Liederivative,Schouten-Nijenhuisbracket.Mostoftheseconstructionswereextendedtofieldtheoryintheprocessofstudyingnonlinearintegrablemodelsduringthelast20years[Olv86].Thisapproachhasbeencalledtheformalvariationalcalculus[GD]becauseitignoresanytermsarisingasaresultofintegrationbyparts.Thisisfullyjustifiedincaseofperiodicboundaryconditionsorfastdecayoffieldsatspatialinfinity,butunfortunately,thismethodisnotapplicableinitsinitialformtomanyotherproblemsinterestingfromphysicalpointofview.Forexample,masslessfieldsareslowlydecayingatinfinityand,asaresult,someimportantcharacteristicsofthesefieldsareexpressedjustthroughsurfaceintegrals(orvolumeintegralsofspatialdivergences).TheyarenecessarytoformcanonicalgeneratorsoftheglobalgaugetransformationsorasymptoticsymmetriesoftheRiemannianmetric.ThegreateffortswerestartedattheendoffiftiestounderstandtheroleofsurfacetermsintheHamiltonianofGeneralRelativity[ADM].Onlyafter15yearsofstudythesatisfactoryexplanationhadbeengiven[RT].Buteventhennotallquestionswereanswered.Forexample,onemightworryhowtoretainthesurfacetermswhicharenecessarytorealizethePoincar´ealgebrainasymptoticallyflatspace[RT],[Sol85]{H(ξ),H(η)}=H([ξ,η])1991MathematicsSubjectClassification:Primary58F05;Secondary70G50,58G20.Thepaperisinfinalformandnoversionofitwillbepublishedelsewhere.[1]2V.O.SOLOVIEVwhendoinglocalcalculationsoftheconstraintsalgebra{H(x),H(y)}=gab(x)Hb(x)δ,a(x,y)−gab(y)Hb(y)δ,a(y,x),{H(x),Ha(y)}=−H(y)δ,a(y,x),{Ha(x),Hb(y)}=Hb(x)δ,a(x,y)−Ha(y)δ,b(y,x).Wewillshowinthefollowingthatallthemainstructuresoftheformalvariationalcalculuscanbeextendedtoincludenontrivialcontributionsofdivergencesthroughin-troductionofanewgradingandnewpairingcompatiblewithit.So,itoccurspossibletopreservethenicegeometricallanguageinthemoregeneralcasethanbefore.AfterallthefieldtheoryPoissonbracketisgivenbyanewformulawhichdiffersfromthestan-dardonebysurfaceterms.Simultaneouslywegettheanswertothementionedproblemofdisappearanceofthesurfacecontributionsinlocalcalculationswithδ-function.ThenaturalwaytotaketheboundarytermsintoaccountistointroducethecharacteristicfunctionθΩ(x)oftheintegrationdomainΩ.ThenrelationslikeθΩ(x)∂∂xi+θΩ(y)∂∂yi=−∂θΩ(x)∂xiδ(x,y),givethesolution.Initsturnthisisconnectedwiththeobservation[Sol92]thattransfor-mationsofthetype(forexample,transformationtoAshtekar’svariables)qA(x)→qA(x),pA(x)→pA(x)+δF[q]δqA(x),infieldtheoryarecanonicalonlyuptoboundarycontributions,becausethestandardEuler-Lagrangevariationalderivativesingeneraldonotcommute[And76],[And78].Weexpectthatboundaryconditionsshouldbetreatedinthisformalismasakindofconstraintsputontheinitialdata,i.e.,theyshouldbeaddedtotheHamiltonianwithsomeLagrangemultipliersandthencheckedforcompatibilitywiththedynamics.TherequirementofcompatibilitymayleadtosecondaryboundaryconditionsortofixingtheLagrangemultipliers.Butnowthissubjectisnotenoughstudiedandourconsiderationispreliminaryandlimitedtooneexample:thenonlinearSchr¨odingerequation.2.NewPoissonbracketformula.BelowweusethelocalcoordinatelanguageandinsteadofthemanifoldwithaboundaryconsideradomainΩinRnhavingasmoothboundary∂Ω.Wedonotexpectthatglobalformulationcouldmeetwithseriousdifficul-ties.Definition1.AnintegraloverafinitedomainΩofafunctionoffieldvariablesφA(x),A=1,...,pandtheirpartialderivativesDJφAuptosomefiniteorderF=ZΩdnxf(φA(x),DJφA(x))iscalledalocalfunctional.Remark1.Incontrasttothestandarddefinitionwedonottreattheselocalfunctionalsasequivalentiftheydifferbyadivergenceterm.AllthefunctionsfandφAaswellastheirvariationsthroughoutthepaperaresupposedinfinitelysmooth,i.e.C∞(Rn).Weusemulti-indexnotationsJ=(j1,...,jn)DJ=∂|J|∂j1x1...∂jnxn,|J|=j1+...+jn,D0=1.DIVERGENCESANDBOUNDARYTERMS3ThederivativeoperatorDwilldenotebelowthefullpartialderivativetakingintoaccountalsocoordinatedependenceoffieldsφA(x).Asthenumberofsumsinsomeformulaeofthispaperislargeenoughwewillwriteonlyasignofsummingwithoutdisplayingtheindicesofsummation.Accordingtothisrule,sumoverallrepeatedindicesshouldbeunderstood.Inthosecases,whereitisnotso,wedisplaythesummationindices.Also,wedonotshowthelimitsofsummation,becausetheyarenatural,i.e.outsidethemthesummandissimplyzero.Usuallyweomitdnxintheintegralsandshowtheargumentsonlywhentheycanbemixed.WedenoteasAthespaceoflocalfunctionals.Itisimportantthatthisspacein-cludesfunctionalswithintegrand