DISTRIBUTION FUNCTIONSINPHYSICS:FUNDAMENTALSM.HILLERYInstituteforModernOptics,Universityof NewMexico,Albuquerque,NM87131,U.S.A.andMax-Planck InstitutfurQuantenoptik,D-8046GarchingbeiMunchen,WestGermanyR.F.O’CONNELLDepartmentofPhysicsandAstronomy,LouisianaStateUniversity,BatonRouge,LA70803,US.A.M.O. SCULLYMax-PlanckInstitutfurQuantenoptik,D-8046GarchingbeiMunchen,WestGermanyandInstituteforModernOptics,UniversityofNewMexico,Albuquerque,NM87131,U.S.A.E.P.WIGNERDepartmentofPhysicsandAstronomy,LouisianaStateUniversity,BatonRouge,LA70803,U.S.A.NORTH-HOLLAND PHYSICSPUBLISHING-AMSTERDAMPHYSICSREPORTS(ReviewSectionofPhysicsLetters)106,No.3(1984)121—167.North-Holland,AmsterdamDISTRIBUTIONFUNCTIONSINPHYSICS:FUNDAMENTALSM.HILLERYInstituteforModernOptics,UniversityofNewMexico,Albuquerque,NM87131,U.S.A.andMax-PlanckInstitutfurQuantenoptik,D-8046GarchingbeiMunchen,West GermanyR.F. O’CONNELLDepartmentofPhysics andAstronomy, LouisianaStateUniversity,BatonRouge,LA70803,U.S.A.M.O.SCULLYMax-PlanckInstitutfurQuantenoptik,D-8046GarchingbeiMunchen,West GermanyandInstituteforModernOptics, UniversityofNewMexico,Albuquerque,NM87131,U.S.A.E.P.WIGNER*DepartmentofPhysics andAstronomy, LouisianaStateUniversity,BatonRouge,LA70803,U.S.A.ReceivedDecember1983Contents:1.Introduction1234.1.Normalordering1562.Wignerdistribution126 4.2.Symmetricordering1582.1.Properties1264.3.Anti-normalordering1612.2.Associatedoperatorordering1324.4.Examples1622.3.Dynamics1354.5.Distributionfunctionsonfour-dimensionalphase space1632.4.Anexample1425.Conclusion1662.5.Statisticsandsecond-quantizednotation146References1663.Otherdistributionfunctions1504.Distributionfunctionsintermsofcreationandannihilationoperators152*Permanentaddress:DepartmentofPhysics,Joseph HenryLaboratory,PrincetonUniversity, Princeton,NJ08540,U.S.A.Single ordersforthisissuePHYSICSREPORTS(ReviewSectionofPhysicsLetters)106,No.3(1984)121—167.Copiesofthisissuemaybeobtainedat the pricegivenbelow.Allordersshouldbe sentdirectlyto thePublisher.Ordersmustbeaccompaniedbycheck.SingleissuepriceDfl.29.00,postageincluded.0370-1573/84/$14.40©ElsevierSciencePublishersB.V.(North-HollandPhysicsPublishingDivision)M.Hilleryeta!.,Distributionfunctionsinphysics: Fundamentals123Abstract:Thisisthefirstpartofwhatwillbe atwo-partreviewofdistributionfunctionsinphysics.Herewedealwithfundamentalsand thesecondpartwilldealwithapplications.Wediscussindetailthe propertiesof thedistributionfunctiondefinedearlierbyone ofus(EPW)andwederivesomenewresults.Next,wetreatvariousotherdistributionfunctions.Amongthe latterweemphasizetheso.calledPdistribution,aswellas thegeneralizedPdistribution,becauseof theirimportanceinquantumoptics.1.IntroductionItiswellknownthat theuncertaintyprinciplemakestheconceptofphase spaceinquantummechanicsproblematic.Becausea particlecannot simultaneouslyhave awelldefined positionandmomentum, onecannotdefineaprobabilitythat aparticlehas apositionq and amomentump,i.e.onecannotdefinea truephase spaceprobability distributionfor a quantummechanicalparticle.Nonethe-less,functionswhichbearsomeresemblance tophase spacedistributionfunctions,“quasiprobabilitydistribution functions”,haveprovento beofgreatuseinthestudyofquantummechanicalsystems.Theyareusefulnotonlyascalculationaltoolsbutcanalsoprovideinsights intotheconnectionsbetweenclassicaland quantummechanics.Thereasonfor thislatterpointisthatquasiprobabilitydistributionsallowone toexpressquantummechanicalaveragesinaformwhichisverysimilarto that forclassicalaverages.As aspecificexampleletusconsideraparticleinonedimensionwithitspositiondenotedbyq anditsmomentumbyp.Classically,the particleisdescribedbyaphase spacedistributionPc,(q,p). TheaverageofafunctionofthepositionandmomentumA(q,p)canthen beexpressedas(A)c!=JdqJdpA(q,p)Pc1(q,p).(1.1)Theintegrationsinthis equationarefrom—~to+~.Thiswillbe thecasewithallintegrationsinthispaperunlessotherwiseindicated.Aquantummechanicalparticleisdescribedbyadensitymatrix~5(wewilldesignatealloperatorsbya ~)and theaverageofafunctionofthepositionandmomentumoperators,A(4,j3)is(A>quant=Tr(A15)(1.2)(TrOmeansthetraceoftheoperatorO).Itmustbeadmittedthat,givenaclassicalexpressionA(q,p),thecorrespondingself-adjointoperatorAisnotuniquely defined—anditisnotquiteclear whatthepurposeofsuchadefinitionis.Theuseofaquasiprobabilitydistribution,P0(q,p),however,doesgivesuchadefinitionbyexpressingthe quantummechanicalaverageas(A)quant=JdqJdpA(q,p)Po(q,p)(1.3)where thefunctionA(q,p)canbe derivedfromtheoperatorA(4,j3)byawelldefined correspondencerule.Thisallowsone tocastquantummechanicalresultsintoaforminwhichtheyresembleclassicalones.The firstofthesequasiprobabilitydistributionswasintroducedbyWigner[1932a}tostudyquantumcorrections toclassicalstatisticalmechanics.Thisparticular distributionhascometo beknownasthe124M.Hilleryetal.,Distributionfunctionsinphysics:FundamentalsWignerdistribution,tandwewilldesignateitasP~.Thisis,andwasmeant tobe,areformulationofSchrödinger’squantummechanicswhichdescribesstatesbyfunctionsinconfigurationspace.Itisnon-relativisticinnaturebecauseitisnotinvariantunder theLorentzgroup;also, configurationspacequantummechanicsfor more than one particlewouldbedifficultto formulaterelativistically. However,ithasfoundmany applicationsprimarilyinstatisticalmechanicsbutalsoinareassuchasquantumchemistryand quantumoptics.Inthecasewhere P0ineq.(1.3)ischosentobeP~,,then thecorrespondencebetweenA(q,p) andAisthatproposedbyWeyl[19271,aswasfirstdemonstratedbyMoyal[1949].Quantumopticshasgivenrise to a numberofquasiprobabilitydistributions,themostwell-knownbeingthePrepresentationofGlauber [1963a1andSudarshan[1963],whichhavealsofoundextensiveuse.Asfarasthedescriptionoftheelectromagneticfieldisconcerned, thesedoexhibit(special)relativistic invariance.Otherdistribution functionshavealsobeen proposed(Husimi[19401;MargenauandHill[19611;Cohen[1966])buthave found morelimited use,although,morerecently,extensiveusehas been madeofthegeneralizedP representationsbyDrummond,Gardiner andWalls[1980,1981].Inthispaperwewilldiscussthebasicformalismofthesequasiprobabilitydistributionsandillustratethem withafewsimpleexamples.Wewilldeferanydetailedconsiderationofapplicationsto alaterpaper.Wenowproceed to thebasicproblem:howdowegoaboutconstructingaquantummechanicalanalogueofaphase spacedensity?Letusagainconsider,forsimplicity,a one particlesysteminonedimensionwhichisdescribedbyadensitymatrixf.5.Inthispaperwewillwork,forsimplicity,inonedimension;thegeneralizationtohigher