搜索
arXiv:gr-qc/0312048v330Jan2006CovariantdescriptionofparametrizednonrelativisticHamiltoniansystemsMauricioMondrag´on∗andMercedMontesinos†‡DepartamentodeF´ısica,CentrodeInvestigaci´onydeEstudiosAvanzadosdelI.P.N.,Av.I.P.N.No.2508,07000CiudaddeM´exico,M´exico.(Dated:February7,2008)ThevariousphasespacesinvolvedinthedynamicsofparametrizednonrelativisticHamiltoniansystemsaredisplayedbyusingCrnkovicandWitten’scovariantcanonicalformalism.ItisalsopointedoutthatinDirac’scanonicalformalismthereexistsafreedominthechoiceofthesymplecticstructureontheextendedphasespaceandinthechoiceoftheequationsthatdefinetheconstraintsurfacewiththeonlyrestrictionthatthesetwochoicescombineinsuchawaythatanypair(ofthesetwochoices)generatesthesamegaugetransformation.Theconsequenceofthisfreedomonthealgebraofobservablesisalsodiscussed.I.INTRODUCTIONThereiscurrentlyagrowinginterestinthestudyofthefundamentalsofbothclassicalandquantummechanicsmotivated,inpart,byseveraltheoreticalapproachesthattrytobuildaquantumtheoryofgravity[see,forinstance,Ref.[1]].Thevariousconceptualissuesfoundintheconstructionofgenerallycovariantquantumtheoriesfrequentlymakepeopletogobacktothefundamentalsofbothclassicalandquantummechanicstotrytoremovewhatisnon-essentialandgetthegenericaspectsofthemwhichcouldbeimplementedlateroninrealistictheories[see,forinstance,Refs.[2,3,4,5,6]andreferencestherein].InthispaperwefocusinthecovariantdescriptionofHamiltonianmechanics.ThegeometricalstructureunderlyingparametrizednonrelativisticHamiltoniansystemsisobtainedbyusingtheapproachofRef.[7],fromwhichtheextendedphasespace(Γext,Ωext)andthepresymplecticphasespace(Σ,ΩΣ)involvedareobtained[seealsoRef.[8]formoredetails].Oncethisisdone,thedefinitionofphysicalobservablesisimplementedandthisfactallowsustoreachthephysicalphasespace(Γphys,Ωphys)forthesystem.ThisisdisplayedinSubSecs.IIAandIIB.Inspiteofworkingwiththecovariantcanonicalformalism,theusualsymplecticstructureisused.TheimplicationsofchoosingalternativesymplecticstructuresinDirac’sformalismareanalyzedinSecs.III,IV,V,andVIwhereitisshownthattherearemanywaysofchoosingthesymplecticstructureontheextendedphasespaceiftheequationthatdefinestheconstraintsurfaceis,inthegenericcase,1accordinglymodifiedinsuchawaythatthegaugetransformationisnotaltered.Duetothefactthatthegaugetransformationisnotmodifiedthegauge-invariantfunctionsarealsonotmodified,however,the‘algebraofobservables’is,inthegenericcase,modifiedbecauseitdependsontheparticularsymplecticstructurechosen.SectionVIIcontainsageneralizationoftheseresultstogenerallycovariantsystemswithfirstclassconstraintsonly.OurconclusionsarecollectedinSec.VIII.II.THEGEOMETRYANDTHEPHYSICSOFPARAMETRIZEDNONRELATIVISTICHAMILTONIANSYSTEMSLetusbeginbyconsideringtheHamiltonianformulationassociatedwithasystemwithafinitenumberofdegreesoffreedomobtainedfromtheactionprincipleS[qj,pj]=Zt2t1dqjdtpj−H(qj,pj,t)dt,j=1,...,n,(1)subjecttothestandardboundaryconditionsqj(tα)=qjα,α=1,2,(2)†AssociateMemberoftheAbdusSalamInternationalCentreforTheoreticalPhysics,Trieste,Italy.∗Electronicaddress:mo@fis.cinvestav.mx‡Electronicaddress:merced@fis.cinvestav.mx1Insomecases,theequationthatdefinestheconstraintsurfaceisnotmodified[seeSec.III].2whereqjαareprescribednumbers.Itisassumedthatthedynamicalsystemiswell-defined,i.e.,thatthereexistsasolutiontothedynamicalproblemthatmatchestheboundaryconditions.Bydefinition,thecoordinatesqilocallylabelthepointsoftheconfigurationspaceCforthesystem;thecotangentbundleofC,Γ=T∗C,isthephasespacewhosepointsarelocallylabelledbythecoordinatesxa,xi=qiandxi+n=pi,a=1,...,2n.Theequationsofmotionforthesystemhavethecanonicalform˙xa=ωab∂H∂xb,(3)whereHistheHamiltonian,ωabarethecomponentsoftheinverseofthesymplecticmatrix{xa,xb}:=ωab,(ωab)=0I−I0,(4)withIthen×nidentitymatrixand0then×nzeromatrix.Thesymplecticstructureω:=12ωabdxa∧dxb,(5)inducesaPoissonstructureonΓ=T∗Cdefinedby{f(x,t),g(x,t)}:=∂f∂xaωab∂g∂xb,(6)where,asusual,thecoordinatetistreatedasaparameter[9].Parameterizingthesystem.Ifthetimevariableisconsideredasacanonicalvariable,theactionforthisHamiltoniansystembecomesS[qj,pj,t,pt,λ]=Zτ2τ1˙qjpj+˙tpt−λγdτ,γ:=pt+H(qj,pj,t),(7)subjecttothestandardboundaryconditionsqj(τα)=qjα,t(τα)=tα,α=1,2,(8)whereptisthecanonicalvariableconjugatetot,λistheLagrangemultiplierassociatedwiththeconstraintγ=0thatcomesfromthedefinitionofpt,andthedotmeanstotalderivativewithrespecttotheparameterτ2.ThenewconfigurationspaceforthesystemistheextendedconfigurationspaceCext=C×Rwhosepointsarelocallylabelledby(qi,t)whereRstandsforthetcoordinate.Itscorrespondingphasespacewillbeanalyzedbelow.A.Hamilton’sprincipleFollowingtheconventionsusedinRef.[10],let˜δdenotethearbitraryvariationofcoordinates(qj,t),momenta(pj,pt),andLagrangemultiplierλatτfixed˜δxμ(τ):=x′μ(τ)−xμ(τ),˜δλ(τ):=λ′(τ)−λ(τ),(9)where(xμ)=(qj,t,pj,pt),μ=1,...,2(n+1).Theobject˜δis,butnotalways,calledthetotalvariation,otherauthorscallitavirtualvariation.2Itisfrequentlyassertedthattheparameterτhasno‘physicalmeaning.’However,thisassertionisnotcompletelytrue.Forinstance,inthecaseoftherelativisticfreeparticletheparameterτ