photon_wf.pdf

ProgressinOptics,Vol.XXXVI,Ed.E.Wolf

Elsevier,Amsterdam1996

PHOTONWAVEFUNCTION

IwoBialynicki-Birula

CenterforTheoreticalPhysics,PolishAcademyofSciences

Al.Lotnik¶ow32,02-668Warsaw,Poland

and

RochesterTheoryCenterforOpticalScienceandEngineering

UniversityofRochester,Rochester,NY14627,USA

Contents

0.1Introduction.............................. 3

0.1.1COORDINATEVS.MOMENTUMREPRESENTATION4

0.1.2PHASEREPRESENTATION................ 5

0.1.3LANDAU-PEIERLSWAVEFUNCTION......... 5

0.1.4RIEMANN-SILBERSTEINWAVEFUNCTION..... 6

0.2Waveequationforphotons ..................... 8

0.2.1WAVEEQUATIONINFREESPACE........... 8

0.2.2WAVEEQUATIONINAMEDIUM............ 10

0.2.3ANALOGYWITHTHEDIRACEQUATION...... 12

0.3Coordinaterepresentation...................... 13

0.3.1PHOTONSHAVENOANTIPARTICLES......... 13

0.3.2TRANSFORMATIONPROPERTIES........... 14

0.3.3PHOTONHAMILTONIAN................. 15

0.4Momentumrepresentation...................... 16

0.4.1FOURIERINTEGRAL................... 16

0.4.2 INTERPRETATIONOFFOURIERCOEFFICIENTS.. 18

0.4.3TRANSFORMATIONPROPERTIESINMOMENTUM

SPACE............................ 19

0.5Probabilisticinterpretation..................... 20

0.5.1SCALARPRODUCT.................... 20

0.5.2EXPECTATIONVALUES................. 21

0.5.3CONNECTIONWITHLANDAU-PEIERLSWAVEFUNC-

TION............................. 24

0.6Eigenvalueproblems......................... 25

0.6.1MOMENTUMANDANGULARMOMENTUM..... 25

0.6.2MOMENTOFENERGY.................. 26

0.6.3PROPAGATIONINOPTICALFIBER.......... 27

0.7Relativisticinvariance........................ 29

0.8Localizabilityofphotons....................... 30

0.9Phase-spacedescriptionofaphoton................ 31

0.10Hydrodynamicformulation..................... 33

0.11Wavefunctionincurvedspace ................... 35

0.12Wavefunctionasaspinor...................... 36

0.13Wavefunctionsandmodeexpansion................ 37

0.14Summary............................... 39

0.1Introduction

Photonwavefunctionisacontroversialconcept.Controversiesstemfrom

thefactthatphotonwavefunctionscannothaveallthepropertiesofthe

SchrÄodingerwavefunctionsofnonrelativisticwavemechanics. Insistenceon

thosepropertiesthat,owingtopeculiaritiesofphotondynamics,cannotberen-

dered,ledsomephysiciststotheextremeopinionthatthephotonwavefunction

doesnotexist.Irejectsuchafundamentalistpointofviewinfavorofamore

pragmaticapproach.Inmyview,thephotonwavefunctionexistsaslongas

itcanbepreciselyde¯nedandmadeuseful.Manyauthorswhosepapersare

quotedinthisreviewsharethesameopinionandhadnoreservationsaboutusing

thenamethephotonwavefunctionwhenreferringtoacomplexvector-function

ofspacecoordinatesrandtime t thatadequatelydescribesthequantumstate

ofasinglephoton.

Thenotionofthephotonwavefunctioniscertainlynotnew,butstrangely

enoughithasneverbeensystematicallyandfullyexplored.Sometextbooks

onquantummechanicsstarttheintroductiontoquantumtheorywithadis-

cussionofphotonpolarizationmeasurements(cf.,forexample[Dirac[1958]],

[Baym[1969]],[Lipkin[1973]],[Cohen-Tannoudji,DiuandLaloÄe[1977]]),but

inalltheseexpositionsacompletephotonwavefunctionnevertakesonaspe-

ci¯cmathematicalform.EvenDiracwhowrites\Theessentialpointisthe

associationofeachofthetranslationalstatesofthephotonwithoneofthe

wavefunctionsofordinarywaveoptics",neverexpressesthisassociationinan

explicitform.Inthiscontexthealsousesthenowfamousphrase:\Eachphoton

interferesonlywithitself"whichimpliestheexistenceofphotonwavefunctions

whosesuperpositionleadstointerferencephenomena.

Inthetextbookanalysisofpolarization,onlysimpleprototypetwo-com-

ponentwavefunctionsareusedtodescribevariouspolarizationstatesofthe

photonandwiththeirhelpthepreparationandthemeasurementofpolariza-

tionisanalyzed.However,itisnotexplained,whyawavefunctionshouldnot

beusedtodescribealsothe"translationalstatesofthephoton"mentionedby

Dirac.Aftersuchaheuristicintroductiontoquantumtheory,theauthorsgoon

tothestudyofmassiveparticlesandiftheyeverreturntoquantumtheoryof

photonsitisalwayswithintheformalismofsecondquantizationwithcreation

andannihilationoperators.Insometextbooks(cf.,forexample,[Bohm[1954]],

[Power[1964]])onemayeven¯ndstatementsthatcompletelynegatethepossi-

bilityofintroducingawavefunctionforthephoton.

Astudyofthephotonwavefunctionshouldbeprecededbyanexplanation

whatisthephotonandwhyadescriptionofthephotonintermsofawavefunc-

tionmustexist.Accordingtomodernquantum¯eldtheory,photons,together

withallotherparticles(andalsoquasiparticles,phonons,excitons,plasmons,

etc.),arethe quantumexcitations ofa¯eld.Inthecaseofphotons,thesearethe

excitationsoftheelectromagnetic¯eld.Thelowest¯eldexcitationofa given

type correspondstoonephotonandhigher¯eldexcitationsinvolvemorethan

onephoton.Thisconceptofaphoton(calledthemodernphotoninatutorial

reviewby[Kidd,ArdiniandAnton[1989]])enablesonetousethephotonwave

functionnotonlytodescribequantumstatesofanexcitationofthefree¯eld

butalsooftheelectromagnetic¯eldinteractingwithamedium.Conceptually,

thedi®erencebetweenfreespaceandamediumisnotessentialsincethephysi-

calvacuumislikeapolarizablemedium.Itis¯lledwithallthevirtualpairs|

zeropointexcitationsofchargedquantum¯elds.Therefore,eveninfreespace,

photonscanbealsoviewedastheexcitationsofthevacuummademostlyof

virtualelectron-positronpairs([Bialynicki-Birula[1963]],[Bjorken[1963]]).

Eventhough,inprinciple,allparticlescanbetreatedas¯eldexcitations,

photonsaremuchdi®erentfrommassiveparticles.Theyarealsodi®erentfrom

masslessneutrinossincethephotonnumberdoesnotobeyaconservationlaw.

Thereareproblemswiththephotonlocalizationandasaresulttheposition

operatorforthephotonisill-de¯ned,butthesimilaritiesbetweenphotonsand

otherquantumparticlesaresoamplethattheintroductionofthephotonwave

functionseemstobefullyjusti¯edandevennecessaryinordertoachievea

completeuni¯cationofourdescriptionofallparticles.

0.1.1COORDINATEVS.MOMENTUMREPRESENTA-

TION

Innonrelativisticquantummechanicstheterm coordinaterepresentation isused

todenotetherepresentationinwhichthewavefunction Ã (r)isde¯nedasa

projectionofthestatevector jÃi ontheeigenstates j r i ofthecomponents^ x; ^ y ,

and^ z ofthepositionoperator ^ r,

Ã (r)= h r jÃi: (1)

Thewavefunctionincoordinaterepresentation,therefore,becomesautomat-

icallyafunctionoftheeigenvaluesofthepositionoperator ^ r.Theposition

operatorsactonthewavefunctionsimplythroughamultiplication.Inquan-

tummechanicsofphotonsthisapproachdoesnotworkduetodi±cultieswith

thede¯nitionofthephotonpositionoperator(cf. x 0.8).Onemaystill,how-

ever,introducefunctionsofthecoordinatevectorrtodescribequantumstates

ofthephoton.Byadoptingthislessstringentpointofviewthatdoesnottie

thewavefunctionincoordinaterepresentationwiththeformula(1),oneavoids

theconsequencesofthenonexistenceofthephotonpositionoperator^r. In

principle,anyfunctionofrthatadequatelydescribesphotonstatesmaybe

calledaphotonwavefunctionincoordinaterepresentationanditisamatter

oftasteandconveniencewhichonetouse.Itshouldbepointedoutthatin

arelativisticquantumtheory,evenforparticleswithnonvanishingrestmass,

thepositionoperatorandthelocalizationassociatedwithitdonotliveupto

ournonrelativisticexpectations.Thedi®erencesinlocalizationofphotonsand,

sayelectrons,aremorequantitativethenqualitativesincetheyamounttothe

"spillingofthewavefunction"beyondthelocalizationregiongovernedbya

powerlawversusanexponentialdecay.

Thephotonwavefunctionin momentumrepresentation hasnotstirredany

controversysincethephotonmomentumoperator^piswellde¯ned.Itsexis-

tence,asthegeneratoroftranslations,followsdirectlyfromthegeneraltheory

ofrepresentationsofthePoincar¶egroupdevelopedby[Wigner[1939]].Ithas

alwaysbeentakenforgrantedbyallphysicistsworkinginrelativisticquantum

electrodynamicsthatthenotionofthephotonwavefunctioninmomentum

representationiswellfounded.Suchwavefunctionsdescribinginitialand¯-

nalstatesofphotonsappearinallformulasfortransitionamplitudesinthe

S-matrixtheoryofscatteringphenomena(cf.,forexample,[Schweber[1961]],

[AkhiezerandBerestetskii[1965]],[Bialynicki-BirulaandBialynicka-Birula[1975]],

[Cohen-Tannoudji,Dupont-RocandGrynberg[1989]]).Thus,onemaysafely

assertthatthephotonwavefunctioninmomentumrepresentationisawell

de¯nedandfullyestablishedobject.

0.1.2PHASEREPRESENTATION

Thephotonwavefunctionsdiscussedinthisreviewaredistinctfromthe one-

mode wavefunctionsthathavebeenintroducedinthepast([London[1927]],

[Bialynicki-BirulaandBialynicka-Birula[1976]],[PeggandBarnett[1988]])to

describe multi-photonstates .Thesefunctionsdependonthephase ' ofthe

¯eldandwerecalledthewavefunctionsinthephaserepresentationby

[Bialynicki-BirulaandBialynicka-Birula[1976]].Thewavefunctionsª( ' )char-

acterizequantumstatesofa selectedmode ofthequantizedelectromagnetic¯eld

and,ingeneral,theydescribeasuperpositionofstateswithdi®erentnumbers

ofphotons.Allspatialcharacteristicsofthesestatesarecontainedinthemode

functionthatde¯nestheselectedmodeoftheelectromagnetic¯eld.One-mode

wavefunctionsª( ' )describepropertiesofmulti-photonstatesofthequantized

electromagnetic¯eldwithallphotonsbeinginthesamequantummechanical

state.Incontrast,thephotonwavefunctioninthecoordinaterepresentation

canbeidenti¯edwiththemodefunctionitself(cf. x 0.13).Itdescribesastate

ofa single photonandnotastateofthequantized¯eld.

0.1.3LANDAU-PEIERLSWAVEFUNCTION

Theconceptofthephotonwavefunctionincoordinaterepresentationwasintro-

ducedforthe¯rsttimeby[LandauandPeierls[1930]].Thesamefunctionhas

beenindependentlyrediscoveredmorerecentlyby[Cook[1982a],Cook[1982b]],

and[Inagaki[1994]].TheLandau-Peierlsproposalhasnotbeenmetwithgreat

enthusiasmsincetheirwavefunctionisahighlynonlocalobject.

ThenonlocalityoftheLandau-Peierlswavefunctionisintroducedbyoper-

atingonthelocalelectromagnetic¯eldwiththeintegraloperator( ¡ ¢) ¡ 1 = 4 ,

(( ¡ ¢) ¡ 1 = 4 f )(r)= ¼

Z d 3 r 0

(2 ¼j r ¡ r 0 j ) 5 = 2 f (r 0 ) : (2)

j k j oftheFouriertransform

anditchangesthedimensionofthewavefunctionfrom L ¡ 2 ,characteristic

oftheelectromagnetic¯eld,to L ¡ 3 = 2 .Thereforethemodulussquaredofthe

Landau-Peierlswavefunctionhastherightdimensionalitytobeinterpreted

Thisintegraloperatorcorrespondstoadivisionby

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