BAYM=conservation_laws_and_quantum_theory_of_trans

CONSERVATIONLAWSANDTHEQUANTUMTHEORYOFTRANSPORT:THEEARLYDAYSGordonBaymDepartmentofPhysics,UniversityofIllinoisatUrbana-Champaign,1110W.GreenSt.,Urbana,Illinois61801,USAThistalkreviews,fromanhistoricalperspective,achapterinthedevelopmentofquantumtransporttheorywithintheframeworkofself-consistentnon-equilibriumGreen’sfunctions.1INTRODUCTIONNearlyfortyyearshavespedbysinceLeoKadanoffandIworkedinCopen-hagenonunderstandingtheroleofconservationlawsandthedescriptionoftransportinquantummanyparticlesystems.Tosetthestageforthismeeting,Iwouldliketodescribeherethephysicsandhistoricalbackgroundofourworkintheseearlydays.Firstafewbiographicalnotes.Asthephotographsthatgracedthedustjacketofour1962bookQuantumstatisticalmechanics1show,LeoandIwerebothinourmid-twentiesatthetime.Whenthetwoofusstartedlifeinthe1930’s,ourparentsquitecoincidentallylivedjustaboutablockapartinIn-wood,aneighborhoodinupperManhattaninNewYorkCity,mineonPaysonAvenue(no.1onthemap)andhisonSeamanAvenue(no.2).Ourimmedi-ateneighborhoodturnedouttobequiteafertileonefortheoreticalphysicists.RoyGlaubergrewupinthesameapartmentbuildingastheKadanoffs,andShellyGlashowlivedjustablockorsoaway.Weallwenttothesameelemen-taryschool(no.3onthemap),imaginativelynamedPS52(PSfor“PublicSchool”),andallwereatHarvardinthe1950’s.ThetheoristsatHarvardinthelatefiftieswereveryexcitedbyrecentdevelopmentsinthemanybodyproblem,andparticularlybythepaperbyPaulMartinandJulianSchwinger2ontheformalismforfinitetemperaturemanybodytheory.3.EventhoughLeoandIweregraduatestudentsinphysicsatthesametimewedidnotcollaboratetogetheruntillater.LeoworkedwithPaulMartin,writingathesis,Theoryofmanyparticlesystems:superconductivity,andwithRoyGlauberaswell,writingasecondthesis,Accelerationofaparticlebyaquantizedelectricfield;IwasastudentofJulianSchwinger,andwroteonlyonethesis,Fieldtheoreticapproachtothepropertiesofthesolidstate.4Discussingfutureplansatourgraduationin1960,wefoundoutthatwewerebothindependentlyheadedaspostdocsinSeptembertoNielsBohr’sIn-stituteinCopenhagen–thenofficiallytheUniversityInstituteforTheoretical1Figure1:LeoKadanoff(left)andGordonBaym(right),circa1961;fromthedustjacketofQuantumStatisticalMechanics.Physics.Copenhagen,withfarmorebicyclesthancarsonthestreets,wasoneoftheforemostgatheringplacesofphysicistsfromaroundtheworld5–aremarkableexperiencefortwofreshHarvardPh.D.’s.Istayedfortwoyears,andLeoforalittleoveroneyear.2THEPUZZLESSincethemainemphasisattheInstituteatthetimewasnuclearphysicsandourfocuswasoncondensedmatterproblemswehadlittlementoringfromtheseniorpeople,withtheimportantexceptionofGerryBrown;primarilyweworkedonourown.Leo’sandmycollaborationbeganonedayinourfirstSpringinCopenhagen,whenheraisedthequestionofhowtoconstructintheMartin-SchwingerGreen’sfunctionformalismapproximationstothetwoparticlepropagatorsthatpreservedthesimpleconservationlaws.Forexample,theoperatornumberconservationlaw,∂ρ(rt)∂t+∇·j(rt)=0,(1)impliesthatcorrelationfunctionsofthedensityandcurrentwithanarbitraryoperator,O(r
t
),shouldobey∂∂tT(ρ(rt)O(r
t
))+∇·T(j(rt)O(r
t
))=[ρ(rt),O(r
t)])δ(t−t
).(2)2TheBronxManhattanHudsonRiverHarlemRiverBroadway207thSt.500mInwood123DyckmanSt.Figure2:TheInwoodareaofupperManhattan.whereTdenotestime-orderingoftheoperators.Leo’sinterestintheproblemarosefromthequestionofhowtowritetheBCStheoryofsuperconductivityinawaythatwouldyieldcorrectlytheAnder-sonmode–thelongitudinalcollectiveoscillationofaneutralsuperconductor6(nowfamousinhighenergyphysicsastheHiggsboson).Asoriginallyfor-mulatedtheBCStheorydidnotgivetwo-particlecorrelationfunctionsthatobeyedthenumberconservationcondition(2).Equivalently,theproblemwashowtobuildlocalgaugeinvarianceintothetheory.Themoregeneralissuewashowtoconstruct,withintheGreen’sfunctionformalism,consistentquan-tumtheoriesoftransport.Thisproblemwasnaggingmetoo,because,asPaulMartinpointedoutatmydoctoraldefense,Ihadessentiallygottenitwronginmythesis.Intryingtocalculatesoundwavedampinginametalviasimple3Green’sfunctionapproximations,Iwasmissingafactorof4/5comparedwithacalculationviatheBoltzmannequation.Paulimmediatelyrealizedthatthelackofthisfactorindicatedaconceptualerrorintheunderlyingphysics.Itisveryhardtogetafactoroffivefromsimplemistakes–inthiscaseitcomesfrom
P2(cosθ)2,whicharisesfromcorrectlyincludingthescatteringback-into-the-beamterminthetransportequation.Leo’sandmyproblemswerecloselyrelated.xx'(a)(b)(c)Figure3:(a)Theoneloopapproximation,(b)theself-energycorrectionsonthelinesthatareincluded,and(c)scatteringinto-the-beamcorrections.Toseehowapproximationscanfailobeytotheconservationlawsconsidergoingbeyondthelowestorderevaluationofthedensity-densityandcurrent-densitycorrelationfunctionsintheHartree-Fockapproximation,i.e.,asasingleloopofGreen’sfunctions.Thelowestordercalculation,Fig.3a,inwhichthelinesareHartree-FockGreen’sfunctions,obeystheconservationlaws.ThetroublestartswhenonetriestoimprovetheGreen’sfunctionsbeyondlowestorder,includingbetterself-energies,asshowninFig.3b,butnotvertexcorrec-tions.Intheoneloopapproximation,thedensity-densitycorrelationfunction,inimaginarytime,isg