Hamiltonian formalisms for multidimensional calcul

arXiv:math-ph/0212036v111Dec2002HamiltonianformalismsformultidimensionalcalculusofvariationsandperturbationtheoryFr´ed´ericH´eleinFebruary7,2008Abstract—Inafirstpartweproposeanintroductiontomultisymplecticformalisms,whicharegeneralisationsofHamilton’sformulationofMechanicstothecalculusofvari-ationswithseveralvariables:wegivesomephysicalmotivations,relatedtothequantumfieldtheory,andexpoundthesimplestexample,basedonatheoryduetoT.deDonderandH.Weyl.InasecondpartweexplainquicklyaworkincollaborationwithJ.KouneiherongeneralizationsofthedeDonder–Weyltheory(knownasLepagetheories).Lastlyweshowthatinthisframeworkaperturbativeclassicalfieldtheory(analogoftheperturbativequantumfieldtheory)canbeconstructed.1IntroductionThemainquestioninvestigatedinthistextconcernstheconstructionofaHamiltoniandescriptionofclassicalfieldstheorycompatiblewiththeprinciplesofspecialandgeneralrelativity,ormoregenerallywithanyefforttowardsunderstandinggravitationlikestringtheory,supergravityorAshtekar’stheory:sincespace-timeshouldmergeoutfromthedynamicsweneedadescriptionwhichdoesnotassumeanyspace-time/fieldsplittingapriori.Thismeansthatthereisnospace-timestructuregivenapriori,butspace-timecoordinatesshouldinsteadmergeoutfromtheanalysisofwhataretheobservablequantitiesandfromthedynamics.Fromthispointofview,aswewillsee,theLepage–DedeckertheoryseemstobemuchmoreappropriatethanthedeDonder–Weylone.Thisisthephilosophythatwehavefollowedin[21].Hereacaveatisinorder,intheclassicalone-dimensionalHamiltonianformalism:westartwithaLagrangianactionfunctionalL[c]:=Zt1t0L(t,c(t),˙c(t))dt,definedonasetofsmooth1paths{t7−→c(t)∈Y}.HereYisasmoothk-dimensionalmanifoldandLisasmoothfunctionon[t0,t1]×TY(TYisthetangentbundletoY:we1hereweassumeforinstancethatcisC2denotebyqapointinYandbyv∈TqYavectortangenttoYatq).ThecriticalpointsofLsatisfytheEuler–Lagrangeequationddt∂L∂vi(t,c(t),˙c(t))=∂L∂qi(t,c(t),˙c(t)).Forallfixedtimet∈[t0,t1],theLegendretransformisthemappingTY−→T∗Y(q,v)7−→(q,p)=(q,∂L(t,q,v)/∂v),whereq∈Y,v∈TqYandp∈T∗qY.Incaseswhere,foralltimet,thismappingisadiffeomorphism,wedefinetheHamiltonianfunctionH:[t0,t1]×T∗Y−→RbyH(t,q,p):=hp,V(t,q,p)i−L(t,q,V(t,q,p)),where(q,p)7−→(q,V(t,q,p))istheinversemappingoftheLegendremapping.Thenitiswell-knownthatt7−→c(t)isasolutionoftheEuler–Lagrangeequationsifandonlyift7−→(c(t),∂L(t,c(t),˙c(t))/∂v)=:(c(t),π(t))isasolutionoftheHamiltonequationsdcidt(t)=∂H∂pi(t,c(t),π(t)),anddπidt(t)=−∂H∂qi(t,c(t),π(t)).ThusthisconvertsthesecondorderEuler–LagrangeequationsintotheflowequationofthenonautonomousvectorfieldXH,tdefinedoverT∗YbyXH,tΩ+dHt=0.(1)whereΩ:=Pki=1dpi∧dqiisthesymplecticformoverT∗Y,Ht(q,p):=H(t,q,p)and“”denotetheinteriorproduct,i.e.foranytangentvectorξ∈T(q,p)(T∗Y),ξΩisthe1-formsuchthatξΩ(V)=Ω(ξ,V),∀V∈T(q,p)(T∗Y)Insteadofviewingthedynamicsasthemotionofapointinsomespace,likeforinstancethephasespaceT∗Y,wecanuseanotherapproachewhichconsistsindetermininghowanobservablequantity,suchasthepositionorthemomentumofaparticle,evolves.ThisisachievedbyusingthePoissonbracketoperationC∞(T∗Y)×C∞(T∗Y)−→C∞(T∗Y)(F,G)7−→{F,G},where{F,G}:=kXi=1∂F∂pi∂G∂qi−∂F∂qi∂G∂pi.Then,forallHamiltoniantrajectoryt7−→(c(t),π(t)),andforallF∈C∞(T∗Y),wehavedF(c(t),π(t))dt={H,F}(c(t),π(t)).Forexampleintheparticularcasewherethevariationalproblemisautonomous(i.e.Ldoesnotdependont)thenHdoesnotdependontimeandwededucefromtheskewsym-metryofthePoissonbracketthattheenergyisconservedalongthetrajectories,aspecialcaseofNoethertheoremwhentheproblemisinvariantbytimetranslation.Eventuallythisformulationofthedynamicsisagoodpreliminaryformodellingtheevolutionofthequantumversionofourproblem:forinstancebyreplacingthefunctionsinC∞(T∗Y)byHermitianself-adjointoperatorsandthePoissonbracketbythecommutator[·,·]we“guess”theHeisenbergevolutionequationi~dbFdt=[bH,bF],consequentlythecommutationrelations[bpi,bqj]=i~δji,isnothingbutaformalisationofHeisenbergincertaintyprinciple.Allthatleadstoanow“wellunderstood”strategyofbuildingamathematicaldescriptionofaquantumparticlegovernedbyaHamiltonianfunctionsH(withtherestrictionthat,amongotherthings,thecorrespondenceH7−→bHisfarfrombeinguniquelydefined).StartingfromavariationalformulationofNewton’slawofmechanics,itisthuspossibletoformallyderivetheSchr¨odingerequationmoreorlessbyfollowingthestepsdiscussedabove.Nowthemorechallengingquestionistoproduceasimilaranalysisforfieldstheories.Quantumfieldstheory2resultsfromtheeffortsofphysicistsinordertocuretheshortcom-ingoftheSchr¨odingerequation.Indeed,thislatterequationisnotinvariantbythegroup2Startingfromtheframeworkofclassicalphysics,theconceptofafieldatfirstmightevokeideasaboutmacroscopicsystems,forexamplevelocityfieldsortemperaturefieldsinfluidsandgases,etc.Fieldsofthiskindwillnotconcernus,however;theycanbeviewedasderivedquantitieswhicharisefromanaveragingofmicroscopicparticledensities.Oursubjectsarethefundamentalfieldsthatdescribematteronamicroscopiclevel:itisthequantum-mechanicalwavefunctionψ(x,t)ofasystemwhichcanbeviewedasafieldfromwhichtheobservablequantitiescanbededuced.Inquantummechanicsthewavefunctionisintroducedasanordinarycomplex-valuedf