CQO7.pdf

CoherenceandQuantumOpticsVII ,Eds.J.H.Eberly,L.Mandel,andE.Wolf

Plenum,NewYork1996,p.313

THEPHOTONWAVEFUNCTION

IwoBialynicki-Birula

CenterforTheoreticalPhysics,PolishAcademyofSciences

Lotnik¶ow32/46,02-668Warsaw,Polandand

AbteilungfÄurQuantenphysik,UniversitÄatUlm,89069Ulm,Germany

INTRODUCTION

Quantizationoftheelectromagnetic¯eldistraditionallyintroducedatthelevelofsecondquan-

tization:theclassical¯eldvariablesarereplacedby¯eldoperators.Ibelievethatthereasonswhy

a¯rst-quantizedtheoryofphotonshasneverbeenfullydevelopedaremainlyhistorical.HadDirac

discoveredhisrelativisticwaveequation[1]priortohisquantizationoftheelectromagnetic¯eld[2],he

wouldhavenoticedandmostprobablyfurtherexploredagreatsimilaritybetweenthewaveequation

fortheelectron(orevenbetterfortheneutrino)andtheMaxwellequations.Asithappened,this

similaritywasnoticedlater(forthe¯rsttimeapparentlybyMajorana[3])andplayednoroleinthe

developmentofthequantumtheoryofelectromagnetismbecausethequantizedelectromagnetic¯eld

hasbeenintroducedfromtheverybeginningandaccountedforallquantumpropertieselectromag-

neticradiation.Subsequentlyquantumelectrodynamicshasbecomesosuccessfulinexplainingwith

utmostaccuracyallexperimentswithinitsrangeofapplicabilitythattherewasnoneedtosearchfor

analternativeformulationthatwouldemploytheconceptofthephotonwavefunction.Considering

ourtrustinquantumelectrodynamicsandourfamiliaritywithitsformalapparatusonemayevenask

ifthereisanyjusti¯cationatallfor,whatitessentiallyamountsto,areconstructionofthenotionof

thephotonfunctionfromQED,onlytofaceanotsofamiliarobjectwhosepropertiesareyettobe

uncovered.

Icanseethreereasonswhytheintroductionofaphotonwavefunctionisworththee®ort.First

ofall,oneachievesauni¯eddescriptionofallparticles(massiveandmassless)atbothlevels(¯rst

andsecond)ofquantization.Thisisparticularlyattractivefromthepedagogicalpointofviewsince

onecanusethesamemathematicaltoolstostudythequantumpropertiesofmassiveandmassless

particles.Second,variousaspectsofthephotondynamics(forexample,theangularmomentumof

thephoton,evanescentwaves,orthepropagationofphotonsinnoninertialframesofreference)can

bedescribedmuchmoreeasilywithintheframeworkofphotonwavemechanicswithoutbringingin

theformalismof¯eldquantization.Third,armedwiththeconceptofthephotonwavefunctionand

allthetoolsofthestandardwavemechanics,onecancomeupwithsomenewmethodsofdescribing

photons.Forexample,onecanstudyquantummechanicaleigenvalueproblemsforthephotons,one

canintroduceananalogoftheWignerfunctionforthephoton,andonecan¯ndacounterpartofthe

hydrodynamicformulationofwavemechanicsforphotons.MuchofwhatIwillsayhereisnotnew;

itcanbeobtainedby"reverseengineering"fromtextbooksonquantumelectrodynamics.Therewill

be,however,asigni¯cantchangeinemphasisleadingtosomenewconclusions.

Photonwavefunctionsappearindisguiseinthestandardformulationofquantumelectrodynamics

asthecoe±cients(calledmodefunctions)inanexpansionoftheelectromagnetic¯eldoperatorsinto

photoncreationandannihilationoperators.Ittakessomee®orttorecognizegenuinewavefunctions

inthesecoe±cientfunctionssincethemodefunctionsarealwaysassumedtodescribemonochromatic

radiation[4{7].Thus,contrarytothespiritofwavemechanics,arbitrarysuperpositionsofthemode

functionsareforbiddensincethey,ingeneral,leadtononstationarywavepackets.Thisrestriction

may,however,beremoved(seeforexample[8]);theonlyessentialpropertyofthecoe±cientfunctions

beingthattheyformacomplete(andpreferablyalsoanorthonormal)set.

Inthispaper,Ishallarguethatnotonlyonemayreconstructphotonwavefunctionsfromstandard

quantumelectrodynamicsbutthatonecansetupaconsistentwavemechanicsofphotonsthatcould

beusedtodescribevariousquantume®ects, independently oftheformalismofsecondquantization.

Inotherwords,inconstructingafullquantumtheoryofphotonsonemayalsoproceed,asinquantum

theoryofmassiveparticles,throughtwostages.Atthe¯rststageoneintroduceswavefunctionsanda

waveequationobeyedbythesewavefunctions.Atthesecondstageoneupgradesthewavefunctions

tothelevelof¯eldoperatorsinordertodealmoree®ectivelywithstatesinvolvingmanyparticlesand

toallowforprocessesinwhichthenumberofparticlesisnotconserved.

Theveryconceptofthephotonwavefunctionisnotnew,butstrangelyenoughithasneverbeen

systematicallyexplored.Manytextbooksonquantummechanicsstarttheintroductiontoquantum

theorywithadiscussionofphotonpolarizationmeasurements(cf.forexample[9{12])butacomplete

photonwavefunctionnevermakesitsappearance,asifthenotionofawavefunctionwasrestricted

onlytothedescriptionofthepolarizationstatesinasimpletwo-dimensionalHilbertspaceandcould

notdescribethephotonpropagationinfull.Insometextbooks(cf.forexample[13{15])thepossibility

ofintroducingaspace-dependentwavefunctionforthephotonisevenexplicitlyrejected.

Ishallintroduceawavefunctionforthephotonbyrevivingandextendingthemodeofdescription

oftheelectromagnetic¯eldbasedonthecomplexformoftheMaxwellequationsknownalready

toRiemann[16,17].Thecomplexvectorthatappearsinthisdescriptionwillbeshowntohavethe

propertiesthatonewouldassociatewithaone-photonwavefunction,includingofcourseaprobabilistic

interpretation.Additionalpropertiesofthephotonwavefunctionaredescribedinmyrecentarticle

[18]whileitsrelationtofullquantumelectrodynamicsisdiscussedinourbook[8].

TheapproachadoptedhereistobecontrastedwiththatofLandauandPeierls[19]andCook[20].

TheLandau-PeierlsandCookwavefunctionsarehighlynonlocalobjects.Theno n localityinspace

isintroducedbydividingtheFouriertransformoftheelectromagnetic¯eldby

PHOTONWAVEFUNCTIONANDITSTIMEEVOLUTION

MydiscussionofthephotonwavefunctionbeginswithwritingtheMaxwellequationforahomo-

geneousmediuminacompactform

i@ t F(r ;t )= cr£ F(r ;t ) ; (1)

r¢ F(r ;t )=0 ;

(2)

bycombining,asRiemannhavedone,tworealvectorsDandBintoonecomplex-valuedvectorF,

F(r ;t )= 1

p ² 0 + i B(r ;t )

p ¹ 0

: (3)

Thesquarerootsof ² and ¹ areneededtomatchthedimensionsofthetwotermsandanadditional

factorof1 = 2isintroducedforfutureconvenience.

Theevolutionequation(1)forFandthetransversalitycondition(2)canalsobewritteninamatrix

form

s ¢ 1

i@ t F= c

i r

F ; (4)

(s ¢r ) s j F= r j F ; (5)

where s i 'sarethespinmatricesforaspin-1particle

( s i ) kl = ¡i" ikl ;

(6)

k (cf.Eq.(43)).

IthasbeenalreadynotedbyPauli[21],thatthesenonlocalwavefunctionshaveaseriousdrawback

sincetheydonottransformunderLorentztransformationsaswell-de¯nedlocalgeometricobjects;

theirvaluestakenatapointinonecoordinatesystemdependonalltheirvaluesinanothercoordinate

system.

µ D(r ;t )

000

00 ¡i

0 i 0

00 i

000

¡i 00

0 ¡i 0

i 00

000

s x =

A ;s y =

A ;s z =

A : (7)

UponthemultiplicationofbothsidesofEq.(4)by¹ h ,onecanconvertittotheSchrÄodingerform

i ¹ h@ t F= H F ´c (s ¢ p)F ; (8)

wherep=(¹ h=i ) r .Thescalarproduct c s ¢ pplaystheroleoftheHamiltonianinwavemechanicsof

thephoton.Note,that c smayalsobeinterpretedasavelocityoperatorsince( i= ¹ h )[r ;H ]= c s.

Theform(8)ofthewaveequationisquiteuniversal.ReplacingthematricessbythePaulimatrices

oneobtainstheWeylequationforthewavefunctionofaneutrino[22].Withtheappropriatechoice

ofspinmatrices,thewaveequationsforhigher-spinmasslessparticlescanalsobecastintothisform

(cf.forexample[23]).Upontheidenti¯cationofthecomponentsofthevectorFwithcomponentsof

asymmetricrelativisticspinor Á AB

Á 00 = ¡F x + iF y ; Á 01 = F z ; Á 11 = F x + iF y ; (9)

Maxwellequationsbecomeamemberofauniversalsetofrelativisticwaveequationsdescribingthe

propagationofmasslessparticlesofanyspin.Alltheseequationshavetheform[23]

¾ ¹C 0 A r ¹ Á AB 1 B 2 ¢¢¢B n¡ 1 =0 ; (10)

wherefourmatrices ¾ ¹A 0 B representarelativisticextensionofPaulimatrices( ¾ 0 istheunitmatrix

andtheremainingonesareordinaryPaulimatrices).Thisuniversalityprovidesastrongargumentin

favorofinterpretingthesolutionsofEq.(4)asphotonwavefunctions.Afterallonehasnoqualms

aboutacceptingthesolutionsoftheWeylequationasneutrinowavefunctions.

Eqs.(4)and(5)arecompletelyequivalenttotheMaxwellequationsbutwhentreatedasthe

equationsforthephotonwavefunctiontheysu®erfromoneseriousshortcoming:theypossessonly

halfofthepositive-energysolutionsneededtodescribeallpolarizationstatesofaphoton.Thesame

istrueinthegeneralcase;Eqs.(10)describeonlyleft-handedparticles.

Positiveenergysolutionsareidenti¯edasthosesolutionswhosewavefunctionhastheform

exp( ¡i!t )F(r)withpositivevaluesof ! .Thesubstitutionofthisforminto(4)yields

i r )F=¹ h! F : (11)

Thisequationsaysthatforpositiveenergysolutionsthehelicity,i.e.theprojectionofthespinofthe

photononthedirectionofitsmomentum,isalwayspositive.Thesignofhelicitycanbetracedback

tothechoiceofeitherForitscomplexconjugateasthephotonwavefunction.Thus,thechoiceofa

signoftheimaginarypartin(3)isequivalenttochoosingonehelicity(right-handedorleft-handed)

overtheother.Oneneeds,however,wavefunctionsofthephotonsofbothhelicitiestodescribe

bothcircularpolarizations:rightandleft.Thisrequiresadoublingofthewave-functioncomponents:

thetwohelicitieswillbedescribedbyupperandlowercomponentsof onewavefunction .Inempty

spacebothpolarizationstatespropagateindependently;thetwopartsofthephotonwavefunction

satisfytwoseparateevolutionequationswithoppositesignsoftheHamiltonian.Byintroducinga

six-componentwavefunction F madeofF § asitsupper/lowercomponents,

µ F +

F ¡

F =

;

(12)

thetwoevolutionequationscanbecombinedtogether

i ¹ h@ t F = c½ 3 (s ¢ ¹ h

i r ) F; (13)

where ½ 3 isamemberofasetofPauli-typematrices ½ i .Thematrix ½ 3 givesplus/minussignwhen

actingonupper/lowercomponentswhile ½ 1 and ½ 2 (tobeusedlater)interchangethesecomponents.

Thedoublingofthecomponentsoftheelectronwavefunctionintherelativisticcase,ascompared

tothenonrelativisticcase,isjusti¯edbytheexistenceofantiparticles.Antiparticlesaredescribedby

thenegativeenergysolutionsoftheDiracequation.Photons,however,donothaveantiparticlesso

thatthenegativeenergypartofthewavefunctionshouldnotcarryanyadditionalinformation.The

doublingofthecomponentsofthewavefunction(12)requires,therefore,anauxiliaryconditionthat

restorestheoriginalnumberofdegreesoffreedom.Thisisachievedbydemandingthatthecomplex

conjugationofthewavefunctionhasthesamee®ectasaninterchangeofupperandlowercomponents,

c (s ¢ ¹ h

F = ½ 1 F ¤ :

(14)

Asanindependentquantityonemaytakejustthepositiveenergypart F (+) ofthewave¯eld F (called

theanalyticsignalinclassicaltheory).Thispartrepresentsthetruephotonwavefunctionandthat

partistobeidenti¯edwiththematrixelementsofthe¯eldoperatorsbetweenthevacuumstateand

one-photonstatesthatmaketheirappearanceinquantumelectrodynamics.

InwhatfollowsIshallusethesymbolªtodenotethepositiveenergypartofthefunction F .

Owingtothecondition(14),thenegativefrequencypartisobtainedbycomplexconjugationandby

aninterchangeoftheupperandlowercomponentsofthepositivefrequencypart.Inthelanguage

ofparticlephysics,complexconjugationcombinedwithmultiplicationby ½ 1 wouldbecalledcharge

conjugationandEq.(14)istheconditionofinvarianceunderchargeconjugation.

Maxwellequationswrittenintheform(13)exhibitacloseanalogywiththeDiracequationwritten

inthechiralrepresentationoftheDiracmatrices(cf.forexample[24,8]).Inthisrepresentationthe

four-componentwavefunction Ã ismadeoftworelativisticspinors Á and Â |theanalogsofF + and

F ¡ |describingtwohelicities,

µ Á

Ã =

(15)

TheDiracequationforinthisrepresentationtakestheform

¾¢ ¹ h

i ¹ h@ t Ã = c½ 3

i r

Ã + mc 2 ½ 1 Ã: (16)

FortheDiracparticlethemasstermalwaysinducesamixingofthetwostatesofpolarizationwhile

forthephotons,asIshallshowlater,themixingisinducedonlybyanexternalin°uence,bythe

medium.

CONSERVEDQUANTITIESANDCORRESPONDINGQUANTUMOPERATORS

Ishalltrynowtomaketheprobabilisticinterpretationofthephotonwavefunctionªplausibleby

comparingtheexpressionsforbasicobservablesinclassicalandinquantumtheories.Thefundamental

conservedquantitiescharacterizingclassicalelectromagnetic¯eld|theenergy E ,themomentumP,

andtheangularmomentumM|aregivenasthefollowingspaceintegrals

E =

d 3 rH (r) ; P=

d 3 rP (r) ; M=

d 3 r r £P (r) ; (17)

thatinvolvethelocaldensitiesofenergy H (r)andmomentum P (r)

H = D ¢ D

2 ² + B ¢ B

2 ¹ = F y ¢F;P =D £ B= F y ½ 3 s F=c: (18)

Theconservationlawsfor E ,P,andMfollowfromthecontinuityequationssatis¯edby H (r)and

P (r),

@ t H = ¡r¢S; @ t P i = ¡r k T ik ; (19)

where S istheenergy°uxand T ij istheMaxwellstresstensor,

S = cF y ½ 3 s F; T ij = F y ( s i s j + s j s i ) F¡± ij F y ¢F; (20)

providedthewavefunctionislocalizedinspacesothattheintegrationbypartsdoesnotintroduce

anyboundaryterms.

Now,Ishallanalyzetheenergy,momentum,andtheangularmomentumfromaquantummechanical

pointofviewapplyingthesamemethodsasinwavemechanicsofmassiveparticles,byexhibitingthe

correspondingoperators,theireigenvaluesandeigenfunctions.Thequantumoperatorsrepresenting

theseobservablesaretakentobe(cf. forexample[9])thegenerators(uptoafactorof¹ h=i )of

theappropriatein¯nitesimaltransformationsofthewavefunction.Thetimetranslationgivesthe

Hamiltonianortheenergy,spacetranslationsgivethemomentum,androtationsgivetheangular

momentum.

Themomentumoperator^panditseigenvalueproblemhaveexactlythesameformasinquantum

mechanicsofmassiveparticles

^p= ¹ h

i r; ¹ h

i @ i ª=¹ hk i ª ; (21)

where k i 'sarethecomponentsofawavevector.

ThetimeevolutionEq.(13)enablesonetoidentifythequantummechanicalHamiltonianoperator

^ H as,

^ H = c½ 3 (s ¢ ^p) : (22)

Theeigenvalueproblemforthisoperatorhastheform

c½ 3 (s ¢ ¹ h

i r )ª=¹ h! ª : (23)

Thetotalangularmomentumoperatorforspinningparticlesconsistsoftwoparts:theorbital

angularmomentumandthespinangularmomentum,

^ J=^r £ ^p+¹ h s : (24)

Itisworthnotingthatitisthetotalangularmomentum ^ J,de¯nedasthesumoftheorbitalandthe

spinpart,thatcommuteswiththephotonHamiltonian(22)and,therefore,isaconstantofmotion.

Theeigenvalueproblemfortheangularmomentumcontains,asusual,theeigenvalueproblemforthe

z -componentofthetotalangularmomentum

^ J z ª=¹ hM ª ;

(25)

andtheeigenvalueproblemforthesquareofthetotalangularmomentum

^ J 2 ª=¹ h 2 J ( J +1)ª : (26)

ThesolutionsofEqs.(25)and(26)arewellknownvectorsphericalharmonics(cf.forexample[25,26]).

At¯rstlook,theclassicalexpressions(17)fortheenergy,momentum,andangularmomentumare

di®erentfromtheirquantummechanicalcounterparts.InthenextsectionIshallshowthatthesedif-

ferencescanbereconciledbyadoptinganappropriatemetricfortheevaluationofexpectationvalues.

EXPECTATIONVALUESANDTRANSITIONPROBABILITIES

Acorrespondencebetweenquantumandclassicaldescriptionsoftheradiation¯eldrequiresthat

theexpectationvaluesofthequantum-mechanicaloperators ^ H; ^p,orMbeequaltotheclassical

valuesofthe¯eldenergy,momentum,andangularmomentum.Ishallshowthatthiscanbeachieved

byanappropriatechoiceofthescalarproductforthewavefunctionsªusedintheconstructionof

expectationvalues.Tothisend,Iinvokethetransversalitycondition(5)forªtoobtainthefollowing

relations

^ H½ 3 s =c ª=^pª ; ^ H½ 3 ^r £ s =c ª=(^r £ ^p+s)ª : (27)

Withthehelpoftheserelations(afteradivisionby ^ H ),onecanwritetheclassicalquantities(17)in

theformofexpectationvaluesofthecorrespondingquantum-mechanicaloperators

E = hEi; P= h P i; M= h M i; (28)

where

d 3 r ª y ^ H

d 3 r ª y ^p

d 3 r ª y ^r £ ^p+s

^ H ª : (29)

Alltheseexpressionsarebilinearinª,likeallexpectationvaluesinwavemechanicsofmassive

particles,andtheonlydi®erenceisasystematicappearanceoftheHamiltonianinthedenominator.

Therefore,itisnaturaltoassumethatthedivisionbytheHamiltonianshouldbeincludedinthe

de¯nitionoftheexpectationvaluesorinthede¯nitionofthescalarproduct h ª 1 j ª 2 i H oftwowave

functions,

hEi =

^ H ª ; h P i =

^ H ª ; h M i =

^ H ª 2 : (30)

Ihaveaddedthesubscript H inthesymbolofthescalarproductasareminderofthenecessary

divisionby ^ H .Withthisde¯nitionofthescalarproduct,oneobtains

h ª 1 j ª 2 i H =

d 3 r ª y 1

hEi = h ª j ^ H ª i H ; h P i = h ª j ^pª i H ; h M i = h ª j (^r £ ^p+s)ª i H : (31)

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