Note Where is the Commutation Relation Hiding in the Path Integral Formulation Yen Chin - PDF document

2integral,andforthatmatter,mostofquantumeldthe-ory;althoughasaphysicist,Iknowhowtousethemandtowavemyhandsasnecessary,deepdownIamdeeplytroubled.Toseewhypathintegralisproblematic,notethatinEq.(1),theintegrationismoreappropriatelydenotedbyZ�D where :[ti;tf]!Rdisanypathconnectingtheendpoints (ti)=qiand (tf)=qf,and�isthespaceofsuchpaths.HereD shouldbethoughtofasaLebesgue-typemeasureonthespace�ofpaths.Unfortunately,thisLebesgue-typemeasuresimplydoesnotexist.Thisfollowsfromthewell-knownresultinfunctionalanaly-sisthata[nontrivial]translationalinvarianceLebesgue-typemeasurecannotbedenedoninnitedimensionalHilbertspaces.However,evenbeforeFeynman,thereal-readyexistssimilarideasofpathintegration,albeititisformulatedtodealwithBrownianmotioninsteadofquantummechanics.ThisistheWienerintegral,formu-latedbyAmericanmathematicianNorbertWienerwhomademajorcontributionstostochasticandnoisepro-cessesaswellascybernetics[Infact,theone-dimensionalversionofBrownianmotionisknownastheWienerpro-cess,wewillreturntothislater].FeynmanhowevermadenomentionofWiener'sworksinhispaper.TheWienermeasureisnottranslationallyinvariance,andonewondersiftheFeynmanpathintegralcanbeunderstoodinasimilarway.Itturnsoutthattheanswerisno:in1960,Cameronprovedthatitisnotpossibletoconstruct\Feynmanmeasure"asaWienermeasurewithacomplexvariance,i.e.aslimitofnitedimensionalapproximationsoftheexpressionei ~Rt0m 2_ (s)2dsD Rei ~Rt0m 2_ (s)2dsD (3)astheresultingmeasurewouldhaveinnitetotalvari-ation,evenonboundedsetsinpathspace.ThisisnotthecasefortheusualLebesguemeasureonRd,whichhasnitetotalvariationonboundedmeasurablesubsetsofRd.Morediscussionsontheattemptstomakemath-ematicalsenseofthepathintegralformulationcanbefoundintherstchapterof[8].Onerelativelysimplewaytomakepathintegralmoresensibleistodoa\Wick-rotation"byanalyticcontinuationandconsiderinsteadadampingfactore�SinsteadofoscillatoryoneeiS,whereS=Rt0dtL(q;_q).OnethengetspreciselyaWienerpathintegration,whichdoesmakesense.Aftercalculationhasbeenperformed,onecanthenWick-rotatebackandreadothenalanswer.Unfortunately,therearesubtletiesinvolvedinthisapproachandnotallFeynmanpathin-tegralsallowWick-rotation.ItmustbeemphasizedthatFeynmanhimselfwasawareofthelackofrigorinhiswork,asevidencedfromhispaper[5]inwhichhewrotethat:[...]onefeelslikeCavalierimusthavefeltcalculatingthevolumeofapyramidbeforetheinventionofthecalculus.Ioftenfeelthatthisremarkisinasensetoomodest.Amoreappropriateanalogywouldbethatofcalculusinitsearlydays,morespecicallywhenitwasstillplaguedbyinnitesimals{averysmallquantitywhichisgreaterthanzeroyetlessthananypositivenumber,ifyouwill.Sometimeswestillthinkinthisway,especiallyinphysics(butthisisbecausewealreadyknowthatifwewish,wecouldalwaysmakeitrigorous).ThephilosopherBerkeleywastherstonetochallengethefoundationofcalculus.Heremarked:Theyareneithernitequantitiesnorquan-titiesinnitelysmall,noryetnothing.Maywenotcallthemtheghostsofdepartedquan-tities?Itwasduetocriticismlikethisthatnallyledtorig-orousformulationofcalculusintermsofandnowdreadedbybeginningmathematicsstudents[7].Never-theless,calculushasyieldedmanyamazingresultseversinceitwasinventedbyNewtonandLeibniz,despitelackingrigorousfoundationuntilBerkeley'sobjection.Thisispreciselythestatewearecurrentlyinforpathintegrationformulationofquantummechanics.III.REVIEWOFPATHINTEGRALFORMULATIONInviewofthediscussiononthemathematicaldi-cultiesininterpretingFeynmanpathintegration,wewillmakeWick-rotationbysetting=it=~andcalculateinsteadtheEuclideanpropagatorK(qi;qf;)=qfe�^H NNqi(4)=Dqfe�^H


e�^HqiE;= N(5)(6)WecannowinsertN�1copiesofthecompletenessre-lationZRdqijqiihqij=1(7)intothepropagatorandobtain 3K(qi;qf;t)=ZRdqN�1


ZRdq1Dqfe�^HqN�1EDqN�1e�^HqN�2E


Dq1e�^HqiE:(8) NoweachfactorDqi+1e�^HqiE=ZRdpDqi+1e�^HpEhpjqii(9)=e�V(qi)ZRdpe�p2 2m"eip(qi+1�qi) ~ 2~#(10)=e�V(qi) 2~"r 2m e�(qi+1�qi)2 (2=m)~2#(11)=1 2~r 2m e�m 2~2(qi+1�qi)2e�V(qi)(12)N()eL;(13)whereN()1 ~r m 2;(14)andL=�m 2~2qi+1�qi 2�1 2[V(xi+1)+V(xi)];(15)wherewehaveusedthemid-pointprescriptiontothepo-tentialtermdiscretization.Inthesecondequalityabovewehaveusedthefactthathxjpi=1 p 2~eipx ~;(16)whileinthethirdlinewehaveevaluatedtheGaussian-typeintegralviathestandardformulaZR(e�1 2ax2+iJx)dx=r 2 ae�J2 2a:(17)Hence,withq0=qiandqN=qf,wehaveK(qi;qf;)=ZRN�1Yn=1dqnN�1Yn=0Dqn+1e�^HqnE(18)=ZRN�1Yn=1dqnN�1Yn=0N()eL(qn+1;qn)(19)=ZRN�1Yn=1dqnm 2~2N 2ePN�1n=0L(xn+1;xn):(20)Thus,K(qi;qf;)ZDqe�S(21)whereZDqZRN�1Yn=1dqnm 2~2N 2;(22)andS=N�1Xn=0m 2~2qn+1�qn 2�N�1Xn=0V(qn):(23)Takingformallimit!0,S!Z0m 2~2dq d2+V(q):(24)UponWick-rotatebacktoMinkowskitimewenallyob-tainK=ZDqeiS=~;S=Zt0"m 2dq dt2�V(q)#dt(25)IV.WHEREISTHECOMMUTATIONRELATIONHIDING?Wenowbegintotrackdownthecommutationrelation.ThissectionisbasedontheusefulAppendixAof[9]aswellasontheoriginalpaperofFeynman[5].Forsimplicitywerstset~=1.Withoutlossofgeneralitywecantaketi=0andtf=T.Thenintheequationhqf;Tjqi;0i=Zdqhqf;Tjq;tihq;tjqi;0i:(26)Onemaywriteeachoftheamplitudesasapathintegralandthusnds 4Z[dq]qf;Tqi;0eiRT0dtL=ZdqZ[dq]qf;Tqi;teiRTtdtLZ[dq]qf;tqi;0eiRt0dtL;Tt0:(27) Thatis,thepathintegralon[0;T]breaksupintoseparatepathintegralson[0;t]and[t;T]andanordinaryintegraloverq(t).Considernowapathintegralwiththeadditionalin-sertionofafactorofq(t),where0tT.ThenwehaveZ[dq]qf;Tqi;0eiSq(t)=Zdqhqf;Tjq;tiqhq;tjqi;0i(28)=Zdqhqf;Tj^q(t)jq;tihq;tjqi;0i(29)=hqf;Tj^q(t)jqi;0i:(30)Thereforeweseethat:q(t)inthefunctionalinte-graltranslatesinto^q(t)inthematrixelement.Simi-larlywecanshowthatforaproductoftwoinsertionsq(t)q(t0)[which,beingvariablesofintegration,isequaltoq(t0)q(t)],wheret;t02[0;T],wehaveZ[dq]qf;Tqi;0eiSq(t)q(t0)=hqf;TjT[^q(t)^q(t0)]jqi;0i;(31)whereTdenotesthetime-orderedproductT[^A(t)^B(t0)]=(t�t0)^A(t)^B(t0)+(t0�t)^B(t)^A(t);(32)wheredenotestheHeavisideStepFunction.Thatis,Theorderoftermsinamatrixoperatorproductcorre-spondstoanorderintimeofthecorrespondingfactorsinthepathintegral.Indeed,duetothewaythepathintegralisconstructedoutofsuccessiveinnitesimaltimeslices,twoormorein-sertionsinthepathintegralwillalwayscorrespondtothetime-orderedproductofoperatorsinthematrixelement.Now,theequationofmotionisobtainedviatakingthefunctionalderivativeS q(t)=0:(33)Indeed,wehave,viaintegrationbypart,Z[dq]qf;Tqi;0eiSS q(t)F=�iZ[dq]qf;Tqi;0 q(t)eiSF(34)=iZ[dq]qf;Tqi;0eiSF q(t):(35)Thus,forinitialstate tandnalstate f,wehave,uponrestoring~, fF qk t=�i ~ fFS qk i:(36)Therefore,weseethattwodierentfunctionalsmaygivethesameresultforthetransitionelementbetweenanytwostates.Wesaythattheyareequivalentandsymbol-izetherelationby�~ iF qkS !FS qk:(37)HerethesymbolS !emphasizesthefactthatfunctionalsequivalentunderoneactionmaynotbeequivalentunderanother.Now,discretizing,wehaveS=PS(qi+1;qi)sothat�~ iF qkS !FS(qk+1;qk) qk+S(qk;qk�1) qk:(38)Thisequationiscorrecttozeroandrstorderin.InthisequationhidestheNewtonianequationsofmotion,aswellasthecommutationrelation.Werecallfromtheprevioussectionthatintheone-dimensionalquantummechanicalproblem,S(qk+1;qk)=m 2qk+1�qk 2�V(qk+1);(39)soweobtainS(qk+1;qk) qk=�m(qk+1�qk) ;(40)andS(qk;qk�1) qk=m(qk�qk�1) �V0(qk)(41)whereV0isthederivativeofthepotential,i.e.[minusof]force.Therefore,�~ iF qkS !F�mqk+1�qk �qk�qk�1 �V0(qk):(42)IfFdoesnotdependonqk,thisgivesNewton'sequa-tionsofmotion.SincetheLHSisnowzero,weget,upondividingbothsidesby,0S !�m qk+1�qk �qk�qk�1 �V0(qk);(43)i.e.V0(qk)S !�m qk+1�qk �qk�qk�1 :(44)Inotherwords,thetransitionelementofmasstimesac-celerationbetweenanytwostatesisindeedequaltothe 5transitionelementofforce�V0(qk)betweenthesamestates.Now,ifFdoesdependuponqk,sayF=qk,thenweget�~ iS !qk�mqk+1�qk �qk�qk�1 �V0(qk):(45)Neglectingtermsoforder,onehasmqk+1�qk qk�mqk�qk�1 qkS !~ i:(46)Takingextracareofthetimeorderingwhengoingbacktooperatorformulation,thisispreciselythecommutationrelation^p^q�^q^p=~ i:(47)V.AWIENERPROCESSAPPROACHTOWARDSNON-COMMUTATIVITYWenowbrie yexplainanothermethodtoextractthecommutationrelationoutofthepathintegral,whichislargelybasedon[10].InsteadoftheFeynman'spathin-tegral,letusconsidereditsWick-rotatedversion,inter-pretedasaWienerintegral.ConsidertheWienerpro-cess,whichisjustaone-dimensionalrandomwalk,withthe[Euclidean]actionS=�Zdq dt2dt:(48)Thepathq(t)is uctuating,withderivativedenedasthelimitofdiscretedierence:
q
t=q(t+)�q(t) :(49)Theproductq_qisactuallyambiguous:itdependsonthediscretization,sothatitcanbeinterpretedaseitherq(t)q(t+)�q(t) ;(50)orasq(t+)q(t+)�q(t) :(51)Therstcorrespondsto^q(t)^p(t)whilethesecondonerepresents^p(t)^q(t)sincetheoperatororderisthetimeorderingaswehavepreviouslydiscussed.Fromtheper-spectiveofStochasticcalculus,thevelocityisaforwarddierenceintheItosense,andthereforeisalwaysslightlyaheadintime.Thedierenceofthetwoyields(q(t+)�q(t))2 :(52)Inordinarycalculus,thedierencewillgotozerointhelimit!0.However,thisisnotthecaseforStochas-ticcalculus.Inparticular,thedistancearandomwalkmovesisproportionaltop t.[Remark:Onewaytoseethisisasfollows:Theran-domvariabledqinasense,representsanaccumulationofrandomin uencesovertheintervaldt.BytheCentralLimitTheorem,dqhasanormaldistribution.Thevari-anceofarandomvariable(whichistheaccumulationofindependenteectsoveranintervaloftime)ispropor-tionaltothelengthoftheinterval,i.e.dt.Thestandarddeviationofdqisthusproportionaltothesquarerootofdt].Consequently,q(t+)�q(t)p :(53)Thisinturnimpliesthat,wehavefromEq.(52),(q(t+)�q(t))2 p 2 =1;(54)insteadofzero.Wethusobtained[^q;^p]=1;(55)theEuclideanversionofthecommutationrelation.Thisisactuallyinessence,theconsequenceofthecelebratedIto'sLemma.UponWick-rotatingbacktoLorentziansignature,weobtain(~=1)[^q;^p]=i(56)inquantummechanics.WeremarkthattheequalitiesobtainedinEq.(55)andEq.(56)areactuallyonlyweakequality:theyarevalidonlyinthesensetheofsenseofdistributions.ForaBrownianmotion,theresultisactuallysayingthatthepositioniscorrelatedwiththe[innite]valueoftheve-locity[sincethepathsareactuallycontinuousbutnon-dierentiable,whichisclearfromEq.(53),sincetheratiothatdenesthederivativewilldivergeinthelimit!0],sothatthefuturepositionisactuallynitelycorrelatedwiththeaveragevelocitygiventhepastposition.Thepastvalueisofcoursecompletelyuncorrelatedwiththecurrent[forward]velocity.VI.CONCLUSIONWeconcludebymerelyemphasizingthatdiscretiza-tioniscrucialinthepathintegralformulation,foritisviacarefulanalysisonthediscretizationthatonecanre-coverthecommutationrelation[^q;^p]=i~ofquantummechanics.