FERMIONS AND BOSONS INTERACTINGWITH ARBITRARILY STRONG

EXTERNAL FIELDS

JOHANN RAFELSKI

ArgonneNationalLaboratory,Argonne,Ill. 60439, U.S.A.

LEWIS P. FULCHER

Departmentof Physics,Bowling GreenStateUniversity,Bowling Green, Ohio 43403, U.S.A.

and

ABRAHAM KLEIN

Departmentof Physics,Universityof Pennsylvania,Philadelphia,Pennsylvania19174, U.S.A.

1NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

PHYSICSREPORTS(Section C of PhysicsLetters)38. No. 5(1978)227-361.NORTH-HOLLAND PUBLISHING COMPANY

FERMIONS AND BOSONS INTERACTING WITH ARBITRARILYSTRONG EXTERNAL FIELDSt

JohannRAFELSKIArgonneNationalLaboratory. Argonne. 111. 60439. U.S.A.

Lewis P. FULCHER*Department of Physics, Bowling Green State University. Bowling Green, Ohio 43403, U.S.A.

and

AbrahamKLEIN~Department ofPhysics. University of Pennsylvania, Philadelphia, Pennsylvania 19174. U.S.A.

ReceivedDecember1976

Abstract:

The question, ‘What happensto theelectronorbitals as the chargeof the nucleusis increasedwithout bounds?”hasinspired muchof theinterest in thedescriptionof particles bound strongly by externalfields. Interestin this problemand in therelatedKlein paradoxextendsbacknearlyto thebeginningsof relativisticquantummechanics.However,thecorrect interpretationof thetheoryfor overcritical potentials,where thepartsof thecompletesetofsingle particlesolutionsassociatedwith particlesandantiparticlesareno longerdistinct,wasgivenonly recently.Theunderstandingof thespectrumof the Dirac and Klein—Gordonequationsis essentialin order to obtain an appropriatephysicaldescriptionwithquantumfield theory.The strongbinding by morethantwice therestmass of the particlesin overcriticalexternalpotentials leadsto qualitativelynew effects. In thecaseof fermions we find spontaneouspositronemissionaccompaniedby creationof a chargedlowest energystate, i.e. achargedvacuum.The numberof positronsproducedspontaneouslyis limited by the Pauli exclusionprinciple. For bosonswefind that dependingon the characterof theexternalpotential,either neutralor chargedBosecondensatesdevelop.While thequestionsassociatedwith themesonfields seemacademicat the moment,the effectsattributedto thefermion field standa good chanceof beingtestedin anexperimentin the nearfuture. It is expectedthat in heavyion collisionssuchas uranium on uraniumneartheCoulombbarrierovercriticalelectromagneticfields will becreated.

t Works performedundertheauspicesof theUS ERDA. Office of PhysicalResearch.

* Supportedin partby Faculty ResearchCommittee,Bowling GreenStateUniversity.

~Supportedthrough funds provided by contract#AT(ll-l) 3071. Part of this work done during the tenureof a John Simon GuggenheimFoundationFellowship.

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J. Rafeiski et al.. Fermions and bosons interacting with arbitrarily strong external fields 229

Contents:

1. Introduction andoverview 230 5. Spin-zeroparticles in strongexternalfields 3072. Solutionsof theDirac equation 240 5.1. Formsof the Klein—Gordonequation 307

2.1. Klein’s paradoxin one dimension 241 5.2. Approachto critical potential in aCoulombfield 3102.2. Bound states 244 5.3. Approachto criticality for short rangepotential 3122.3. Continuumstates 257 5.4. Limit of the singleparticle description 313

3. Quantumfield theoryof spin one-halfparticlesin strong 5.5. Conventionalquantization 3l6externalfields 262 5.6. The ground statefor small Z~Zcr, long range3.1. The quantizedDirac field 263 potential 3183.2. The Fermienergy 266 5.7. Scatteringresonances 3193.3. The reducedHamiltonian 267 5.8. Decayof theunstablevacuum 3203.4. The instability of the neutralvacuain over- 5.9. Self-consistentchargedcondensateequations 321

critical externalfields 271 5.10. Solutionsof the condensateequations 3233.5. The Green’sfunction and its analyticstructure 275 5.11. Quasi-particleexcitationsandthe stability of the3.6. Realandvirtual vacuumpolarization 277 condensate 3263.7. GeneralizedHartree—Fockequations 280 5.12. Quantizationbeyondthe critical point Acr for short

4. Laboratory testsof the theory of electronsin overcritical rangepotentials 329fields 281 5.13. Mean field approximationfor the neutral conden-4.1. Overview 281 sate 3314.2. Adiabatic approximationin heavy-ioncollisions 283 5.14. Modificationsnecessaryfor weakinteractions 3344.3. Solution of thetwo-CoulombcenterDirac equa- 6. Applicationsandextensionsof thetheory 336

tion 284 6.1. The charged vacuum in the Thomas Fermi4.4. The critical distance 290 approximation 3374.5. Inducedand spontaneouspositron productionin 6.2. Resolutionof Klein’s paradox 341

heavy-ioncollisions 292 6.3. Neutralizationof nuclearmatter 3444.6. Treatmentof thetime-dependenceof heavy-ion 6.4. Gravitationalfield, gravitational collapse 349

scattering 296 6.5. The role of supercriticalfields in field theory 3524.7. Other sourcesof positronsin heavy-ioncollisions 300 References 3564.8. Coulomb excitationof inner electronicshells in Note addedin proof 359

heavy-ioncollisions 303

230 J. Ratelski et al., Fermions and bosons interacting with arbitrarily strong externalfields

1. Introduction and overview

In this paper,we review andconsolidatesomeof the recentwork that dealswith the spontaneouscreation of particles and antiparticlesby strongstatic externalfields. An example of this kind ofphenomenon,whichhasstimulatedmuchof the currentinterest,would be the spontaneousproductionof electron—positronpairs by the electric field surroundinga superheavynucleuswith a chargeZ> 170. The positron escapesto infinity and the electronremainstightly bound to the nucleus, incontrastto the usual pair productionprocessby energeticphotonsin the vicinity of nuclei. Anotherexample of spontaneouspair productionshould occur during the collision of two very heavy ions,suchas the collision of a uraniumnucleuswith anotheruraniumnucleus,providedthat the two nucleicomeclose enoughtogetherfor their combinedelectric fields to exceedthe critical strength.In thiscase,the spontaneousproductionmode is only oneof severalwhich contribute.The othersinvolvetime dependentelectromagneticfields. Thus,experimentswith superheavynuclei would allow us toscrutinize the basic physics of spontaneouspair production more cleanly than experimentswithcolliding heavy ions. However, the likelihood of doing such experimentswith superheavynuclei isvery small, andthe heavy ion experimentsdiscussedabove will soonbe done at GSI in Darmstadt.

Much of the recentwork hasbeendonein Frankfurtandin theSoviet Union andis a consequenceof interestin superheavynuclei [1] andheavyion collisions. Interestin the spontaneousproductionofparticles and antiparticles,however,extendsback as far as the beginningsof relativistic quantummechanics,apoint that we wish to emphasize.

Soonafter the inventionof quantummechanicsandthe realizationthat the introductionof electronspin was necessaryto accountfor the observedlines in atomic spectra,Dirac [2] appreciatedthat thisconceptwould enterthe theorynaturally if he soughtto write the equationfor the wave function in aform consistentwith the special theory of relativity. However, from this synthesisemergedwavefunctionswith four componentsinsteadof the two that sufficed to describethe spin in earlierwork.The increasednumber of solutions of the wave equationis a consequenceof the fact that therelativistic relationshipbetweenthe energyand momentumof a free particle dependsonly on thesquareof the energy.Negativeenergysolutions*hadalso beenencounteredin earlierefforts [3,4] toreconcilequantummechanicsandthe specialtheoryof relativity. Thesesolutions,which arerequiredfor completeness,must haveseemedmysteriousuntil the discoveryof the positron [5,6], althoughDirac [7] hadtakena majorstepin interpretingthetheory by proposingthat thenegativeenergystateswere filled with electronsin accordwith the Pauli exclusionprinciple. Thus,Dirac’s theorybecameamany particle theory in which positronswere viewed as holes in the negativeenergysea.The fillednegativeenergyseais not an essentialpart of modernrelativistic quantummechanics,but electronsandpositronsmustbe treatedsimultaneously.

An interesting,andperhapsunexpected,featureof Dirac’s relativistic theorywas reportedby Klein[8]. He found that an attempt to confine an electronwith a strongstatic externalfield may lead towave functionswith an oscillatory behaviorin regionsof spacewhere,in analogywith nonrelativisticquantummechanics,onewould expectdecayingexponentialbehavior.In particular,electronsincidentupon a strongrepulsivebarrier seemedto havesomeprobability of penetratingit. In the contextof asingle particle theory,this would seemto be a paradox.However, in the contextof relativistic Dirac

* In the body of this paperwe are careful to makethe distinction betweenobservableenergies,which emergefrom the secondquantizedformulationand arealways boundedbelow,and theenergyeigenvaluesthat emergefrom thesolutionof theDirac equation.Thesemay assumearbitrarily large negative values.Perhapsa more appropriatedescriptionof the solutionsof the lower continuum would be negativefrequencysolutions.

J. Rafelski et al., Fermions and bosons interacting with arbitrarily strong external fields 23 I

theory [9] one realizesthat this oscillatory behavioris associatedwith the appearanceof positrons.Chargeconservationrequiresthat therebe correspondingelectronsproduced.Klein’s work gaveriseto a number of papers[10—14],but interest subsidedafter a number of years. We also record anoteworthycontribution[15] to the questionsraisedby Klein’s work.

Although Klein’s work set the stagefor the studyof electronsbound by arbitrarily stronglocalizedstatic potentials,no paper dealing with the subject appeareduntil the work of Schiff, Snyder andWeinberg [16], with the exceptionof a paper dealing with the relativistic harmonicoscillator [17].Theseauthorsstudiedelectronsboundto a strongsquarewell potential.As depictedin fig. 1.1, theyshowedthat theenergyof the is statedecreasedmonotonicallywith increasingpotentialstrengthandthat at somevalue of the potential strength,which we call the critical potential strength,the ls statejoined the lower continuum.The correctphysical interpretationof this phenomena,as we discussindetail below, is that positronsappearat infinity, provided that the is statewas originally unoccupied.

In 1945,PomeranchukandSmorodinsky[18,191 consideredtheproblemof an electronboundto anextendednuclearchargedistribution. Their work was motivatedby the knowledgethat the solutionsof the Dirac equation [20—231become singular when the nuclear charge is greater than 137. Anexampleof the singular behavioris foundwith the expressionfor the Is energy* E1~= m\/l —

which becomesimaginary if analytically continuedbeyond Za = 1. Pomeranchukand Smorodinskyshowedthat the singularbehaviordid not persistwhenthe effect of finite nuclear size was included.This is eigenvalue(asa function of Z) decreasedsmoothlythrough zero and continuedto decreaseuntil E1~= — m, wherethe lower continuumbegins.

WernerandWheeler[24] studiedthe problemof an electronboundto a uniformly chargedsphere,a more realistic (presumably)charge distribution than the uniformly chargedspherical shell ofPomeranchukand Smorodinsky.The numericalsolutionsof the Dirac equationwere obtainedforZ = 137 and Z = 170. It was found that the Is state was near the lower continuum for Z = 170.AlthoughWernerandWheelerdid not pursuethe questionof the critical potential strengthin detail,from their calculations,it is apparentthat thecritical chargeis near 170. This differs significantly fromthe resultof PomeranchukandSmorodinsky(Zc. = 200).Someof the disagreementbetweenthesetwovaluesis a consequenceof the differencein the nuclearchargedistributions,but the largerpart resultsfrom a poor choice for the electron radial functions outside the nucleus by Pomeranchukand

.1

Fig. 1.1. Energyeigenvaluesfrom theDirac equationfor a squarewell potential.The discretenessof thecontinuumis introducedby normalizingthewave functionsin a large box. Both � and V0aregiven in units of m, theDirac particle mass.

* Units aresuchthat h = c = 1.

232 J. Rafeiski et al., Fermions and bosons interacting with arbitrarily strong external fields

Smorodinsky.D. Rein [25, 26] also examinedthe is eigenvaluesfor electronsbound to superheavynuclei. He studiedthe dependenceof the eigenvalueson nuclearsize and made contactwith Case’searlier work [22] on point chargeswith Z> 137. He may havebeen awareof the correctphysicalinterpretationof the theory for Z> Zcr.

The investigationthat led ultimately to the solutionof theproblemsbeganwith the work of Pieperand Greiner[27]. They consideredin detail the solutionsof the Dirac equationfor an electronboundto a uniformly chargedsphere.Their resultsfor the energyeigenvaluesas functionsof Z areshowninfig. 1.2. The generalbehaviorof the curvesof fig. 1.2 is in agreementwith what one might expectonthe basis of the isolatedpoints calculatedin the earlier work. Pieperand Greinerfound the criticalvalueof Z to be 169. The differencebetweenPieperandGreiner’svaluefor Z~.andthe result that canbe inferredfrom WernerandWheeler’swork is due to a differentchoiceof the nuclearradius.

The correctphysical interpretationof the theory for Z> Z~.is thata vacantis stateleadsto thespontaneousproductionof two positronsat infinity, as shown in fig. 1.3. This processoccurswithoutany energy input; in fact, energy is given up as the kinetic energyof the positrons. In order toconservecharge,the region surroundingthe nucleusmust carry the chargeof two electrons.Forreasonsdiscussedbelow in section3, we refer to this stateas the charged vacuum[28] andprefertocharacterizethe positronproductionprocessas the spontaneousdecayof the neutral vacuum.* Theabovediscussedchangeof the ground stateas a functionof the strengthof the externalpotential canbethoughtof assomewhatsimilarto aphasetransitionfrom aneutralto achargedelectronvacuum.Thisexciting realization hasbeen the driving force behind most of the recenttheoreticalwork and theexperimentalefforts.

The chargedensityassociatedwith the chargedvacuumis mostly confined to distancessmallerthan the electron’s Comptonwavelengthfrom the nucleus.To calculate this chargedensity, it isnecessaryto analyzein somedetail [28] the group of statessurroundingthe resonancein the lowercontinuum,which is the analyticcontinuationof the bound is state.If Z is not muchlargerthanZcr,

then the chargedistributiondoesnot differ greatlyfrom the chargedistributionof the bound is statein the undercriticalregion. Only two positronscanbe producedwhen the is state joins the lowercontinuum becauseof the Pauli exclusionprinciple which prohibits more than two electronsfromoccupying the is state.More positronscan be producedwhen higher boundstates,such as the 2p

z<z z>zCr cr

500 ~/IIII~7~ ~1IIIII~7~

____ ~EUS ~us

C \ ~I/2 \ \\~ e~

-50C’-____ _______________/////// 7////7/ /_/7////7/ ///,POSITRON EMISSION

Fig. 1.2. Z dependenceof theeigenvaluesfor electronsboundto a Fig. 1.3. Schematic representationof the fate of the vacuumuniformly chargedsphere.The dashedcurvesrefer to resultsfor a surroundingcompletely ionized nuclei in the undercritical andpoint nucleus.From PieperandGreiner[271. overcritical cases.Only one spontaneousproductionis shown.

* In theopeningparagraph,we have loosely spokenof theprocessasthespontaneousproductionof electron—positronpairs.

J. Rafeiski et al., Fermions and bosons interacting with arbitrarily strong external fields 233

states,join the lower continuum,provided that thesearenot filled with electrons.The 2P112 level doesnot join the lower continuumuntil Z is near 180. If only one electronoccupiesthe is statewhenZ< Zcr, then only onepositronwould be producedas Z is increasedadiabaticallythroughZ = Z~,..Iftwo electronsoccupy the Is statewhen Z< Zcr, then positronsare not produced.The effects of afilled Is statewith Z> Zc. could be observedas a resonancein the positron scatteringcross section.

Pieperand Greineralso suggestedthat for a strongfield with somewhatlessthan the critical fieldstrength,oneshould be able to regarda vacancyin the ls stateas a boundpositron state.In this view,which is mostreadily understoodby a new choice of the vacuumstate[28—30],the appearanceof apositron at infinity when Z is adiabatically increasedthrough Zc. is viewed simply as the delo-calizationof a bound state.Binding of a positronto a superheavynucleusis a consequenceof purelyrelativistic effects, which have no analoguein the nonrelativistic limit, where the positron—nucleusinteractionis repulsive.One can gainsomeinsight into how this arisesby writing the relativisticradialequationsin a form similar to the Schrôdingerequationand identifying an effective potential.Thispotentialhasboth attractiveandrepulsiveparts (as shownin fig. 1.4). For largeZ, the attractivepartsdominate.

After Pieper and Greiner’s initial work, there was a search for a mechanismto stabilize thevacuum. Questionswere raised about the importanceof vacuumpolarization corrections,and aself-consistentschemefor the study of field correctionswas proposedby Reinhardand Greiner[31,32]. A preliminary investigationof these effects [33] did not suggest that anything unusualhappenedto the vacuumpolarization correction as the is level approachedthe lower continuum.Furtherwork at Frankfurtbasedon phenomenologicalmodificationsof the shortdistancebehaviorofMaxwell’s equationsand nonlinearterms in the Dirac equationfailed to uncoversucha mechanism[34—36].Experimental measurementsof inner-electronbinding energiesin heavy elementsrule outsubstantialeffects in superheavyelementsdue to thesephenomenologicalmodifications [37—39].Amore careful study of the vacuumpolarization question [40,41], also based on the formalism ofWichmann and Kroll [42], has shown that the vacuum polarization correction remains small asZ”9Zcr. Someof the graphsincludedin this calculationareshownin fig. 1.5. At present,thereare noindicationsthat otherfield corrections[43—45]becomelargeenoughto havea significanteffect on theenergy eigenvaluesof fig. 1.2. The questionof whether the Dirac equationprovides a suitabledescriptionof such astrongly interactingelectronnucleussystemhasalso beenstudied.Dittrich [46]showedthat useof the Dirac equationis justified as long as oneof the interactingparticleshasa massthat is small in comparisonwith the other, andan approachbasedon the Bethe—Salpeterequationisnot necessary.

Therewas a moreor less simultaneousrealizationthat positronswouldbe producedspontaneouslyif a vacant is statejoined the lower continuumby Gershtein,Zeldovich andPopov [47—Si].Much of

log (r.m)

Fig. 1.4. An approximateeffectivepotential for positronsnearsuperheavynuclei. No spin dependenteffectsareincluded.

234 1. Rafelski et al., Fermions and bosons interacting with arbitrarily strong external fields

Fig. 1.5. Someof thevacuumpolarizationgraphsincludedin Kroll andwichmann’scalculation.The crossesrepresentpointswherethenuclearCoulombfield acts.

their work was basedupon the effective potentialapproach,wherethe productionprocessis viewedas the penetrationof a particle through a barrier. The first calculationsof the rate of spontaneouspositron productionwerecarriedout by Popov[50] andMUller, RafelskiandGreiner[52,53]. Popov’swork was based on an analytic expressionfor the is energy eigenvalue. As this expressionisanalytically continuedbeyondE10 = — m, the eigenvaluebecomescomplex, and the imaginarypartgives the lifetime of a vacancyin the is shell. Becauseof the simplifications made in obtainingtheanalyticexpression,Popov’sresultwas valid only nearZ = Zcr. By carefully examiningthe solutionsof the Dirac equationnear Zcr, Popov also found that the Is wave function remainedlocalizedatZ = Z~r.Thus,positronproductiondid not occuruntil Z> Zcr.

The work of MUller et a!. was basedupon a judicious separationof the overcritical potentialintotwo parts. One of thesewas chosento be an undercriticalpotentialfor which the is eigenvaluewasbarely above the lower continuum.Under the action of the secondpart of the potential, the lseigenvalueis pushedinto the lower continuumand the boundstate is transformedinto a resonance.The width of this resonanceis relatedto the lifetime of a vacancyin the is state.Resultsobtainedinthis manner are valid for a much larger range of Z than those obtainedwith Popov’s analyticexpression.The approachof MUller et a!. is closely akin to that developedby Fano [54] for boundatomic statesembeddedin the continuum,and it hasbeenshown [55] that resultsobtainedwith thisapproachare the sameas those that emergefrom an exacttreatmentof the resonancein the lowercontinuum.Furtherwork [28—30]hasdealtwith the interpretationof the theory from differentpointsof view and the calculation of the chargedensity of electronsbound to superheavynuclei withZ> Zcr. Brodsky [56]hasgiven a short accountof the work up to this point, andMeyerhof [57] andHamilton andSellin [58] havewritten popularscientific accounts.

After the calculationsof the spontaneouspositron productionrate, the emphasisshifted to thestudyof the electronicmotion during the collision of two very heavyions, suchas U andU (seefigs.1.6 and 1.7). Theseelectronsmovedin molecularorbitals aboutthe two centersof positive charge.The earlier work wasbasedon the adiabaticapproximation,wherethe orbitalswere assumedto adjust

J. Rafeiski et a!., Fermions and bosons interacting with arbitrarily strong external fields 235

Fig. 1.7. Positronproductionin heavyion collisions.

Fig. 1.6. The collision of two heavy ions with chargesZ, and .Z~.When the two ions arecloser togetherthan thecritical separationRcr, spontaneouspositronproductionsoccurs.

gradually to the nuclear motion. This picture was supportedby a comparisonof the velocities ofelectrons in lower bound states(very relativistic) with the velocities of the two nuclei, typicallyone-tenthas large. The earliestwork at Frankfurt[59,60]addressedthe questionof the magnitudeofthe critical separation,that is, the distancebetweenthe two ions at which the combinedelectric fieldswere just sufficient to pull the Is level into the lower continuum.These calculationsof the positronspectrum found it to be sharply peaked.The initial calculations were based upon a monopoleapproximation,in which a central potentialwas chosento replacethe electrostaticpotentialof thetwo heavy ions. The effects of the ion motion were included by an averageover the scatteringtrajectories,and reasonableassumptionsaboutthe probability of the occurrenceof a vacancyin theis stateweremade.The positronspectracalculatedwith the aid of theseapproximationsweresharplypeaked,a fact that would aid the experimentalsearchfor thesepositrons.

An important advance occurred when the Frankfurt school [61,62] solved the problem of arelativistic electronboundto two nuclei separatedby adistanceR, the so-calledtwo centerCoulombproblem(seefig. 1.8). Thesebasisfunctionsand energyeigenvaluesarea necessaryingredientfor allproblemsrelated to the electronicmotion during heavyion collisions.To solve this problem,it was

~ Z~Z~92> -5( . R~R~,,~36.5fm -

l00~POINT NtJCLEI -

z[fm]

Fig. 1.8. A profile of theelectrostaticpotentialenergyof anelectronmoving neartwo uraniumnuclei separatedby 36.8 fm.

236 J. Rafelski et al., Fermions and bosons interacting with arbitrarily strong externalfields

necessaryto diagonalizea ratherlargematrix,whose elementshadbeencalculatedwith a setof basisfunctions chosenso that all of the matrix elementscould be calculatedanalytically. An interestingfeaturethatemergedfrom thesecalculationswas that, for mediumheavy ion collisionssuchas Br onBr, the energyeigenvalueswerealmostconstantfor valuesof the nuclearseparationbetween1000fmand Ofm. For much heavierions this featuredid not persistfor the lower atomic orbitals,but thehigheratomicorbitalsdid behavethisway. For collisions betweentwo U ions,as shownin fig. 1.9, the Islevel joined the lower continuum,and the critical separationwas 36fm. Including finite nuclearsizeeffectsreducedthis valueby about2 fm. Calculations[61—68] of the critical nuclearseparationandthepositron spectrumwerealso done in the Soviet Union by Marinov, Popovet al., who againrelied onthe effective potentialapproach.New techniquesfor calculatingthe critical nuclearchargewere alsodeveloped[69—72].Initially, the resultsof thesecalculationsseemedto agreewith thoseobtainedatFrankfurt, provided that one rememberedto correct for the different values of the ionizationprobability used in the two calculations.Soviet work [73—75]basedon avariational approachto thesolution of the Dirac equationled to significant differenceswith their earlier work becauseof thesensitivityof all resultsto the valueof the critical radius.Thesedifferencesled to a critical reviewofthe variational approachto the solution of the Dirac equation [76] and a more precise computerprogram[77] to solve the Dirac equationfor colliding heavy ions. A comparisonof resultsbasedonthe variationalapproachwith the moreprecisecalculationsis shown in fig. 1.10.

K. Smith, MUller andGreiner[78,79]studiedcorrectionsto the adiabaticapproximation,which hadbeenthe starting point for all previouswork. They consideredthe productionof positronsinducedbythe time dependenceof the nuclearelectric fields, which maytransferenergyto the electron—positronfield. Thus electron—positronpair productionoccurs even if the Is level does not join the lowercontinuum.Of course,positronemissionalsooccursafter the is level hasjoined thelower continuum.In fact, one mustcoherentlyaddthesetwo amplitudesto computethe positronproductionspectrum.This spectrumis very braodin comparisonwith the spontaneousproductionspectrum[80].

More recently somebackgroundpositronproductionhasbeenconsidered.An exampleof suchaneffect is the de-excitationof a Coulomb excited nuclear statewith a subsequentdirect electron—positronpair production,as shownin fig. 1.11. An unusuallyhighly excitednuclearstate If) is reached

15 30 50 tOO 300 500 lO~ 3x103 10~

10 I I I I I R[frn]— 3spd

3pd3/2— ~ =~:

I500ls U-+-U relativistic molecular orbitals92U

—E[keVJ for extended nuclei

Fig. 1.9. Energyeigenvaluesasa function of nuclearseparationfor two U ions. From Muller, RafelskiandGreiner[61].

J. Rafeiski et al., Fermions and bosons interacting with arbitrarily strong external fields 237

I ‘ I100 — -

If>

~ __~~ EXACT Fig. 1.11. Pair production from conversion of internal nuclear

86 90 94 98 02z

Fig. 1.10. Critical radius in symmetricheavy ion collisions. FromRafelskiandMuller [76].

in a multistepprocessvia anintermediatestateJi). If the energydifferencebetweenIf) andthenuclearground state10) is largerthan2me,the pair productionshownin fig. 1.11 mayoccurprovided that thesuperselectionrules are satisfied. Oberacker,Soff and Greiner[81] havestudiedthis process.Sincethe lifetime of the nuclear statesgreatly exceedsthat of the supercriticalvacuumcreatedin theheavy-ioncollision, pair production stemmingfrom the separatedions dominates.However, thelifetime of the Coulomb excited nuclear state is not yet large enough to make an experimentalseparationof the cross sectionspossible.The calculationsof Oberacker,Soff and Greiner indicatethat a separationof the differentmodescanbe madeby observingthe positronspectrumas afunctionof energyandscatteringangle.

Someother effects havebeenalso consideredand found to be neglibible in comparisonwith theprocessdiscussedabove.In particular,we refer to the work of Reinhardt,Soff and Greiner [82] fordiscussionof the pair productionfrom conversionof transversenuclearbremsstrahlungphotons.

In closing our surveyof electronsbound to two differentnuclei, we wish to discussbriefly somerecentdevelopmentsin the X-ray spectroscopyof heavy ion collisions* [84—87].Much of this efforthas been directed towards the observationof X-rays that are not characteristicof the target orprojectile, presumablyoriginating from the transitionsbetweenquasimolecularstatescreatedin thejoint Coulombpotentialof thecolliding ions. Forexample,Mokler et al. haveseenevidencefor X-raysthat probably originatedin the 4f—3d transitionsof unitedatomsin collisions of I ions with Au, Th andU targets.The chargesof thesetransientintermediatenuclear systemsare Z = 132, 143 and 145.MUller, Smith and Greiner [88, 89] have studied the X-ray spectrum from a point of view of acoordinatesystemthat rotateswith an axis passingthrough the two heavy ions. In this coordinatesystem,it is easyto separatethe spontaneousX-ray transition probabilities from the inducedones,which reflect the nonstationarycharacter of the electronic motion. The induced radiations arepredictedto be asymmetricabout the ion-beamaxis. Observationsof this asymmetryin Ni—Ni andI—Au collisions are importantevidencefor the existenceof transientintermediatemolecularstates[90,91]. A further importantdevelopmentis relatedto the firm evidencefor the observationof K-shellX-rays in Br—Br collisions by Meyerhof et a!. [86, 92]. From theseX-ray yields one may infer theprobability of producingvacanciesin the K-shell duringthe collisions.Furtherexperiments[93] haveconfirmed andrefined theseresults.

The intensestudyof the problemof electronsboundto strongpotentialshasstimulatedinterestin a

* W. Meyerhof [831has just written anexcellentreview of this area.

238 1. Rafelski et al., Fermions and bosons interacting with arbitrarily strong externalfields

numberof similar problems.Much of this effort hasdealtwith the theoryof bosonsboundto strongpotentials.Clearly, the conceptsheremustbe very differentsinceit was the Pauli exclusionprinciplethat stabilizedthe chargedvacuumin the caseof fermions in overcritical fields.

Although the superboundbosonsystemis theoreticallya very interestingproblem, theredoesnotseemto be a prospectof experimentaltestsin atomiccollisions in the nearfuture. This can beseenbynoting that we needan externalpotentialcomparablein strengthto the massof the lightestmeson,thepion (m,r = 139.6MeV), which requiresnuclearchargeof the order of Z = Rm,,/a 2000. Here Rdescribesthe nuclearsize. The aboveformula is valid only if Rni> 1 andthereforedoesnot apply toelectronsboundin the nuclearCoulombfield.

One of the earliestinvestigationsof the solutions of the Klein—Gordon equationwith a strongexternalpotentialwas carriedout by Schiff, Snyderand Weinberg[16,94]. They solved the problemof the squarewell potential [16] and found that the spectrumbehavesqualitatively differently fromthat of the Dirac equationas the potential strengthis varied. They discoveredthat for a given state,there are two critical points; at a value V0 an antiparticle statewith the samequantumnumbersemergesfrom the negative energycontinuum while at Vcr the particle and antiparticle statesmeeteachother. No particle or antiparticle statewith theseparticular quantumnumbersis found aboveVcr. This situation is illustrated qualitatively in fig. l.i2. Although this behaviorof the spectrumissuggestivefor any potential, a different result was found later by Bawin and Lavine [95] forlong-rangepotentials.In particular, they found that a cut-off Coulomb potentialhasan eigenvaluespectrumsimilar to that of the Dirac equation.

Snyder and Weinberg [94] successfully introduced a second quantizationof the theory forpotentialssmallerthan V~,but madeno attemptto treatthe overcritical case.The theory remainedatthis point for a long time, with the exceptionof the technicaldevelopmentof first-orderequationsbyFeshbachand Villars [96].

The moderndevelopmentof the subjectwhich led to the understandingof the supercriticalstatewas initiated by Migdal [97],who was stimulatedby the work on the overcriticalDirac equation.Herecognizedcorrectly that to stabilize the vacuum in the over-critical case,somehigher-ordereffectmust be included in the Hamiltonian. He, in particular,chose to considera residualA~

4term in theboson field Hamiltonian. His physical picture was based on the following considerations:as VapproachesVcr, the energynecessaryto make a mesonpair vanishesallowing an infinite numberofpairs to be produced.To stabilize this situation,apositivedefinite part in the Hamiltonianis needed

-m i—-~ I

Fig. 1.12. Energyeigenvaluesfrom the Klein—Gordonequationfor a squarewell potential. From Klein and Rafelski[98].

I. Rafelski et al.. Fermions and bosons interacting with arbitrarily strong external fields 239

that would stop the production of the condensateswhen a certain mesondensity is reached.In asubsequentdevelopment,Klein andRafelski[98—100]showedthat it is possibleto considerthe chargeof the mesonsas the stabilizing mechanism.They demonstratedthat the repulsionwhich is alwayspresentin anychargedistributionof the mesoncondensatessuffices to stabilize the condensate.Theirtreatmentwas formally complete for both the under and overcritical cases.A fully relativisticquasi-particleformalismwas developedfor V ~ Vcr and equationsdeterminingthe Bose condensatesgiven. The self-consistenttreatmentwas simplified by the use of the first-order Feshbach—Villarsrepresentationof the Klein—Gordon equation.In the original work of Klein and Rafelski, the loweststate of the supercritical system was consideredto consist of a neutral Bose condensate.Assubsequentlypointed out by Bawin and Lavine [95], the long-rangecharacterof the Coulombpotentialallows the is state to join the lower continuum.This led Klein and Rafelski [101]to theconjecturethat the neutral condensatemust be replacedby a chargedcondensatein the long-rangeCoulombfield. In accordancewith the chargeconservation,a largenumberof free antimesonswill beproducedin this process.Furthermore,in view of the possibility of weak decayof chargedvacuumothermechanismsleadingtopion condensatesinexternalfieldshavebeenconsidered(c.f. sections5,6ofthis paper).

In anotherapplicationof the overcritical theory,Muller andRafelski[102]haveestablisheda modelof real vacuum polarization with the help of the Relativistic Thomas Fermi (RTF) model. Theprincipal new stepwas the recognition of the importanceof properuseof the Fermi surfacein theRTF equations.After theinitial studyof the spontaneousneutralizationof anovercritical backgroundcharge,the modelhasalso beenapplied to prove the stability of the chargedvacuum[103].

Somesimilar developmentshavetakenplace in the studyof gravitationalphenomena.It hasbeenknown for sometime [104]that the singularfields associatedwith blackholes give rise to significantproblems in treating the quantum mechanics of particles moving in external gravitational fields.Hawking [105—108]hasshown that particlesarecreatedin the vicinity of collapsingstarswith energyspectrathat are similar to the thermalradiationspectrum.This result hasbeenstudiedfrom variouspointsof view [109—ill].AlthoughHawking’s work is widely accepted,problemsariseif oneattemptsto identify the actualprocessesresponsiblefor the production.It may bethat the time dependenceofthe collapse is the essentialingredient. Perhapsthe emission of particlesis a consequenceof theexistenceof a sufficiently stronggravitationalfield, analogousto spontaneouspositron productioninovercritical fields. In the formercase,the fact that the radiation continuesbeyondthe time at whichthe eventhorizon is formed may be understoodin termsof the length of the time requiredfor theparticlesto reachthe observer.

The difficulties encounteredwhen the gravitationalfield is treatedas an external field led [112]MUller, Greiner and Rafelski to the study of a self-consistentapproach, in which the particlesproducedwere also consideredsourcesof the gravitationalfield. In particular, they consideredabosonfield in interactionwith the gravitationalfield and found thata Bosecondensatewas produced.This condensatehasa significant influenceon the metric,but its full implications arenot yet clear.

At presentthe consensusof the theoristsseemsto be that for quantumelectrodynamicsmostof thenew qualitativefeatureshavebeenisolatedandpointedout. Thus onewould expectthat theemphasiswill shift to the detailedcalculationsneededto comparetheoryandexperiment.Of particularinterestin this regard is the fact that the first energeticU beam has already been obtained at GSI inDarmstadt.It is hopedthat the initial positron experimentscan be done in a year or two. There istheoreticalwork in progresswhich dealswith refinementsof theproblemof atomicelectronsboundtocolliding heavy ions, such as electron—electronrepulsionin the relativistic Thomas—Fermimodeland

240 J. Rafelski et a!.. Ferrn ions and bosons interacting with arbitrarily strong external fields

the useof electronic wave functions obtainedin this approximation in the calculation of the innershellionizationprobabilities.

Our review is organizedin the following manner.Section 2 deals with solutions of the Diracequationfor boundstatesand alsofor scatteringstates.In section3 the secondquantizationof theDirac theory andapplicationsto physicalprocessesarepresented.Heavy ion collisions is the subjectof section4. Section 5 is devotedto the problem of bosonsin strongfields and section6 dealswithapplicationsandextensionsof the theory.

2. Solutionsof the Dirac equation

The Dirac equationfor a particle of momentump andmassm takesthe form,

(2.1)

wherethequantitiesa and/3 arematricesthat operatein afour-dimensionalspinorspace.The energyand momentum operatorsof eq. (2.1) must satisfy the same relation as the eigenvalues,that is

= p2+ m2,which leadsto the following relations,

a1a3+a1a~2ô~1, a1f3+13a~—0, /32i (2.2)

where i or j mayhaveanyvaluefrom 1 to 3. A generalrepresentationof the operatorssatisfyingeqs.(2.2) is

a=S6k®o, (S±l)13=6~®l, 1�k (2.3)

wherethe quantities6 andu satisfy the samecommutationrelationsas the Pauli matrices,1 denotesthe unit matrix in two dimensionsand ® denotesa direct product of two two-dimensionalspinorspaces.For the present,it will suffice to choosethe standardrepresentationof the Dirac matrices,which maybe generatedfrom eq. (2.3) by choosingk=1, I = 3, S = i andthe usual representationofthe Pauli matrices.This yields

a=[0 ~],I3=[~ 01]. (2.4)

The two solutionsto eq. (2.1) which correspondto the positive energyeigenvaluee = Vp2+ m2 are

given by

r x±1i/’±(x,t) = a~~ p exp{i(p . x—Et)}. (2.5)

L~+ m XsjThesetwo solutions do not spanthe four-dimensionalspace. In order to have a complete_setoffunctions,onemustalso includetwo solutionswith a negativeenergyeigenvalue� = — + m2. Theeigenvaluespectrumis shown in fig. 2.1. Thereis a positiveenergycontinuumbeginningat � = m anda negativeenergycontinuumbeginning at � = — m.

The implication of the existenceof two parts of the spectrumis that electronsandpositronsmustbe treatedsimultaneously.The precise relation betweenpositron statesand the statesof the lower

J. Rafelski et a!., Fermions and bosons interacting with arbitrarily strong external fields 241

C

________ POSITIVE FREQUENCY

IN U

0

~NEGATIVEFREQUENCY

~CONTINUUM~

Fig. 2.1. Energyspectrumfor free-particlesolutionsof theDirac equation.

continuum requires the quantizationof the theory,which is treatedin detail in section 3. Beforereviewing the secondquantizationformalism and developingthe parts that are relevantfor criticalfield phenomena,we discussin detail the behaviorof solutionsof the Dirac equationin strongfields.An intimate knowledgeof thepropertiesof thesesolutionsis a prerequisitefor understandingcriticalfield phenomena.We beginwith adiscussionof the Klein paradox,which furnishessomeinsightintomuchof what follows.

2.1. Klein’s paradoxin onedimension

Let usconsideran electronof momentump andenergy� = ~~‘p2+ m2 with spin up that is incidentfrom the left on an electrostaticsquarewell barrier suchas that shownin fig. 2.2. The kinetic energyof the electron K is less than the height of the barrier.* The discontinuousform of the potentialrequires that the two regions I and II be treatedseparately.In region I the solution of the Diracequationis

1 1

ipz 0 ~ 0= a e + b e , (2.6)p/(�+m) —p/(�+m)

0 0

VIz)

Fig. 2.2. Electrostaticsquarewell barrier.

* Our treatmentis similar to that of ref. [9].

242 J. Rafelski et a!., Fer,nions and bosom interacting with arbitrarily strong external fields

wherethe secondpart of the wave function describesthe reflected wave. The form of the wavefunction in region II dependsupon the magnitudeof the potentialstrength.We first considervaluesofV0 that arenot too large.Then

= c e~[i/3/(e — + m~1’ (2.7)

where /3 = \/m2— (�— V

0)2. It is not necessaryto add terms to eqs. (2.6) and (2.7) that describe

particleswith spin down sincethereis no probability of a spin flip. Requiring that the wave functionbe continuousat z = 0 leadsto the relations

a + b = c,

(a — b)p/(�+ m) = i/3c/(e — V0+ m). (2.8)

From thesetwo equationswe mayfind the ratios b/a and c/a, which takethe form

c/a = 21(1 + ii’),b/a (1—iF)/(1+if), (2.9)

where I’ = [(� + m)(m — � + V0)]u2 x [(� — m)(m+ � — V0)]112 is real. The incident current may be

calculatedfrom j~= i/4a3t~~1,wherethesubscripti denotesonly thefirst part of thewave function of eq.

(2.6). Thiscurrentis equalto 2p1a12/(�+ m). The ratio of thereflectedcurrentto theincidentcurrentis

I bI2/IaI2, which is equalto onebecauseof the form of thesecondof eqs.(2.9). Thetransmittedcurrentis equaltozero.Thusall of theincidentcurrentis reflected,andthesituationis analogusto nonrelativisticquantummechanics.

Let usnow considerwhat happensto f32 = m2— �2 + 2�V0— V~as ~ is increased.It increasesuntil

V0= e, where it takes on its maximum value m2. As V

0 is increasedfurther, the quadratictermbecomesincreasinglyimportantand/32 decreases,until at V0 = � + m it is zero.In this rangeof V0 anincreaseof the potential strengthhasthe effect of allowing further penetrationinto the barrier.Theassociatedcurrent is zero. As V0 is increasedbeyond� + m then/3 becomesimaginaryandthe wavefunctionmustbe written

~11(z)= d ebo2[l/( +~ — V0)]~ (2.10)

wherep’ = \7(V

0— ~)2 — m2. Now applyingthe continuity conditionleadsto

a + b = d,(a — b)p/(e+ m) = dp’/(�+m — V

0). (2.11)

Solving theseequationsfor the ratios b/a and d/a yields

d/a = 2/(1 —F’),b/a =(l+F’)f(l—F’), (2.12)

where I” = [(� + m)(V0—� + m)V’2X [(e — m)(Vo—� — m)]_U2. The transmitted current is equal to

J. Ratelski et a!., Fermions and bosons interacting with arbitrarily strong external fields 243

O.4~~3rn

1.0 1.2 1.4 .6 I.6 2.0 2.2 2.4 2.6�

Fig. 2.3. Transmissioncoefficients asfunctionsof energy.

2p’Id~2I(�+ m — V0), which is negative,andthe magnitudeof the reflectedcurrentis largerthanthat of

the incidentcurrent.Thusthe transmissioncoefficient, whichis the ratio of the transmittedcurrenttothe incidentcurrent,is given by

T = —4f’/(l —f’)2. (2.13)

In fig. 2.3 the negativeof the transmissioncoefficient is shown as a function of energyfor %T0 = 3m

andfor Sf0 = lOm. It is clearthatthe transmittedcurrentmaybe much largerthantheincidentcurrent.Thesephenomenaare examplesof the kinds of featuresnoted by 0. Klein [8] shortly after the

invention of relativistic quantummechanics.Only in the context of a single-particleinterpretationdothey appearparadoxical. When one appreciatesthat electronsand positronsare inextricably con-nected in the Dirac theory, it is natural to identify the negative current in region II with theappearanceof positrons.The increaseof the reflected current over the incident is necessarytoconservecharge.In fact, onecan showthat the reflectedcurrentplus thetransmittedcurrentis alwaysequalto the incidentcurrent.

The essentialpoint that we wish to makeis that a given solution of the Dirac equation(with adefiniteenergy)in astrongexternalpotentialmaybe usedto describeparticlesin oneregion of spaceand antiparticlesin another.The details of this descriptionrequire the formalism of secondquan-tization, since this is necessaryfor the connectionbetweenthe statesof the lower continuum andpositrons.Anotherway of viewing the Klein paradox,which servesto underscorethis essentialpoint,is presentedin fig. 2.4. Therethe potentialundergoesan abrupt changefrom zero to V0 in a smallintervalcenteredaboutz = a. Far to the left, thewave functionsare the sameas thoseof eq. (2.6), andanobserverwould interpretanywave functionwith e> m as that of a particleandany wave functionwith � < — rn as that of an antiparticle.Far to the right, the wave functionsare the sameas thoseofeq. (2.10), andan observerwould interpretany wave function with �> m + V0 as a particle stateandany wave function with e < V0— m as an antiparticle state.(The coefficients a, b and d would notsatisfythe relationsof eqs. (2.11) sinceeffectsassociatedwith tunnelingbetweenthe two continuum

244 J. Rafe!ski et a!., Fermions and bosons interacting with arbitrarily strong external fields

~ CONTINUUM

______ ~

/ A ANTIPARTICLE — ~ \ ~ PARTICLE STATECONTINUUM 1~\ /~ EMBEDDED IN

/ -— -- / ANTIPARTICLE-m____________ — -- -- AN’flPAWflCLU /~~c0NTINuUM

- — ~ ~~-— -- -- --

ANTIPARTICLE CONTINUUM z~

zFig. 2.4. Mixing of particle and antiparticle statesby a strong Fig. 2.5. Mixing of particle andantiparticlestatesby a potentialofelectrostaticpotential step. finite range.

regionsmust be included.) Thus, if �> V0+ m, both observerswould agreethat the wave functiondescribedparticles,and, if � <— m, both observerswould agreethat the wave function describedantiparticles.If the energyis in the interval, m <� < V0—m, however,then the observeron the leftwould interpretthe wave function as that of a particle andthe observeron the right would interpretthe wave function as that of an antiparticle.Sucha wave function is schematicallydrawnin thefigure.Decreasing V0 so that it becomessmaller than m would eliminate this mixing of particle andantiparticlecontinuumstates.A similar phenomenonoccurs in the caseof an over-critical potentialwith finite range,as shownin fig. 2.5.

If the barrier of fig. 2.2 is replacedby a potential that transforms as a world scalar, then noanalogueof the Klein paradox[113]canbe found, regardlessof the strengthor sign of the potential.The crucialdifferenceis that both electronsandpositronsrespondto the scalarpotentialin the samemanner;whereasan electrostaticpotentialthatis repulsiveto electronsis attractiveto positronsandvice versa.

2.2. Boundstates

In this subsectionand the next, we discussin detail two examplesthat haveplayed an importantrole in the developmentof our presentunderstandingof critical field phenomena,the squarewellpotentialandthe Coulombpotentialof an extendednucleus.Our considerationswill be restrictedtothe caseof sphericallysymmetricpotentials,and thuswe write the Hamiltonianin the form

V(r)+m ~y!(r.p+iu.l)or . V(r)—m , (2.14)—~-- (r . p + ~ . I)

after exploiting someof the propertiesof the Pauli matrices[114].In this equationI is the angularmomentumoperator. In obtainingeq. (2.14), the potential V is assumedto transform as the timecomponentof a four-vectorunderLorentz transformations.The solution of the Dirac equationmaybe written,

~ ~g(r)(~) 1 (215)~ r, - Lif(r)~~~(~)]’

where ~ is an eigenfunctionof the total angularmomentumoperator.It is a combinationof

J. Rafeiski et a!., Fermions and bosons interacting with arbitrarily strong external fields 245

sphericalharmonicsand Pauli spinors. In eq. (2.15) the quantities K and ~ are the usual angularmomentumquantumnumbers.Substitutingeq. (2.15) into the Dirac equation,wherethe Hamiltonianhasthe form of eq. (2.14), andusingthe relations

~ —(K+ i)x~~ (2.16)

r~= — ~ (2.17)

onecan derivethe coupledradial equationsfor the functionsf andg, namely,

(~_— ~—-J)~r~+ (�— V(r) — m)g(r)= 0,

(~-+~_±--i)g(r)+ (V(r)— m — �)f(r)= 0. (2.18)

We first considerthe squarewell potential[16]of fig. 2.6. Insidethe squarewell the solutionto eqs.(2.18)that is regular atthe origin involvessphericalBesselfunctionsof the first kind [115].Thus, thewave function in this region is given by

r j~(p’r)~~(~) 1~4r(r)=A~ ip’S~ .~ , , (2.19)

LVo+m+�~t~rJx_~(r)jwherep’ = V(�+ V0)

2 — m2, S,., = ,~III~I,I is the orbital angularmomentum,1= 1 ±I dependingon thesign of K and A is a normalizationconstantto be determinedbelow. Outside the well the wavefunctionis given by

r k1(/3r)~~(r~)1

i/i(r)=CJ —i/3 k- ~‘ (2.20)

L + I(I3r)x_K(r)J

where /3 = \/m2—�2 and k, is a modified sphericalBessel function. For large argumentsk1(z)-+

IT e_z/2z. Requiring the wave function to be continuousat r = R leads to a transcendentalequationwhich determinesthe energyeigenvalues,namely,

F( )=— S~p’(�+m)kJ(f3R) j,(p’R)0 221)K � — 13(�+m+Vo)kr(13R) jr(p’R) (

V(r)

R r

Fig. 2.6. Profileof thesphericallysymmetricsquarewell.

246 .1. Ratelski et a!., Fermions and bosons interacting with arbitrarily strong external fields

For s statesthis reducesto

I(~ R) F1(�)= (�+ m+ V0)(1 + f3R)— 1— p’R cotp’R =0, (2.22)

after setting 1 = 0 and 1= 1. In fig. 2.7 the is eigenvaluesdeterminedfrom eq. (2.22) aredepictedasfunctions of the potential strength for threedifferent values of the range. All energiesthereareexpressedin termsof the particle’smass,andthusthe resultsapply to otherspin one-halfparticlesaswell as to electrons.One of the rangesis chosento be the Comptonwavelengthof the particle.Thecurvaturenearthe upper continuumis much more pronouncedin this case.It is barely noticeablewhenR = lOm’ andimperceptiblefor the short rangecase.The behaviorof all threecurvesnearthelower continuumis linear. The smallestvalueof the rangecorrespondsto that chosenby PieperandGreiner[27].(R = 8 fm for electrons.)We note that eq. (2.22) maybe solvedanalytically for the pointat which the various s levels join the lower continuum.Setting� — m thereyields the requirementp’R = flIT, wheren is a nonzeropositiveinteger.This requirementmaybe satisfiedby

Vocr = m -t- V~ir/R)2+ m2, (2.23)

wherethe positiveroot is the appropriateonefor attractivepotentials.Substitutingthe values of therangeusedin fig. 2.7 into eq. (2.23),one obtains(Vocr)is = 2.048,4.296, 152.77 and(V~r)

2s= 2.181for

\\ RrIO~’ m ~ ••. I

::LI~i?~~1_ 2.0

(a) (b)

R=O.O2O7m~

50

(c)

Fig. 2.7. is eigenvaluesasfunctionsof the potential strengthfor a squarewell potential.Resultsfor the2s level arealsoshown in onecase.

J. Rafeiski et a!., Fermions and bosons interacting with arbitrarily strong external fields 247

fig. 2.7a. One canalso developapproximatetechniqueswhich suffice to determinethe Is eigenvaluesfor rangesR that arelargeor small in comparisonwith the particle’sComptonwavelength.Thesearebasedon expansionsof eq. (2.22) in powersof (mr)’ or mR,whicheveris appropriate.In the caseofa short range, it is straightforward to show that the eigenvalues� = m, 0 and — m occur whenV0= ITIR — 3m, rr/R — m and IT/R + m. Thesevalueswereobtainedby PieperandGreiner[27].Two ofthe threevalues listedon page119 of Akhiezerand Berestetskii[19] areerroneous.

Next, we considerthe normalizationof the radial functionsand the determinationof the constantsA andC. The normalizationconditionstatesthatf d

3rt/it(r)i/i(r) = 1. Substitutingthe expressionfor t/i

from eq. (2.15) yields

fr2dr(g2+f2)= 1, (2.24)

after using

fdi~~’,y~= 6KK6wM. (2.25)

The quantity Pv = + f2, which occursin the integrandof eq. (2.24),will be referredto as the vectordensitybelow. Using eqs. (2.19) and (2.20) for the radial functionsthe normalizationconditiontakesthe form

A2 JR r2 dr[J~(p’r) + (V0+m + ~)2J~(p’r)]+ C

2f r2 dr[k~(/3r)+ (� )2 k~(/3r)]= 1. (2.26)

It is straightforwardto integratethe sphericalBesselfunctionsin eq. (2.26). In order to determinetheconstantsA and C, it is necessaryto usethe relation

Aj,(p’R) = Ck,(f3R), (2.27)

which results from the continuity requirementat r = R. Then one may calculatethe probability oflocatingthe particle inside the potentialwell P

1 = f~r2 dr(g2+ f2), which is equalto the first term of

eq. (2.26). This probability is shownas a functionof potentialstrengthin fig. 2.8 for R = m’. As thepotentialstrengthincreases,this probability first increases,in accordwith one’sexpectationsbasedonnonrelativisticquantummechanics,reachesa maximum near ~ = 3.5m and then decreasesuntilV

0 = 4.30m, wherethe bound level joins the lower continuum.Another instanceof this behavioris

‘II

C_L-~.-- . .11.5 2.5 3.5 4.0v~[m]

Fig. 2.8. The probabilityof locatingtheparticleinside the well asa function of potential strength.

248 J. Rafelski et al., Fermions and bosom interacting with arbitrarily strong external fields

v0= 4.25 r ~m1 2

v~-3.0 v0K2.O Fig. 2.10. The sum of the squaresof the radial functions for0 I 2 3 4 5 6 V0= 1.3m, 2.Om, 4.Om and4.295m.

[rri~]

Fig. 2.9. The sum of thesquaresof theradialfunctionsoutsidethesquarewell for variouspotential strengths.

depictedin fig. 2.9, where values for Pv outside the squarewell are plotted for various potentialstrengths.Again, when the potential strength is such that the Is eigenvalueis near the lowercontinuum,a further increasein its magnitudeleads to a wave function that is less localized.Thedashedcurve in fig. 2.9 representsa casewherethe Is eigenvalueis very nearthe lower continuum(�~~= —0.9989m).Carefully taking the limit �—~— m in eqs. (2.20) and(2.27) yields p~—*Dir

4 outsidethe well. We fit the valueof D so that Pv hasthe correctvalueat R= m~andobtainD = 0.42m~.Thus at� = — m, the Is statecan still benormalized,althoughthereis no exponentialdampingfactor.In fig. 2.10, the quantityPv is given for V

0 = 1.3m,2.Om,4.Om and4.295m.In fig. 2.11, valuesfor Pv aregiven for a rangeR muchsmallerthanthe Comptonwavelength.This exampleis chosento emphasizethat the radial functionscanbe localizedto distancesthataresmall in comparisonwith the particle’sComptonwavelength.

Before discussingthe Coulomb problem, we wish to discusssome of the implications of thepropertiesof the squarewell wave functionsfor variousaspectsof the interpretationof the theory.The first of theseis relatedto the limit of the radial functionsas � —~— rn. This meansthat at � = —

the positrondoesnot escapeto infinity. Spontaneouspositron productionat infinity occursonly forV> Vcr and not at V = Vcr. A similar phenomenonwill occur in the caseof the Coulombpotential,but the limits of the radial functionsareverydifferent.The secondis relatedto the increasein thesizeof Pvas the Is eigenvaluenearsthe lower continuum.As statedin the introduction,the absenceof anelectronin this statemaybe interpretedas apositron.Thus,in thisregion, the positronwave functionis becoming less and less tightly boundas the potential strength increases,in anticipationof thedelocalizationof the wave functionfor V> Vcr.

Next, we considerthe solutionsof the Dirac equationwhen the externalpotential is a Coulombpotential V(r) = — Za/r, wherea is the fine structureconstant.Thesesolutionsare essentialfor thethreedifferent chargedistributions treatedhere,which are the point charge,the uniformly chargedsphereand the uniformly chargedsphericalshell (seefig. 2.12).Making the substitutionsW = rg and

.1. Rafelski et a!., Fermions and bosons interacting with arbitrarily strong external fields 249

Fig. 2.11. The sumof thesquaresof theradial functionsfor a very 0 4 8 12 16 20 24 28short rangepotential. r ~fmJ

Fig. 2.12. Coulomb potentials for a point charge (full curve), auniformly chargedsphere(dashedcurve),anda uniformly chargedsphericalshell (dottedand dashedcurve).

U = rf in eqs. (2.18), we have

dU / Za\ K dW / Za\ K—=im—�——--iW+-U, —=im+�+----jU——W. (2.28)dr \ rj r dr \ rj r

To solve thesecoupledequations,it is convenientto reduce them to a second-orderdifferentialequation [114]. To this end we make the substitutions x = 2/3r, U = \/m — �(çt.~— &) and W =

+ 42),which leadsto a setof coupleddifferentialequationsfor ~ and&. This latter set isvery readily uncoupled,and the result may convenientlybe expressedin terms of 4~2= x”2~

2asfollows:

2 1 2

dc12 1 ~1 Za�~1~—y ,~

where ~2 = K2 — Z2a2. Equation(2.29) hasthe form of Whittaker’s equation[115,116, 117]. Thus the

most generalsolutionof eq. (2.29) canbewritten

12(x)= AMk~(x)+ BWk7(x), (2.30)

whereMk~(x)is the Whittakerfunction that is regularat the origin, W57(x)is the Whittaker functionthat decaysexponentiallyfor large x, and k = + Za�/3~.The Whittaker functionsmay beexpressedin termsof [115]hypergeometricfunctions,that is,

Mk,,(x)= e’2x’2~’M(~— k+ y, 2y + 1; x),

Wk7(x)= e’

2x”’2~”U(~—k+ y, 27 + 1; x). (2.31)

Let us first treatthe caseof the point charge.Theneqs. (2.28) apply for the entire rangeof r, andthe solutionmustsatisfyboundaryconditionsatboth r = 0 andr = ~. In order to satisfythe conditionof regularityat the origin, thecoefficient B of eq. (2.30) mustbe zero sinceWk~(x)is irregularat theorigin. For large x, the hypergeometricfunction M behavesas ~ ex unless the seriesterminates.Thus, to satisfy the condition of finitenessat infinity, the series must terminate.This

250 J. Rafeiski et a!., Fermions and bosons interacting with arbitrarily strong external fields

implies ~— k+ ‘y = — n, where n = 0, 1,2,3,..., which leads to an expressionfor the energyeigen-values

2 2 —1/2Za�=mii+ I . (2.32)L (n + VK2 — Z2a2)2i

For the is state (n = 0, K = — 1) eq. (2.32) reducesto �~= rn’s/i — Z2a2. The normalized is wavefunction [9] for spin up is given by

çfi(r) = (2mZa)312 1 + ~ (2mZar)~em~ 0 , (2.33)\/4IT 2F(i + y) i(1 — y)(ZaY’ cos0

i(i — y)(Za)’ sin 0 e’~’

where1’(x) denotesthe usualgammafunction.Now weconsiderthe limit Za—~ 1. Fromthe analyticexpressionfor the is eigenvaluegiven above,

it is apparentthat �~—~0.Further,d�10/dZis infinite at this point. The waveof eq. (2.33) is singularat

the origin, but it can still be normalized.In fig. 2.13, the quantity r21t112 is plotted as a functionof r.

This quantityhasa nonzerovalue at the origin for Za = 1.Let us now examine the point chargesolutions when Za> 1. The analytic expressionderived

above for ~ gives imaginaryresults,which are not compatiblewith the Hermiticity of the DiracHamiltonian. Actually, eqs. (2.32) and (2.33) do not furnish an acceptablestarting point when Za> Isince the assumptionZa < 1 is necessaryto obtain them. The readershould not confusethe issuesthat arisefor the point chargecasewhenZa> 137 with thosethat arisewhenthe nucleushasa finitesize.

One methodof obtainingsolutions to eqs. (2.28) when Za> 1, which is due to Case[22], beginswith a seriessolution,namely,

U=r°(a0+a1r+~•.), W=r~(b0+b1r+.. .) (2.34)

Substitutingthesetwo expressionsinto eqs. (2.28),one can determinethatp = ±VK2 — Z2a2,where

the two roots correspondto the two linearly independentsolutions.Thus,nearthe origin

U = a0r ~

2~2(i+~. .) + a~r~~(i +~..) (2.35)

If Z2a2< 1, then there is a qualitative difference in the behavior of the two solutions,and the

normalizationconditionsufficesto choosebetweenthe two solutions(at leastthis is clear for Za ~ 1).

C o 0.4 0.8 .2[m’]

Fig. 2.13. The probability densityasa function of r for the pointcharge.

J. Rafelski et al., Fermions and bosom interacting with arbitrarily strong external fields 251

However, for Za> 1 and IKI = I, the quantity p is purely imaginary and thus the behavior of bothsolutionsnearthe origin is similar. The normalizationconditiondoesnot suffice to choosebetweenthesolutions. To proceed further, it is convenient to write eq. (2.35) in the form, U =

a cos[(Z2a2 — 1)1/2 In r + 6] ~ ., wherea and 6 are arbitraryconstants.If one also requiresthat allof the s statesbe mutually orthogonal, then thereis only one arbitraryconstant6 for all s states.Examiningthe solutionsat larger allows one to determinea complicatedtranscendentalequationforthe energy.The energyeigenvaluesdeterminedfrom this equationarenot unique;theydependon thevalueof ô, but theyarereal. The arbitrarinessassociatedwith the parameter6 alsomeansthat the setof basis functions is not unique. Thus, although the Dirac Hamiltonian is hermitian, it is notself-adjoint whenZa> 1 unlessthe parameter6 is specified [118].Herewe havean exampleof thedistinction between hermiticity and self-adjointness,which arises becauseadditional informationbeyondthe form of the hermitian operatoris neededto specify uniquely the completeset of basisfunctions. Oncea completesetand the associatedspectrumhasbeenspecified,it is referredto as aself-adjoint extensionof the operator.*

The questionssurroundingthe solutionsfor the point chargecasewhenZa> 1 areinterestingfromthe pointof view of mathematics,but they bearalmostno relationto the questionsthat arisewhenthenucleushasa finite size. For example,thereis no questionabout the uniquenessof the eigenvaluesafter the nuclearchargedistributionhasbeenspecified.Equation(2.28) no longer describesthe wavefunctionnearthe origin, and thusthe oscillatorybehaviorof eq. (2.35) doesnot occur.Equation(2.28)doesdescribethe wave function outsidethe nucleus,however,and one might justifiably wonderifpurely imaginaryvaluesfor y in eq. (2.30) lead to any unusualfeatures.This is not the case.

When the nucleushasafinite size,the Coulombpotentialno longer becomesinfinite nearthe origin(seefig. 2.12). Instead,in this region,the potentialmaybe written as apower series,

V(r)= V0+ V1r+ V2r

2+~ .. . (2.36)

The valuesof the coefficients dependupon the natureof the chargedistribution. If the nucleusistakento be a uniformly chargedsphericalshell, then V

0 = — Za/R,whereR is the nuclearradiusandall the remainingcoefficientsare zero.If the nucleusis takento be a uniformly chargedsphere,then

— ~Za/R, V2 = ~Za/R3and the remainingcoefficients are zero. In thesetwo cases,the power

seriesexpressionis exactthroughoutthe entire nuclearvolume. Someof the resultsdiscussedbelowarebasedon a Fermi distribution

( ~.. Po (237)P T) ~+exp(4ln3(r—R)/t)’

wherePo is a normalizationconstant,t = 2.5 fm and R plays the role of the nuclearradius.The newfeaturein eq. (2.37), which arisesas a consequenceof the parametert, is a diffuse nuclearsurface.This featureis necessaryto reproducethe resultsof electronscatteringexperimentsandmuonic atomexperiment[119].Of course,thenuclearradiusfor suchlargevaluesof Z is a matterof conjecture.Ifone assumesthat the density of superheavynuclei is the same as that of heavy nuclei, thenR 1.2A”3 fm, whereA is the total numberof nucleons.For superheavyelectronicatoms [27], therelationbetweenA andZ is usuallytakenas A = 0.00733Z2+l.3Z+63.6. Somestudiesof superheavymuonic atoms [45]arebasedupon the relationA = 2.5Z.

252 J. Rafelski et a!., Fermions and bosons interacting with arbitrarily strong external fields

We seekseriessolutionsto the radial equationswhich apply inside the nucleus,namely

U = ~ A~r”~, W= B~r”~. (2.38)

Inserting theseexpressionsinto the radial equations,we find that either p = K, B0 = 0 and A0 is

arbitrary (to be determinedlater by normalization) or p = — K, A0 0 and B0 is arbitrary. Theconditionof regularityat the origin allows us to choosebetweenthesetwo casesfor agiven valueofK. Further,

A V1B0+(V0+m—�)B1A1— 1+p—K 0’ 2 2+p—K

B — (�+ m — V0)A B — — V1A0 + (�+ m — V0)A1 (2 39)~ l+p—K 0’ 2 2+p—K

For the is state,the resultsare

U=B0(V0+rn—�)r2/3+~,

W=Bor[l+~m VoXV0+rn~~�)r2+...] (2.40)

and it is clear thatg = W/r andf = U/r arenot singularat the origin, as in the point chargecase.Toobtain eq. (2.40) we haveset V

1 = 0. This doesnot influencethe first two terms in the expressionsabove,which decidethe questionof regularityat the origin.

In PieperandGreiner’soriginal work, theradial functionsfor r < R weregeneratedfrom a powerseries that is an extensionof eq. (2.40). The radial functions outside the nucleus, the Whittakerfunctions, were calculatednumerically. (To obtain the relevantWhittaker functions, onemust setA = 0 in eq. (2.30).) The eigenvalueswere determinedby solving the transcendentalequationwhichresultsfrom the continuity condition on the wave function at r = R. Most of the recentcalculations,however,haverelied on numericalintegrationtechniquesboth insideandoutsidethe nucleus.Pieperand Greiner’s resultsfor the energyeigenvaluesare shown in fig. 1.2. In the introduction,we havealreadyremarkedthat all of the eigenvaluesdecreasemonotonicallyas the chargeincreasesand thatnoneof the eigenvaluesor the wave functionsexhibit anyunusualbehaviorat Za = I. The pointsatwhich the levelsjoin the lower continuumarewell isolated.The critical valueof Z is 170, wheretheis level joins the lower continuum,and the

2P112 level joins the lower continuumat aboutZ = 183.Thus,if oneimaginesasuperheavynucleussurroundedby no atomicelectrons,increasingthe chargebeyond169 would lead to the productionof two ~l/2 positronsandincreasingthe chargebeyond182would lead to the productionof two Pl/2 positrons.As statedin the introduction,the Pauli principlelimits the numberof positronsproduced.Largenumbersof positronswould not be producedevenforZ near300sincethe degeneracyof all thelevels thatjoin thelower continuumbelowZ = 300is rathersmall.

PieperandGreiner’sresultsfor r2ItfrI2 are shownin fig. 2.14. For Z = 169, the ls probability densitypeaksat 25 fm, which is much closer to the nucleusthan one might expecton the basis of a naiveargumentaboutthe Z dependenceof the exponentialfactor in the Whittakerfunction of eq. (2.30).Animportant question,which was first answeredcorrectly by Popov [49], is the nature of the wave

J. Rafe!ski et a!., Fermions and bosons interacting with arbitrarily strong external fields 253

~,—Is,Z~I69

Z~140

a~ ~

~100 200

Z~I8I

0.001 f”\.~1,—2p2, Z~I4O

fl 2=255 -

C ~R0 00 200r[fm]

Fig. 2.14. Low-lying sandp stateprobabilitydensitiesfor severalvaluesof Z. The dashedline markstheapproximateextentof thenuclearchargedensity(l0fm).

functionat Zcr. One can arriveat Popov’sresult by carefully taking the limit � —* — rn in the Whittakerfunctions.The key stepin this processis the evaluationof

1/am \ --1/21h(r)=lirntf(\—j__2~r)Wk.~(xX2f3)‘ J, (2.41)

which appearsin eq. (2.30). Usingthe secondof eqs. (2.31) and letting a = ZamA’, we have

h(r) = �i-~~m{,~1/2e2~~Tf(a—2y)U(a, 2y+ 1;2Zarnr)} (2.42)

The limit of the hypergeometricfunctionmaybe found in Abramowitzand Stegun[115],and

h(r) .!—~m2(2Zma)~r”2K2~(V8Zamr), (2.43)

whereK is a modified Besselfunction. For large r, h(r) -~ r”4 exp{ — \/8Zami}. Thus, spontaneous

positronproductionat infinity doesnot occur atZ = Zcr, but only when Z> Zcr. Below, we shall findthat the rate of spontaneouspositron productionvanishesas Z —* Zcr, which is consistentwith thisresult. Furtherresults[35]for the is stateprobability densitiesareshown in fig. 2.15.

In fig. 2.16, a comparisonof Is eigenvaluesfor a uniformly chargedsphereandauniformly chargedsphericalshell is carried out. The radius of both chargedistributions is the same.NearZ = 170, thedifference in binding energy is about 100 keV. This makesa differencein the value of Zcr of onlyabout 3 units becauseof the steepnessof the slopesof the curves in this region. The differencebetweenthe two curves of fig. 2.16 is probablyan exaggerationof the effectsof uncertaintiesin thenuclearchargedistribution since the uniformly chargedsphericalshell representsan extremecase.The differencesbetweenthe eigenvaluesfor a Fermi distribution and auniformly chargedsphereshouldbe considerablysmaller.

The effect of increasingthe density of nuclear matter is studied in fig. 2.17. The density forr0 = 0.2fm is more than 200 times greaterthan the densityfor r0 = 1.2fm. As the densityof nuclearmatterbecameinfinite, andR—*0, one would expectZcr~*137. Thus,if one could makesuperheavy

254 J. Rafeiski et a!., Fermions and bosons interacting with arbitrarily strong external fields

SC I

Z=I00 Z~II4 Z=164

xl/8

0 A II0~ IO~ IO~ d2 0’ IO4 I0~ 02 0’ I0~ I0~ 02 1& 10

R

Fig. 2.15. Probabilitydensitiesfor the Is statefor Z = 100, 114 and169. The unit of theabscissais the Bohr radius a0 and that of the ordinateis

a~1.

nuclei with sufficiently small radii, then spontaneouspositron production at infinity would beobservablenearZ = 150 or 160. In the limit R—*0, the peakof the is probability density in fig. 2.14movestowardthe origin. At R = 0, it shouldbealmostidenticalin shapewith thecurveof fig. 2.13 forZ = i37. It is interestingto note thatonewould probably neverbe able to infer the correctphysicalinterpretationof the theory from the above treatmentof the point chargecasewith Za> 1 (seeeq.(2.35) and the discussionfollowing).

The exactlocation of Zcr is an importantquestionfor the experimentalverification of the theory.The effects of other atomic electronsas well as the effects of vacuumpolarizationand electromag-netic self-energywill modify the resultsof Pieperand Greiner. It hasoften beenstatedthat quantumelectrodynamicalcorrectionsmight be considerable.That this is not so can now be stated withconfidencefor vacuumpolarizationeffects,as we will discussin somedetail in section3 below. Thestudyof theelectromagneticself-energycorrectionshasnot progressedto the samestage,but thereisno indication that thesebecomevery large. The sum of all field correctionsis probably less than

I I I

~ ~TELL ~ ________________________IO0 120 140 160 ISO 00 120 140 60 ISO

z z

Fig. 2.16. A comparisonof the Is eigenvaiuesfor a uniformly Fig. 2.17. The sensitivityof the is eigenvalueto thenuclearradius.chargedsphereand auniformly chargedsphericalshell. The nuclearradiusis given by R = r0A”

3.

I. Rafelski et al., Fermions and bosons interacting with arbitrarily strong external fields 255

20 keV nearZ = 170. If this assertionis true,then anyuncertaintyattendingthe field correctionsgivesrise to an uncertaintyin Zcr of one unit or less.

Fricke,Greinerand coworkers[120,12i,122] havestudiedextensivelythe effects of the electronicmany-body problem on the eigenvaluesof fig. 1.2. Their calculations have been based upon therelativistic Hartree—Fockapproachor the relativistic Hartree—Fock—Slaterapproach[123].The basicworking equationof theseself-consistentcalculationsis

[Ho+ e2~ f d~r’i/i~(r’)~(r’)— e2~ f d3r’ ~frt~4~,(y1 JIIIfr) = E~l/J~(r), (2.44)

whereH0 includesthe Dirac kinetic energyoperatorandthe nuclearCoulombpotentialandthe indexk runsover the quantumnumbersof all the occupiedatomicorbitals.The exchangeterm of eq. (2.44)has beenwritten in a form which is a convenientstarting point for the Slaterapproximation.Thisconsistsof replacingthe third term in the bracketsof eq. (2.44) by a term that dependsupon theelectronicchargedensity.The differencesbetweenthe relativistic Hartree—Fockand Hartree—Fock—Slatercalculationsareimportant for detailedagreementbetweentheory [122]andexperiment[124]inheavy atoms such as fermium, but are not important when viewed on the scale of fig. i.2. Acomparisonof the resultswith and without the effectsof atomic electronsis carriedout in table 2.1and fig. 2.18. The differencebetweenthe two results is considerablenear Z = 170, about 120keV.However,this differenceshifts the critical chargeby only aboutfour units from 170 to 174. In carefulcomparisonsof the two, it should be rememberedthat Fricke and Soff’s work is basedon a FermichargedistributionandPieperandGreiner’swork on a uniformly chargedsphere.The total electronicchargedensitiescalculatedby Fricke et al. [121]areshownin fig. 2.19.

Soff, MUller and Rafelski [45] havecalculatedthe is eigenvaluesfor superheavymuonic atoms.Sincethe muon’s Comptonwavelengthis about i.9fm, finite nuclearsize playsa very differentrole inthe determinationof the eigenvalues.This fact is, of course,well known in the heavy elements,andthere,muons bound in atomic orbitals are often usedas probesof the nuclear chargedistribution[i19]. Someof the eigenvaluescalculatedby Soffet al. areshownin fig. 2.20. Therethe nuclearcharge

Table 2.1Is eigenvaluesand some corrections.The rela-tivistic Hartree—Fock results are taken fromFrickeandSoff, andthe finite nuclearsizeresultsfrom Pieper and Greiner. The point nuclear 2

+mcresults are listed for comparison.The vacuumpolarization correction is takenfrom Pieperand — — — — —— —

Greiner. —•_... ~-.. ——

I mYT~140 124.3 85.1 —2.00 Z 160 170150 4.2 —48.8 —3.89160 — 168.7 —238.7 —7.11 Fig. 2.18. Energyeigenvaluesfor the inner electronshells.The full169 —381.3 curvesarebasedon theThomas—Fermimodel, the dashedcurves173 —495.2 — 12.00 on theself-consistentpotentialandtheremainingcurveson Pieper___________________________________________ and Greiner’swork.

256 J. Rafeiski et a!., Fermions and bosons interacting with arbitrarily strong external fields

I0~ ID3 IO~ 0’ I 10 -mc ~ ~ ~ —. —--- — —.--- .~ .— — —~----~---- -.----,.----

r [ac]

Fig. 2.19. Totalelectrondensitiesfor Pb, Z = 114 and Z= 164.The Fig. 2.20. The Is energyeigenvaluefor muonsas afunction of Z.

abscissais given in atomicunits.

distributionwas takento be a Fermi distributionwith diffusenessparametert = 2.2fm. The equivalentradiusandthe numberof nucleonsare givenby Req= 1.2A113(fm) and A = 2.5Z.Thecritical valueof Zis near 2200. It is an easyexerciseto obtain an estimatefor Zcr. If one assumesthat the Coulombpotentialof the Fermi chargedistributiondoesnot differ too muchfrom that of a uniformly chargedsphericalshell, then V(0) = — ~Za/R.Substitutingthe abovevaluesfor R andA into this expressionyields V(0) = — ~aZ213/(2.5)”3r

0, wherer0 = 1.2 fm. Further,assumingthat the conditionfor the criticalpotentialis not grosslydifferent from that of the squarewell (eq.(2.23)),one has that

~aZ213/(2.5)’t3r

0 = m + ‘s/rn2+ (ir/R)2. (2.45)

The lastterm in the squareroot is small in comparisonwith the first term andthuswe neglectit. Theneq. (2.45) yields Zcr~2000. In fig. 2.21, the dependenceof the is eigenvalueon the nuclearradiusis

m c2

!~~~~IRn,m1

— mC2

Fig. 2.21. The Is energyeigenvaluefor muonsasa function of nuclearradiusfor Z = 184.

J. Rafelski et a!., Ferm ions and bosons interacting with arbitrarily strong external fields 257

shownfor Z = 184, which is the largestchargestatethat could be manufacturedin U—U collisions.The resultingsuperheavynucleusmusthavearadiusof lessthan 1 fm for the spontaneousproductionof muons.The positivemuonswould escapeto infinity, just like the positrons.

Forcontrastwith theabovebOundstateeigenvalues,which areall basedon electrostaticpotentials,we discusson exampleof a particle of massm boundby a potential 41 that transformsas a scalarunderLorentz transformations.Thus the Dirac Hamiltoniantakesthe form

Hra.p+f3(rn+41). (2.46)

In this example, the potential is given by 41 = —g/r. Then one can find an analytic expression[106,125, i26] for the energyeigenvalues

�±= ±m[l —g2/(N+ .~/~Sj)2]iI2, (2.47)

where N = 0, 1,2 The dependenceof several of the lower eigenvalueson the strengthof thepotential g is shownin fig. 2.22. In this figure and eq. (2.47), the symmetrybetweenpositive andnegativeenergiesis apparent.Fromeq. (2.47) it is clearthat �~—*0 for all statesas g—* ~ After secondquantization,the negative energystatesare associatedwith antiparticles.Thus,unlike the caseofstrongelectrostaticpotentials,the parts of the spectrumassociatedwith particlesandantiparticlesarealways separatedfor finite g. Spontaneousproduction doesnot occur since the energyrequiredtocreatea particle—antiparticlepair is alwaysgreaterthanzero.The resultsdiscussedin connectionwiththis exampleserveto underscorethe remark madeatthe endof the discussionof the Klein paradoxaboutthe crucialdifferencebetweenelectrostaticpotentialsand world scalarpotentials.

2.3. Continuumstates

As the final part of this section,we discussthe continuumsolutionsof the Dirac equationfor asquarewell potential and a Coulomb potential. Our main interest will be confined to the lowercontinuum,althoughmany of the expressionslisted beloware valid for the uppercontinuumas well.The central issue of this discussionis the appearanceof a resonancein the lower continuumas thepotentialstrengthis increasedbeyondthe critical value. This resonanceis the analyticcontinuationofthe boundstateinto the lower continuum.Knowledgeof its propertiesis crucial in understandingtheshape of the energydistribution of the spontaneouslyproducedpositronsand the nature of theresultingchargedvacuum.

We first considerthe caseof the squarewell potentialandrestrict the discussionto s states.The

~:~~3/2

Fig. 2.22. Energyeigenvaluesbasedon a scalarpotential with strengthg.

258 1. Rafe!ski et a!., Fermions and bosons interacting with arbitrarily strong external fields

radial functionsinside the well aregiven by_____ — Bp’ Icosp’r sinp’r]g(r) — p’r f(r) — � + m+ V0 L p’r — p’2r2 ~ (2.48)

Theseexpressionsarevery similar to thoseof the boundstatecase(eqs.(2.i9)) andp’ is definedin thesameway. Outsidethe rangeof the potentialthe radial functionsmay be written

— B’ sin(pr + 6) ,, — B’p Icos(pr+ 8) sin(pr+ 6) 2 49g(r)— pr ‘ j(r)_+ L pr — p2r2 ( . )

wherep = \/~2_ m2 and8 is the phaseshift [127,128]. Using the conditionof continuity of the wavefunctionat r = R, we maydeterminethe phaseshift 8. After somerearrangement,this conditiontakestheform

cot(pR+ 8) = cotp’R ~pR(� +m + 17~) ‘ (2.50)

which maybe readily solvedfor 6. Phaseshiftsdeterminedfrom this equationareshown for severalvaluesof thepotentialstrengthin fig. 2.23. For V

0 = 4.29m, a valueslightly lessthan the critical value,the phaseshift is negativeand exhibits no sharpdependenceon the energy.For the other valuesofV0, which aregreaterthan V0cr, the phaseshifts beginatzero andincreasesharplythrough the point8 = ir/2, which is characteristicof a resonance.Thus,below the critical potential strength,the phaseshift behaveslike that for a repulsivepotentialandabovethe critical potential strengththe phaseshiftbehaveslike that for a strong,attractivepotential.The contrastbetweenundercriticalandovercriticalpotentialsis especiallyapparentin fig. 2.24 wherethe energydependenceof sin

2 6 is shown.Theresonantpeak,which appearsonly in the overcritical cases,movesfurther into the lower continuum

3.0 • I I I

fT J~I46rn

~~0=4.29m -1.2 -1.4 .6

e Em] �[mI

Fig. 2.23. Phaseshifts for squarewell potentialswith strengths Fig. 2.24. The energydependenceof ~ S for severalsquarewellsnearthecritical potential, with strengthsnear thecritical potential. R = m’.

I. Rafelski et al., Fernsions and bosons interacting with arbitrarily strong external fields 259

andbecomesbroaderas the potentialstrengthis increased.The positionof the resonancein the lowercontinuumis shownin fig. 2.25.Near theboundaryof thelower continuum,the curveis almostlinear,but as the resonancebecomesmore deeply embeddedin the lower continuum,deviations fromlinearity becomemoreapparent.The slope of the curve near � = —1.0 is —0.7, which is nearly thesameas the slope of the correspondingcurvefor boundstatesin fig. 2.7. In fig. 2.26the sin2(8— 6~)isshown for severalovercriticalpotential strengths.The quantity 6~is the phaseshift computedatthesamevalue of � for V

0 = 4.29m, an undercriticalstrength.Thus, a backgroundphase is subtractedfrom thephaseshiftsof the overcriticalpotentials.The reasonsfor doingthis will becomeapparentinour discussionof real and virtual vacuumpolarization. Loosely speaking,the subtractionof thebackgroundphase serves to emphasizethe new featuresassociatedwith the appearanceof theresonancein the lower continuum.One can seethe influenceof the backgroundphaseby comparingthe resultsof figs. 2.24 and2.26.

Now we discussthe continuumsolutionsof the Dirac equationfor the electrostaticpotential,

V(r)=—~~(l_~j~),0<r<R(2.51)

V(r)=—~, R<r<c~

which is the appropriateexpressionif the nucleusis takento be a uniformly chargedsphere.Thesolutionsto the radial equationsinside the nucleusare exactly the same as the bound statecase,exceptfor the normalizationconstant,andone mayobtainthem in the samestraightforwardmanner[27].The calculationof thesefunctionsallows oneto computethe ratio of the radial functionsat thenuclearsurface,which is necessaryto calculatethe phaseshifts.

To find the solutions outside the nucleus, it is convenientto make the substitutions, W =

+ m)(411+ 412), U = iSEVSE(E— rn)(411 —412)and x = 2 ipr, whereS~= �/~�~.Thenonecan derivea setof coupleddifferential equationsfor 41~and 412, whose form is such that 41~= 44 can alwaysbechosen[55,114]. This condition ensuresthat both U and W are real for both positive and negativevaluesof �. The coupledequationsfor 41~and412 maybe readily uncoupled,andwith the substitution

~1)

‘a r— w as 0

4~44!6 ___V0[m] CErn]

Fig. 2.25. Position of the resonanceas a function of potential Fig. 2.26. The energydependenceof sin2(5 — ö~)for severalsquare

strength. wells with strengthsnearthecritical potential.

260 1. Rcjfel~’kiet a!., Fermions and bosons interacting with arbitrarily strong external fields

= x~24

1, reducedto Whittaker’sequation,that is,

~‘_ [~+(~+~)!+~ = 0, (2.52)

wherey = Zae/p and ~y= — Z2a2 as above. In the caseof nearcritical and overcritical fields y

will alwaysbe purely imaginary.Thus we write ‘~‘ = iy’. The mostgeneralsolutionto eq. (2.52)maybewritten

11(x) = a±Mk., 1,..(x) + a.Mk., ,~.(x) (2.53)

where k’ = — (~+iy) and Mk.,,..(x) is defined as in the first of eqs. (2.31). Calculating412 from thecoupledequationsfor 4,~and4’2 and rememberingthat 4 = 41’ onemayderive [55] therelations,

= i(’y’— y)a+* e’~’/(K+ iZam/p),

a÷= — i(y’ + y)a*_ e’~’J(K+ iZamJp), (2.54)

which aresatisfiedwith the choice

— N ~ — . N e~e~’(y’— 2 55a÷— e , a—i (+iZ/)

The quantitiesN and ~ are both real, N is the normalization constantand i~ plays the role of thephaseshift. Thus

= Nx 2[e~sM (x)+ i e’~e~’~ (2.56)

The matchingphase ,~may be determinedfrom the continuity of the wave function at r = R or,equivalently,the continuity of the ratio of the radial functions,that is,

--~ (257)W(r) r=R-E W(r) r=R+�

To work out the implicationsof this relation for the matchingphase i~ it is convenientto write eq.(2.56) in termsof the functions

B±(x)= x~’2[Mk.

1.,,(x)±i e”(y’ — y)(K + iZarn/p)’Mk.~,,.(x)]. (2.58)

Thus,eq. (2.57) yields, after somealgebraicmanipulation

tan = Re[B+(2 ipR)] + S�V’(�— m)f(�+ m)( W1(R)/U,(R)) Im[B+(2 ipR)] (2.59)Im[B.(2 ipR)] — S�V(�— m)I(�+ rn)( W1(R)/U1(R))Re[B_(2ipR)}

The quantity t~ doesnot give the entire phaseshift of the wave functionobservableat infinity. Wedefinethe phaseshift i~with the condition

41 1 ~ (2 60)~ 2Virp

The relationbetween~ and i~ maybe determinedby taking the limit of large r in eq. (2.56).Using the

J. Rafelski et a!., Fermions and bosons interacting with arbitrari!y strong external fields 261

asymptoticforms for the behaviorof the Whittaker functionsthere[117],one finds that

r ~ ii ..i_ ~ —In —,ry~’ ‘ \ (1 —

i~— 12 + 1e ~ ty;+.e e ~y y,~ ly 261—y n pr ~ I (K+iZam/p)F(l—iy’+iy)

where the first term gives the usual logarithmic phaseshift at large distancescharacteristicof theCoulombpotential.

Muller et al. [55] have noted that the experimentalphaseshifts are determinedfrom differentialcross sectionsand thusare undeterminedup to an overall constantphase.In order to eliminatethisambiguity in their calculatedphaseshifts, the Frankfurt group introducedthe physicalphaseshift 6suchthat

(2.62)1J-=0

This conditionis not satisfiedby the quantity~ of eq. (2.61),insteadit satisfies

_..+ &tog = y(ln ~ i) + ~. (2.63)

Thus theydefinedthe physicalshiftsas 6 = — 6iog.

Muller, RafeiskiandGreiner [55,59] generatedall of thesefunctionsnumericallyandobtainedtheresultsfor sin

2(8— 6~)shownin fig. 2.27. The backgroundphase8~was calculatedusinga nucleuswiththreeless protons.The resonancein fig. 2.27 is centeredat � = — 926 keV and the full width at halfmaximumF = 4.8keV.Their resultsfor the positionsof the Is,

12 and1Pi/2 resonancesas functionsof

the nuclearchargeare shownin fig. 2.28. The slopesof the curvesnearthe uppercontinuumare thesameas the slopesof the correspondingcurvesin fig. 1.2 nearthe critical valueof the potential.TheFrankfurtschoolalso showedthat onecould parametrizethe position andwidth of the resonanceasfollows:

— (Z — Zcr)ô— (Z — Z~~)2r, ~R (Z— Zcr)27, (2.64)

providedthat Z is not too muchgreaterthanZcr. The secondof theseexpressionsis applicableonly ifZ> Zcr + 3. For values of Z nearerZcr, it is necessaryto include a damping factor [50] whichconsidersthat the probabilityof finding low energypositronsnearthe nucleusis smallwhenZ Zcr.

2 (5~8) Is,12- resonanceat Z 184

-900 -910 -920 -930 -940

e [key]

Fig. 2.27. The energydependenceof sin2(S— S~)in an overcritical electrostaticpotentialZ = 184.

262 J. Rafeiski et a!., Fermions and bosons interacting with arbitrarily strong external fields

10 ~N2\ P~ametersf~he2~s1I2and 2p,,~

70 ISO 190 200 210z

Fig.2.28. The positionsof the Is112and2p112 resonancesasfunctions

of thenuclearcharge.

Valuesfor 8, r and y arelisted in table2.2. The motivation for writing the resultsin the form of eqs.(2.64) was to makecontactwith an earlier approach[52,53] for the calculation of the resonanceparametersandto makesimpleparametricequationsavailablefor calculations.

3. Quantumfield theory of spin one-half particles in strong external fields

This is the centraltheoreticalsectionof our treatmentof the spin ~field. Herewe deal in earnestwith the secondquantizationof the theoryandthe specialdifficulties thatattendthe caseof electronsmoving in strong external potentials.We hope to convince the reader of the validity of all theassertionsconcerningthe interpretationof the theorymadein the two previoussections.In particular,wedemonstrate:

(a) In overcritical externalfields, it is impossibleto haveastable,neutralvacuum[27—30,53,591.(b) The neutral vacuumdecaysby emitting positronsinto a new groundstate— a chargedvacuum

[28].(c) The chargedvacuumis stable[28].(d) It maybe convenientto introducea chargedvacuumas the stateof reference,evenif thefield

doesnot quite havethe critical strength*[29,30].To illustrate the main physical distinction between the case of undercritical and overcritical

potentialslet usexaminethe energyrequiredto makean electron—positronpair as a function of thestrengthof the externalpotential.

The minimum energyof a photonrequiredto createsuch a pair is the energydifference betweenthe mostdeeplybound state(the Is,,

2state in the casestudiedin section2) and the lower continuum,which is always lessthan2m.The electronremainsnearthe nucleusin abound state,andthe positronescapesto infinity. Of course,this presupposesthat the ls state was not already filled with twoelectrons.Thus,as illustratedin figs. 1.1 and1.2, theenergydifferencebetweenthe lowestbound stateandthe lower continuumdecreasesas the strengthof the potential increases,andthe minimum energyrequiredto createelectron—positronpairs becomessmaller and smaller. Finally, at the critical field

* With respectto this stateof reference,a vacancyin theIs stateis a boundpositronstate.

J. Rafe!ski et a!., Fernsions and bosons interacting with arbitrarily strong external fields 263

strength,where the lowest bound level joins the lower continuum,this energy is zero. Theretheneutral vacuum, the one-electron/one-positron,and the two-electron/two-positronstate are alldegeneratein energy.Statescontainingmoreelectronshaveahigher energysincethe Pauli principlerestrictsthe numberof electronsin the is stateto two. If the field strengthis increasedbeyondthecritical value,thenthe statecontainingtwo positronshasthe lowestenergy,and the neutralvacuumwill spontaneouslydecayby the emissionof positrons.Since thesepositronsescapeto infinity, theyare observable,andthe observationof their energyspectrumwould serveas a signal that the criticalstrengthhadbeenexceeded.This changeof the qualitativenatureof the groundstateas the potentialincreasesthrough its critical strengthis analogousto a phasetransition,as is shownschematicallyinfig. 3.1. There, the total energy including that of the positron of zero kinetic energy is plottedqualitatively as a function of the chargesurroundingthe nucleus.While in fig. 3.la the minimum isfound for neutralvacuum,in the caseof the supercriticalpotential strengthit shifts in fig. 3.Ib to afinite value, chosenhereto correspondto the supercriticalls,,2 shell only.

Most of the material presentedhere was developed in refs. [28—321.We include some sup-plementarymaterial to make the discussionself-contained.After review of someof the ideas of thesecondquantizedformalisma reducedHamiltonianis introducedwhich containsthe essentialphysics.We then discussat length the properchoice of the ground statethrough the selection of a Fermisurfacefor undercritical and supercriticalpotentials.Subsequentlywe prove the instability of theneutralstate in supercriticalfields. We also discussthe theory from the point of view of the singleparticleGreen’sfunctionandusethis to makecontactwith a self-consistentformulationof the theory.Then we deal with vacuumpolarizationand emphasizethe differencebetweenthe real and virtualvacuumpolarization. Throughoutthis section,we shall rely on material presentedin the previoussection.

3.1. The quantizedDirac field

Before writing our equationsin a chargesymmetrizedform, we discussthe quantizationof thetheorybasedon the following Hamiltonian:

= J d~x~(’x)[a. p + f3rn + V(x)]~(x), (3.1)

wherethe circumfiexesare used to denotefield operators.Thesefield operatorsmaybe expandedintermsof the completesetof solutionsof the Dirac equation,

[a . p + 13m + V(x)]4i~(x) �~t/i~(x), (3.2)

0 I 2 3 4 5 <qIQIq> 0 I 2 3 4 5 6

II’~iI<II ~ IJ’~~’II>1 VI~ <q}QIq>

Fig. 3.1. The relationbetweenenergyand chargefor (a) undercriticaland (b) overcritical potentials.

264 J. Rafe!ski et a!., Fermions and bosons interacting with arbitrarily strong external fields

whichwe tentativelywrite as follows:

= ~ £~4i,,(x), (3.3)

where the sum includesterms that belongto both positive and negativevalues of �. Inserting thisexpressionandthe correspondingonefor l//(x) into eq. (3.1) yields

H = ~ �~b~b0+ ~ ~ (3.4)

wherethe sumoverall of the stateshasbeenseparatedinto two categories.As we shall seebelow, thefirst is associatedwith particles and the secondwith antiparticles.The secondcategory containsalmost all* of the negativevalues of �. If eq. (3.4) is valid, then the eigenvaluesof H may becomearbitrarily large and negativesince the quantities�. are not boundedbelow. The path out of thisdilemma is provided by the postulatethat the operatorsbn and b~,satisfy the canonicalanticom-mutation relations[129],namely,

+ &b~,,= 0, 6X. + = 0,

+ b~.b”,,= ~ (3.5)

The last of theseequationsallows us to write eq. (3.4) in the form,

H =~ �~b,~b0—~�nbnb~+Eo. (3.6)

The first two termsof this expressionare boundedbelow sincealmostall of the negativevaluesof �

are included in the second term. The final step in the procedureis the recognition that theanticommutationrelationsof eqs. (3.5) are invariantunder the interchangeof b~and b,~.It is thissymmetry that allows one to interchangethe role of the creation and annihilation operatorsfornegativevaluesof �; thatis, onemaydefine d~= b~for all of the termsin the secondterm of eq. (3.6)andhence

= ~ — ~ �~d~d~+ E0. (3.7)

In termsof the operators6,, and ci,,, the vacuumis definedby the equations6,,Io) = d,,~O)= 0, andtheinfinite constantE0 is not observable.A convenientmeansof discardingE0 is the normal-orderingprescription[130,131]. After this is done,the eigenvaluesof H are boundedbelow,and the vacuumexpectationvalue of H is zero. In an older version of the theory, the constantwas viewed as theenergyof the occupiednegativeenergysea.(We note that if the operatorsfor negativevalues of �

were not renamedthen the definition of the vacuum would involve the statementb~I0~= 0, whichdescribesan occupiednegativeenergysea.)The field operator4i maythusbe written

iI~’(x)=~6,,~p,,(x)+~d~fi,,(x). (3.8)

The first termrepresentsthe annihilationof particlesandthe secondterm the creationof antiparticles.

* We usethis phraseto meanall with afinite number(includingzero)ofexceptions.Thus,in thecaseof weakfields,almostallmeansthesamething

asall.

J. Rafelski et al., Fermions and bosons interacting with arbitrarily strong external fields 265

Using the anticommutationrelationsandthe completenessof the solutionsof the Dirac equation,wemayobtainthe canonicalanticommutationrelations

+ ~!i~(x’)~(x)= 0,+ ~(x’)~!i~(x)= 0,

+ x’)i/i,(x) = 6,,,~6(x— x’). (3.9)

Now wewritetheHamiltonianin aform thatis hermitianandinvariantunderchargeconjugation[132],

H = ~fd3xf[~t(x),(a . p + $m + V)~(x)]+ [~/(x)(a p + ~m+ V), ~(x)]~, (3.10)

wherethebracketsdenotecommutatorsandtheoperatorexpressionwithin the parenthesesof thesecondterm actsto theleft. Insertingtheexpansionof ~ andci’ in termsof thesolutionsof theDiracequation,wehave

i~= ~ �,,[6;, 6,,j — ~ �n[cP~dn]}. (3.11)

As before,it is necessaryto subtractthe zeropoint energy,which maybe convenientlywritten as thevacuumexpectationvalue,

E0= (OIñ’IO)= —~[~ç, —~�~J. (3.12)

The symmetrizedform of the total current densityoperatoris

1~(x)= ~[~‘(x),y~(x)], (3.13)

andthe associatedtotal chargeoperatoris

= ~fd3x[~(x),~(x)]. (3.14)

In thesetwo equationse is the electroniccharge.This operatorcommuteswith the Hamiltonianandthus thereexist simultaneousstatesof the total chargeoperatorand the Hamiltonian. This fact doesnot meanthat thestateof lowestobservablechargeis the stateof lowestobservableenergy.Insertingeq. (3.8) and the correspondingexpressionfor cit~into eq. (3.14),we havethat

O = e{~6,~6,,— ~ + — (3.15)

wherethe differenceof sign in the first term accountsfor the differenceof chargebetweenparticlesand antiparticles.The last term, which is equalto the vacuumexpectationvalue of the total chargeoperator,may be interpretedas zero as long as the critical strengthof the potentialis not exceeded.The vacuumpolarizationchargedensity

p~(x)= (0lj~(x)l0)= tP~(x)4’,,(x)— ~ cir(x)ifr~(x)], (3.16)

is, of course,observableandhasbeenstudiedextensivelyelsewhere[9, 130, 131].

266 J. Rafelskj et a!., Fermions and bosons interacting with arbitrarily strong external fields

3.2. The Fermi energy

Fromthe discussionof thequantizedDirac field above,it is apparentthat particlesandantiparticlesmustbe treatedsimultaneously.However,the distinctionbetweenthe two is not as simpleas it mightappear.For the free particlesand the weak field case,the distinction is straightforward.All positiveenergystatesmaybe identified as particle states,and all negative energycontinuumstatesmay beidentified as antiparticlestates.When thepotentialbecomesstrongenoughso that someof the boundstateeigenvaluesbecomenegative,as in figs. 1.1 and 1.2, then onehasto makea decisionas to whichcategorycontainsthesestates.The theory can be treatedwithout furthermodificationsby identifyingtheselocalizednegativeenergyeigenstateswith either particles or antiparticlesas long as the fieldstrengthis subcritical.

As shownin fig. 3.2 we usethe Fermi energyto distinguishbetweenparticlesandantiparticles.Forundercriticalfields, the choiceof the Fermi energyis to someextentarbitrary.If all of the eigenstatesarenearthe uppercontinuum,as in fig. 3.3a,thenone is free to choosethe Fermi energyanywhereinthe gap betweenthe lowestboundstateandthe lower continuum.The conventionalchoice is ~F = 0.

Increasingthe strengthof the potential so that the lowest eigenvaluehasthe value zero (fig. 3.3b)posesno problem.Onecan simply lower the Fermi surfaceuntil ~F — m,as shownin fig. 3.3c. At thecritical potential strengthand beyond(fig. 3.3d),however,the Fermi energymustremainat EF = — msince it cannotbe lowered into the lower continuum. Thus, the resonance,which is the analyticcontinuationof the boundstate,must be identified with antiparticles.The choice~F = — m, which weshall usethroughoutmostof this section,can lead to eigenvaluesof the Hamiltonianof eq. (3.7) thatarenegative.Thereis nothingmysteriousaboutsuchstates.Electronsare simply boundto the nucleusby anamountthat is greaterthantheir restenergyandanystatecontainingonly suchelectronshasanenergylessthanthevacuum.Chargeconservationpreventsany spontaneouschangein the numberofelectronsas long as �> — m.

Let us considernow in moredetail the questionof the choiceof the Fermi energyin the undercriticalcase where some of the boundstate energyeigenvaluesare negative.In this case there are severalpossiblechoicesfor a stable ground statewith different charges.The neutralvacuumis in fact thestateof highestenergyamongthese,while the chargedstate characterizedby ~F = 0 is the lowestenergystate.With this choiceof the Fermi surface[28,29] all the negativeenergystates,both bound

C

CONTINUUM~~

U !! J—/~JV\uj\J\AJ~,/v\j~\~F

-in

NEGATIVEFREQUENCY_____ {n}CONTINUUM

Fig. 3.2. The Fermienergyandthedistinctionbetweenparticies andantiparticles.

J. Ratelski et a!.. Ferm ions and bosons interacting with arbitrarily strong external fields 267

H*~:,~kEF _Fig. 3.3. The Fermi energyfor undercriticaland overcritical potentials.

and continuum are consideredto be antiparticle states. Then, the operatorsci~in eq. (3.8) areassociatedwith the boundnegativeenergyeigenvaluesas well as the negativeenergycontinuum.Thechargeof sucha ground statedependsupon the numberof bound stateswith negativeenergyelgen-values.With this definition of the vacuum,then the observableeigenvaluesof eqs. (3.7) and (3.11)(disregardingthe zero-pointenergy) are alwayspositive. A vacancy in the ls state thusbecomesaboundpositron. The neutral vacuummay be viewed as an excited but stablestate(in view of thechargeconservation)of the new vacuum containing a numberof boundpositrons.The excitationenergyof the neutralstate is given by

E=— ~ �,,. (3.17)

0> �, > —

Both the neutralvacuumand the chargedvacuumare stablefor undercriticalpotentials since theenergyrequiredto createan electron—positronpair is greaterthanzero.The choicebetweenthetwo isonly a matter of convenience,unlike in the overcritical casewhere one must choosethe chargedvacuum.

The chargedvacuumdoesoffer an opportunity to view the spontaneousproductionof positronsdifferently. Let us consider the fate of a positron bound in the is state in a slightly undercriticalpotential as the potential strengthis increased;the positron becomesmore and more loosely boundand becomesunboundas the potential strengthpassesthe critical point. Hence,the useof the newvacuumallows one to view spontaneouspositron productionsimply as the delocalizationof a boundstate.

In closingthis subsection,we would again like to emphasizethat the useof the chargedvacuumisimperative for the overcritical potential. The essentialdifferenceis that in the undercriticalcase,energyis requiredto producean electron—positronpair, whereasfor the overcritical case,energyisgiven up when the neutral vacuumdecaysto producepositronsand the chargedvacuum. In theundercriticalcase,a boundelectronmay be annihilatedby a positronto give a stablephotonstate.However,the chargedvacuumcannotbe annihilatedby apositron since it will spontaneouslydecayemitting a positron with the energy of the initial positron. The effects of this processwill beobservableas aresonancein the positronscatteringcrosssection.

3.3. The reducedHamiltonian

We consideran overcritical potentialwherethe resonanceis not so deeplyimmersedin the lowercontinuumas to havebecometoo diffuse. In the first part of this subsection,we shall show thatonecanobtainthepropertiesof the resonanceby expandingthe eigenfunctionsof the overcriticalproblem

268 J. Rafelski et a!., Fermions and bosons interacting with arbitrarily strong externa! fields

in termsof a reducedsetof the basis functionsof the critical potential [28,30,521. This reducedsetincludes only the boundstate at � = — m and the lower continuum.Apart from its appearanceasreasonable,the ultimate justification for the approximationlies in the fact that oneobtainsresultswithit which arevery similar to the exactresultspresentedat the endof section2.

We write the overcriticalpotentialas the sumof the critical potentialand a piece V’ as follows:

V Vcr+ V’, (3.18)

andintroducethe Diracoperatorassociatedwith Vcr, that is

Hcr = a p + J3m + Vcr. (3.19)

The operatorHcr hasa completeset of eigenfunctions,

HcrIIJ~= �ifr~, (3.20)

whichincludesthe boundstatewith the eigenvalueat �,, = — m. The continuumsolutionsof eq. (3.20)may be normalizedas follows:

Jd3x~c~~(~,s)~r~(x,s’) = 8(�— �‘)6~~’, (3.21)

wherethe spin degreesof freedomhavebeenmadeexplicit.

As explainedabove,weexpandthe overcritical functionswith � < — m in the reducedbasis,= a(�)~+ J d�’h~(�)~, (3.22)

wherea(�)andh~(�)areexpansioncoefficientsthatmay be formally definedas the projections,

a(�)= Jd3xqi~(x)i/i~(x), h~.(�)= fd3Xci’~(x)ci’E(x). (3.23)

At this pointwe will not makethe dependenceon theangularmomentumdegreesof freedomexplicit.To do so is straightforward.Our major concernin this sectionis with centralpotentials,andthusonlythe matrix elementsbetweenstateswith the sameangularmomentumquantumnumbersare non-vanishing.The effect of the spin degeneracywill be includedlater. To determinethesecoefficients,werecall that ‘/‘~ satisfiesthe Dirac equation.

(Hcr+ V’)IIIE = �iJi~, . (3.24)

Substitutingthe expansionof eq. (3.22) into eq. (3.24)andtaking the scalarproductof the resultwith

q,~randthenwith I/I~leadsto the equations

(�— ë)a(�)= f d�’V.h6.(�), (3.25)

(�— E’)hE.(E) = V~.a(�)+ f d�”U~..h~,(�), (3.26)

where V~recordsthe influenceof the boundstateon the continuum,

V~= Jd3xci~~~r(x)V’(x)~!,~(x), (3.27)

J. Rafeiski et a!., Fermions and bosons interacting with arbitrarily strong external fields 269

andU~includesthe effectsof mixing amongthe continuumstates,

U~.= J d3x~/i~(x)V’(x)ifr~(x). (3.28)

Thequantity ~ includesthe diagonalshift in the energyof the formerboundstate,namely,

= �,,+f d3x~/,,~(x)V’(x)~~~(x). (3.29)

In solving for the coefficientsa(�)andh~.(�)we alsorequirethe relation,

8(�—�’)=a*(�)a(Ei)+ J d�”h~(�)h~.(�’), (3.30)

which arisesfrom writing the normalizationconditionfor ~r. in termsof the reducedbasis.The solution of eqs. (3.25), (3.26) and(3.30) is straightforward,andthe detailsof the solution are

given in the appendixof ref. [28], assumingthat the influenceof the matrix elementsU~can beneglected.This approximationis valid if the resonanceis not too deeply embedded(— m ~ — 2m)in the lower continuum,which is the region relevantfor experiments.Alternatively,one mayincludethe effectsof ~ by a prediagonalizationof the lower continuum[52].Thenthe wave functions~ arereplacedby ~ which includethe effectsof ~ The resultsare

a(�)= V~/(�— i — F(�)), (3.31)

h~(�)= Ve.a(�? + 8(�— e’), (3.32)��+lfl

where t~ is a smallpositivenumberandF(e) is given by

F(�)= JdE’~t. (3.33)

Many of the results discussedin section 2 may be obtainedfrom theseequations.In fact, theirstructureis almostidenticalto that of a particleresonancein the uppercontinuum[54].

We now study in more detail the resonancedescribedby eqs. (3.31) and(3.32). In particularat thecomplexvalue �, determinedfrom the equation

(3.34)

both amplitudes a(�),h~(�)havea pole. Consideringthe real and imaginarypart of eq. (3.34) we

obtain the following equations= e+ f d�’ ~ (Er �‘), (3.35)

= - f ~2d�’. (3.36)

270 J. Ratelski et a!., Fermions and bosons interacting with arbitrarily strong external fields

A very useful approximation called the narrow resonance approximation is obtained noting therelation

ir~5(x)= urn x2 t2 (3.37)

which whenapplied to the right handsideof eq. (3.36)for small �~gives:

— i~Iv~,(2. (3.38)

Then,in the sameapproximationthe real part of the resonantfrequencyis given by the equation,—

2

~ I d�’ ~‘~‘ ,. (3.39)J �~�

Herethe quantityP denotesthe Cauchyprincipal value.We note in passingthat the exponentiallyvanishingbehaviorof the positron productionformulas

nearthe thresholdZ=Z~. (and hencei—~—mand �res_*m), which was first discussedby Popov[50]andby MUller, Rafelski andGreiner[59],is carriedby the low energybehaviorof V~.As apparentin eq. (3.27) this matrix elementincludesthe scatteringwave functionfor low energypositrons,whichhaveavery smallprobability of beingfound nearthe nucleus.

Armed with thesesolutionsfor the overcritical case,we shall now write the overcritical Hamil-tonianin thereducedbasis.We beginwith the expansionof ~‘ in thecompleteset of basisfunctionsofeq. (3.20), which is separatedinto threecategories,for convenience,

~(x) = ~ 6r(X) + fir(x) + J dEa:rt~(x) (340)p � ts

Substitutingthis relationinto the normal-orderedHamiltonian[130]yields

= ~ �,,6;~6~+ �~6~t6~— J d��ci:rta:r+ d3x: ~t(x) V’(x)~(x):, (3.41)

p*~ts

wherethe last term arisesbecausethe basisfunctionsarenot eigenfunctionsof the Dirac operator.Ifthe resonanceis not too deeply embeddedin the lower continuum,then the essentialfeaturesarecontainedin a reducedHamiltonian,

HrH0+IIi’ (3.42)

where

= e6~6~— J deEa:Ita:r, (3.43)

is the diagonalpieceand

= f dE(6~ta:rtV~+ h.c.), (3.44)

.1. Raf’elski et al., Fermions and bosons interacting with arbitrarily strong external fields 271

is the nondiagonalpart. In obtainingeqs. (3.42)—(3.44) we have neglectedall matrix elementsthatrecord the influenceof statesof the first term of eq. (3.40). The first term of eq. (3.43) includes thediagonalpiece of the last term of eq. (3.41), and the effects of the matrix elements UE�’ haveagainbeenneglected.It is clear that H1 representsthe effects of creatingor destroyingan electronand apositron.

3.4. Theinstability of the neutral vacuain overcritical external fields

The distinction betweenthe neutral state in the presenceof overcritical fields and the ordinaryneutralstatein undercriticalpotentialis subtle.In theundercriticalregion,onemayobtainthe neutralvacuumstatefrom the statecontaininga single Is electronwith the following projection:

JO) = fd3xçb~s(x)c~(x)Ils), (3.45)

where ~(x) is a field operatorthat destroysa chargeequalto the electronicchargeat point x andci’,~(x)is the wave function for the Is state.Both the ls state and the vacuumare stable. In theovercritical case,the neutralstatemaybe obtainedfrom the statecontainingthe electronchargeasfollows:

0) J d�a(�)f d3x~(x)~(x)J4), (3.46)

where tfr~(x)is a (overcritical) lower continuum eigenfunction and a(�) is the amplitude (3.31)associatedwith the resonance.Only the chargedvacuumstateJc~)in eq. (3.46) is stable.*The stateJO)

is not an eigenstateof the Hamiltonian sincea superpositionof solutions of the Dirac equationisinvolved in its description.Let us considera gedankenexperimentin which the neutralstate(3.46) ispreparedat time t = 0. Subsequentlytwo positronswill be producedspontaneouslyandthe chargeofthe region surroundingthe nucleuswill equalthat of two electronsin order to conservecharge.Onemight be tempted to view this process as pair production, but we feel characterizing it as thespontaneous decay of the neutral vacuum is much more appropriate [28,53,59].

Now we considerthe fate of the neutralstateof eq. (3.46) in more detail, usingthe undercriticalbasis.Supposethat at time t = 0 we prepare the neutral state in the presence of an overcriticalpotential.To describethis statefor all times,we makethe following ansatzfor its time dependence:

Jt) = y(t)JO)+ J d�w~(t)d~t6~tJO), (3.47)

where Jy(t)J2 represents the probability of finding the neutralvacuumand w6(t)1

2 gives the probabilityof finding a positronwith energy�. (We will discuss the spin degeneracyfurtherbelow.)Thefunctionsyandw~satisfythe initial conditionsthat we prescribe:

y(O) = 1, w~(O)= 0. (3.48)

* Thevacuumstate i~) is only stableif thespindegeneracyis not introduced.Otherwise,only thestatecontainingtwo electronicchargesis stable,

and it is necessaryto usetwo destructionoperatorsto obtain theneutralState.

272 J. Rafelski et a!., Fermions and bosons interacting with arbitrarily strong external fields

The time evolutionof Jt) is governedby the SchrOdingerequation

(H0+ I~1)Jt)= i~Jt). (3.49)

Substituting the relations of eqs. (3.43), (3.44),(3.47) into thisexpressionandtaking the scalarproductswith (01 and (OJb~d

1yields the coupleddifferentialequations,

— Jd�v~w~(t)~ (3.50)

i(c9w~(t)/8t)= (i— �)w~(t) — V~y(t). (3.51)

We search for solutions of eqs. (3.50) and (3.51) that havethe integralrepresentations,

y(t) = J d�’e’~’~~’b(�’), (3.52)

w~(t)= J d�’ e~’~’g~(�’). (3.53)

Substitutingtheseexpressionsinto eqs. (3.50) and(3.51) we find that

(�‘—�)g~(�’)=b(�’)V~, (3.54)

(�‘—~)b(�’)=J d�V~g~(�’), (3.55)

which are identical to the complex conjugatesof eqs. (3.25) and (3.26), after the effects of U~areneglectedthere.Thus the solutionsare

b(�’)= f(�’)a*(E’), g~(�’) = f(&)h~(�), (3.56)

wherea(�’)andhE(E’) aredefinedby eqs. (3.31) and(3.32), andf(�’) is a functionwhich is chosensothat y and w satisfy the boundaryconditionsstatedin eq. (3.48). The correctchoice is f(�’) = a(�’).

Hence the integralrepresentationsof y and w~takethe form

y(t) = e~’J d�’e~’ia(E’)a*(�/), (3.57)

WE(t) = e~1~’J d�’ e’~”a(�’)h~(�’). (3.58)

The most straightforwardmethod of evaluatingthese integrals is to use the methodsof complexanalysis.Closing the contoursof the integral (3.57) in the upperor lower complex �‘ planewe findonly contributionsfrom the polesat

= ~r ± ff72, (3.59)

J. Rate!ski et a!., Fermions and bosons interacting with arbitrarily strong external fields 273

whereF = 2irJ V�r12 in thenarrow resonanceapproximation,see eqs.(3.35—3.38)and�~is determinedby

eq. (3.39). In the caseof eq. (3.53) thereis alsoanimportantcontributionfrom the deltafunctionpartof h~(�’).Thus,the solutionsare

y(t) = e~[9(t)e~tt+ O(— t) elEuu], (3.60)

w~(t)= ~ O(t)[e~~’— e1~_~~]+ ~ �0(_t)[e’~~’— e’~~’]. (3.61)

wherewehavewritten theequationsin a form which showsthat we haveobtainedsolutionsfor whichtheoriginof time is acenterof symmetry.It is apparentthattheamplitudey decaysaccordingto thelaw

y(t) = e’~ e~’1’1”2, (3.62)

andthustheprobability of finding the neutralvacuumdecaysexponentiallywith time as Jy(t)J2= e~’1.

The spectrumof positronsis given by

Jw~(±~)I2= - F2/4 = Ja(�)I, (3.63)

which hasthe usual Breit—Wignerform since the variation of I V~J2nearthe resonanceis unimportant.This establishesour claim that the neutralvacuumspontaneouslydecaysby emitting positrons.Oursolution shows that it is possible to preparethe neutral stateat t = 0 with an incoming beam ofpositronsat time t = — with an energydistribution given by eq. (3.63). We further note that thefunction w~(t)and its derivativearecontinuousat t = 0, while only y(t) is continuous.

It is straightforwardto include theeffectsof the degeneracyassociatedwith spin. Sinceneither Vcr

nor V’ dependson the spin, the positron probabilities associatedwith each spin state are equal.Adding thesetwo equalprobabilitiesamountsto multiplication of eq. (3.63) by afactorof two.

Let us now considerthe chargedstate~q)= b~tJO). Its time dependenceis also determinedby theHamiltonianof eqs.(3.43) and (3.44).However,in this case,the nondiagonalpart of the Hamiltonianwill not have any effect, all the relevantmatrix elementsbeing zero. Thus the time dependenceisgiven by

q(t)) = e_~Et6~tlo), (3.64)

which verifies our claim that the chargedstateis stable.(If spin is consideredthen both spin statesmust be occupiedand the stablestate is J2q)= b~b~JO).)No dramaticconsequencesensueuponlowering a fully occupiedstate into the lower continuum.The total energyof the chargedstate is(qJHrIq) = ë, which is lower than the energyof the neutral vacuum (OIHrIO) = 0, and in fact exceedsitby more thanthe restmassof the positron.

With this we concludeour proof that in order to have a stable referencestate,we must choosethechargedstate and henceforthrefer to it as the charged vacuum [28]. This means that the Fermisurfacemustalwaysbe chosenjust abovethe lower continuum.Thenwe usethe freedomimplicit inthe anticommutationrelations (eqs. (3.5)) to introduced~t b~,and the state of charge zero iswritten in termsof the charged ground stateas

0) = ci~rtIq). (3.65)

274 1. Rafelski et al., Fermions and bosons interacting with arbitrarily strong external fields

The field operator is thuswritten

= ~ 6~~r(~)+ dt~’~(x)+ J d�d:’~tci~::(~). (3.66)p~Is

The normal-orderedreducedHamiltoniantakesthe form

= — — J d��d:~ci:r— J d�(d~d~V~+ h.c.). (3.67)

The zero-pointenergiesof eqs. (3.42) and (3.67) differ by E~.With this choice of referencestatetheenergyof the chargezero state is given by (OIHrIO) = —~ and the energyof the chargeone state iszero. To obtainthe time dependenceof the statevectorswith respectto this stateof reference,onemustmultiply the vectorsof eqs. (3.47) and(3.64) by e’~’,which eliminatesthe factore~tthat appearsin eqs. (3.60), (3.61) and(3.64).

The entire discussioncould havebeencarriedout in the Hilbert spacespannedby the overcriticalbasisfunctions.Thesearegeneratedby the Dirac operator

(a . p + f3m + V)~= �~, (3.68)

where V is the overcriticalpotentialof eq. (3.18). Thenthe expansionof the field operatoris

~ 6,,~,,(x)+Jd�d~�(x)~ (3.69)p�ts

where the secondterm includes the resonancein the lower continuum.In this basis the reducedHamiltonianis diagonal,that is,

= — J dE�d~. (3.70)

Now we establishthe relationbetweenthe creationandannihilationoperatorsdefinedwith respectto the two differentbasis sets. Substitutingthe expansionof the overcritical wave functionsin termsof the reducedbasis of eq. (3.22) into eq. (3.69),we have

~ 6,,Ip~(x)+f d�d~a(�)t/i~(x)+J d�d~J d�’h�’(�)cir~(x). (3.71)p�Is

Comparisonwith eq. (3.40) revealsthat

= Jd�a(�)d, (3.72)

= J d�h~(�)d~, (3.73)

are the linear transformationswhich connect the two sets of operatorsdefined with respect todifferentbasissets.In eq. (3.71) wehaveonly usedthe reducedbasis set,but eqs. (3.72)and(3.73) are

J. Rafeiski a’ a!., Fermions and bosons interacting with arbitrarily strong external fields 275

of general use. The vacuum of the reducedHamiltonian may be defined with referenceto theovercriticalbasis functions

60Iq)=O, d~Jq)=0, (3.74)

or with referenceto the basisfunctionsof the critical potential6cr~q)= 0, d~Jq)= 0, ci~jq)= 0, (3.75)

where the subscript p appliesto all statesabovethe Is state.

3.5. The Green‘s functionand its analytic structure

Now thatwehavedealtwith the Hamiltonianformulationof thetheory,we arepreparedto discussthe theory from the poi;t of view of the Green’sfunction.Many of the observationsmadebelow,ofcourse,havecounterpartsin the Hamiltonianapproach,and we shall try to point theseout as theyoccur.The presentdiscussionleadsrathernaturally to the treatmentof vacuumpolarizationandtheself-consistentformulationbelow.

Let us considerthe caseof a particle moving in a time-independentpotential V~.The Green’sfunctionsatisfiesthe equation

(~y. p°~,— y~V— m)G(x,x’) = 64(x — x’). (3.76)

Becauseof the time independenceof the potential, the Green’s function must be invariant underdisplacementsin time. Thus,the Green’sfunctionmay be representedas the Fouriertransform,

(1) -i,o(t---t’)Gc(x~x)j ~—e G(x,x ;w). (3.77)

The choiceof the contourC is relatedto boundaryconditionssatisfiedby G(x, x’) as t -+ ±co~It playsthe samerole as the choiceof the Fermi energy(figs. 3.2and3.3) in the Hamiltonianapproach,whichmakesthe distinction betweenparticlesand antiparticles.The conventionalchoiceof C, which leadsto the Feynman—StUckelbergboundaryconditions[9, 131] is shownin fig. 3.4. There, the two branchcuts beginning at w = ± m as well as the poles associatedwith the bound statesare shown. Theintegrandof eq. (3.77) maybe representedas a sumover the entire spectrumof eigensolutionsof theDirac equation,namely,

G(x,x’; w) = ~ cic(x)iIi�(x’) (3.78)�

Im()

Re(E)

CF V<Vcr

Fig. 3.4. The conventionalchoiceof thecontourin thecomplexw plane.The contourC crossestherealaxisat the FermienergyCF.

276 J. Rafeiski et al., Fermions and bosons interacting with arbitrarily strong external fields

Substitutingthis expressioninto eq. (3.77) and usingthe contourof fig. 3.4 leadsto the representation

G(x, x’) = — iO(t — t’) ~ ci’�(x)ci’�(x’)e~~t’~+ iO(t’ — t) ~ ci’�(x)ct’�(x’) ~ (3.79)

Our treatmentof the Green’sfunction is so far identicalto the usualdiscussions[130,131] of theFeynmanpropagator,which maybe convenientlydefinedas

S(x,x’) = —i(OITO,1(x),I/J(x’))JO), (3.80)

where T denotesthe time-orderedproduct,that is,

T(ô(t), 6(e))= O(t — t’)a(t)b(t’) — O(t’ — t)b(t’)a(t). (3.81)

Substitutingthe expansionsof tfJ(x) and ci’(x’) in termsof the solutionsof the Dirac equation(seeeq.(3.8)) leadsto the expressionon the right-handside of eq. (3.79) for S(x,x’), which establishesthat Sand G are identical in the caseof weak fields, provided that one makes the distinction betweenparticlesandantiparticlesthat is consistentwith the contourof fig. 3.4.

As discussedin section3.2, in the overcritical case,it is importantfor the Fermi energyto remainat EF = — m in order to havea stablestateof reference.In evaluatingthe Fourier integral represen-tation of y(t) in eq. (3.52) above,weencountereda poleat acomplexvalueof �, which was associatedwith the resonantbehavior in the lower continuum.The behavior of G(x, x’; w) in the complexwplaneis similar. As the potentialstrengthis increasedfrom an undercriticalvalue to an overcriticalvalue, the pole associatedwith the lowestboundstatein fig. 3.4 movesoff the real axisand into theupperhalf of the complexplaneas shown in fig. 3.5. It is importantto appreciatethat this singularityis on the secondsheet[128] andthat the contour C is not deformed(into the contourC’) so as tocontinueto embracethepole. Insteadit is necessaryto choosethe contourD, wherethe Fermi energyremainsat— m.ThepathC’ correspondsto thechoiceof theneutralvacuumasthe referencestate,whichis not stable.

Now we show that the choiceof contour D leads to a reasonableresult and that the choice ofcontourC’ doesnot. Substitutingthe first term of eq. (3.32) into eq. (3.78) for G, we havethat

____________ Cr Cl IGD(X,X ;o)— j d� I/

0(x)ciIo(x). (3.82)

whereonly the interestingpart of G hasbeenkept* and~ is negativewhen w <— m andpositivewhen

____ 34~ Re~)V > V~

Fig. 3.5. The correctcontour in theovercritical case.

* The modificationsof eq. (3.82) due to thecontinuumpartof eq. (3,22) aresmall. This is most readily seenby comparingthechargedensities

of fig. 3.8 below with thoseof thecritical potential.

J. Rateiski et a!., Fermions and bosons interacting with arbitrarily strong external fields 277

w> m, as required by the choice of contourD. From eq. (3.31) it is apparentthat a(�)carries thesingularityassociatedwith the resonance,that is, the poleshownin the upperhalf planeof fig. 3.5. Thispole,however,occurson the secondsheetandthe only contributionto the integralof eq. (3.82)arisesfrom the poleat � = w — i~ (providedthat w <— m). Thus the resultof the integrationis

( m — w) Cr ~cr

GD(x, x ; o) -~ I — ~ )2 + F2/4 cit0 (x)ifr0 (x), (3.83)

wherewe havetreatedthe resonanceapproximatelyas discussedin section3.2. A verydifferent resultwould havebeenobtainedif we had chosenthe contourC’. Then the pole at � = Err. + iF/2 makesacontributionof the form

G~.(x,x’; w) — (3.84)

whichis characteristicof acomplexeigenvalue,areflectionof thelackof stability of thestateof referencedefined by the choiceC’. To summarize,everytimea bound statedescendsinto the negativeenergycontinuum,we mustredefine the Green’sfunctionsoas to includeonly the remainingpoleson the realaxis.This is donebymaintainingthefixed shapeDof thecontour.As describednext,thisimpliesachangein the chargeof the vacuumeachtime a polecrossesthe fixed integrationpath D.

3.6. Realand virtual vacuumpolarization

It is perhapseasiestto appreciatethe distinctionbetweenreal andvirtual vacuumpolarization[28]byexaminingtwo different vacuumpolarizationchargedensitiesdefinedwith referenceto the two Fermisurfacesshownin fig. 3.6.Thenuclearchargesare thesamefor parts(a) and(b),andthefirst Fermienergyis belowthe ls levelandthesecondFermienergyisabovethe Is level. Thechargedensitiestaketheform

Pv(X) = ~ cii~(x)iI’~(x)— ~

p~x)= ~ cii~(x)ifr~.(x)— ~ 41J1~(x)lfrp.(x)}. (3.85)

Taking the differencebetweenthesetwo yields

Pv(X) — p Ux) = e4’I~5(x)I/II5(x), (3.86)

~~::~,,,

(a) (b)

Fig. 3.6. Two choicesof theFermi surface.

278 J. Rafelski et a!., Fermions and bosons interacting with arbitrarily strong external fields

andthusthe two chargedensitiessatisfythe conditions

Jd3xp~,(x)=e, fd3xp~x)=o. (3.87)

(This integral is equalto 2e if the effectsof spin degeneracyareincluded.)In the overcritical casethe vacuumpolarizationchargedensitymustbe definedwith respectto the

Fermi surfaceof fig. 3.6b. We write this chargedensityas the sum pv(X) = p~x)+ p~.(x),wherethereal andvirtual vacuumpolarizationdensitiessatisfythe relations

Jd~xp~~x)= e, f d3xp’~(x)= 0. (3.88)

A meansof unambiguouslyidentifying the realpart of the vacuumpolarizationdensitybeginswith the

relationbetweenthe vacuumpolarizationchargedensityandthe Green’sfunction,namely,Pv(X) = — ie urn Tr(

70G(x,x’)), (3.89)

whereonetakesthe limit by taking the averageof letting x—* x’ from the future andthe past[133].Itis easyto establishthis relationin the undercriticalcaseusingeqs. (3.16) and (3.79).For undercriticalfields the implication of eq. (3.89) is that the vacuumpolarizationmaybe written as a contourintegral

Pvx=—~f~Gx~x’;w), (3.90)

after usingeq. (3.77).The contour Q is shown [42]in fig. 3.7. It is obtainedfrom the contourof fig. 3.4by carryingout the deformationsdictatedby the limiting procedureof eq. (3.89) and consistentwiththe singularitiesof G(x, x’; w).

As the potentialstrengthis increasedfrom anundercriticalvalueto an overcritical value, thenthevacuumpolarizationchargedensitychangesdiscontinuouslyfrom p~(x)to the sump’~(x)+ p~(x).Thevirtual vacuumpolarizationcorrectionfor V> V~.doesnot differ significantlyfrom that for V< Vcr.All of the effects of the discontinuity are includedin the real part. As we have seenabove,theresonancein the lower continuumgivesrise to a singularity on the secondsheetin the complex to

plane(fig. 3.5). The real vacuumpolarization chargedensitymay in principle be calculatedfrom theexpression:

p~(x)= — iel ~ G(x,x’; to), (3.91)

whereR is the closedcontoursurroundingthe singularity on the secondsheetshown in fig. 3.8. Theorigin of this contributionis simply the changein the chargedensityarisingfrom the changein thedefinition of the Green’sfunction whenthe is orbit becomessupercritical.Thus p~x)is intimatelyrelatedto the residueof the Green’sfunction at the poleon the secondsheet.It is easyto verify theconsistency of this definition for the approximate treatment of the contribution of the resonance to theGreen’s function carried out above. Inserting eq. (3.83) into eq. (3.91) one finds that

p~(x) ei~t(x)~(x), (3.92)

J. Rafelski et al., Ferm ions and bosons interacting with arbitrarily strong external fields 279

lm(E)

lm(e)

~ ~ ~ x (i~~m mRe(e) ...j, )( )( I

V <Vcr

Fig. 3.7. The contourof Wichmannand Kroll. Fig. 3.8. The contourR surroundingthepole on the secondsheet.

after carrying out the to integration.Integratingthis resultover all spaceyields e, as requiredby thefirst of eqs. (3.88).

The first calculationsof thereal vacuumpolarizationdensitywerecarriedout at Frankfurt [28,55].

Their resultsfor severalovercritical potentialsare shown in fig. 3.9. It is interesting to comparetheresultsfor Z= 172 and Z = 184. The result for Z = 184 suggeststhat the real vacuumpolarizationchargedensitycontinuesto shrinkas the nuclearchargeis increased.Thesecalculationswere carriedout approximately.The Frankfurt school beganwith the exactexpressionfor the s statecontributionin the overcritical basis

Pv(X) = e f d�~(x)~~(x)- ~f de~(x)~(x)- ~ ~(x)~(x), (3.93)# ts

—~

wherethe first term includes the effects of the resonance.Initially, they chosean energyintervalcentered on the resonance and computed

p~x) ~ Jd�[~(x)~~(x)- ~i�(x)~~~(x)1, (3.94)

9r2 [i33fm~ I

100 200 300 r lfml

Fig. 3.9. Realvacuumpolarizationchargedensitiesfor severalovercriticalpotentials.From Rafelski,MOller andGreiner[28].

280 J. Rafelski et at., Fermions and bosons interacting with arbitrarily strong external fields

where �+ = �res±SI’, thus incorporatingthe symmetrybetweenpositive and negativevalues of �. Abetter method [80]for isolating the contribution of the real part is based on

p~(x)= 2(p~(x)— Pv_av(X)), (3.95)

whereSVis chosensothat the potential V— ÔV doesnot generatea resonancein the interval (�, �~).

Thus, the secondterm amountsto a subtractionof the effects of virtual vacuumpolarization. MUller[134]was able to showthat the chargedensitydefinedin this mannersatisfiedthe condition

Jp~(x)d3x = L~/7r, (3.96)

whereA6 is the changeof the phaseshift between�~and �_. It differs from i~ by aboutonepercentsincethe approximatemethodof subtractingthe effects of virtual vacuumpolarizationin eq. (3.95)isvalid to this accuracy.

Pieper and Greiner [27] calculatedthe first-order virtual vacuumpolarizationcorrectionsfor thesuperheavyelements shown in fig. 3.10. This correction, which is calculated from the Uehlingpotential,is neverlargerthanabout12 keV andthusis smallerthanthe changein thebinding associatedwith changingthe charge by one unit (about 30 keV near Zcr). Higher-ordervacuum polarizationcorrectionsare muchsmallerthan the first-ordercorrectionin heavyelectronic[37]andmuonic atoms[135]. Gyulassy[41] showedconclusivelythat the contributionsof an importantclassof higher-orderdiagrams(fig. 1.5) doesnot becomeanomalouslylargeas Z~ZCr.Gyulassyalsocalculatedthe chargeddensitiesof the overcritical vacuumusingthe connectionbetweenthe Green’sfunctionandthe chargedensity.His resultsagreedwith thoseof the Frankfurtgroupandconfirmedthat the size of the regionoccupiedby p’~(x)continuedto shrinkas Z is increasedbeyondZCr. Similarcalculationswerecarriedoutby Wilets andRinker [40]who appliedthe expression(3.93) as basisof their investigations.Their earlyresults helped establishthe insignificance of higher order vacuumpolarization contributions.Theself-energycorrectionshavebeenobtainedup to Z = 160 by Mohr andby ChengandJohnson[431.Nosignificantcontributionis expectedfor Z—’ 175.

3.7. Generalized Hartree—Fock equations

One of the important formal developmentsthat was stimulatedby the attemptsto stabilize theneutralvacuumis a self-consistentformulationof quantumelectrodynamicsby Reinhard,GreinerandArenhOvel [31,32]. The work proposeda self-consistentschemefor incorporatinga large numberof“important” field effects. Originally it was believedthat theseeffects might becomesufficiently large

00 120 140 160

zFig. 3.10. The first-ordervirtual vacuumpolarizationcorrectionasa function of nuclearcharge.

J. Rafelski et at., Fermions and bosons interacting with arbitrarily strong external fields 281

to screen a significant part of the nuclear Coulomb potential and prevent the ls state from joining thelower continuum.As we discussedin the previoussubsection,this is now known not to be the case.This self-consistentapproachwas very similar to a formulationof quantumelectrodynamicsexploredby Pratt[136],andGomberoffandTolmalchev[137]also studiedthe self-consistentmethod.

The approachof Reinhardandcollaboratorswas basedon the Schwinger—Dyson[19] equationsforthe electronpropagator.The self-consistentsystemof equationsis obtainedby neglectingradiativecorrectionsto the vertex function and the photonpropagator— a procedurethat is equivalentto theHartree—Fockapproximation in non-relativistic many-bodytheory. Using the representationof theGreen’sfunction of eqs. (3.77) and (3.78) one mayobtain

(-ia ~ V~’+~m~

— fd~z~y 1~.(x)lIJ~.(z)yci’~(z)F~(Jx_zJ)}o (3.97)J ,, Jx—zJ

where

F~(jx- zJ) = - ~ I ~qsin(qJx- zJ) (3.98)~TJ �~—E~+trq

0

In this equationu = + I for �> ~F and — 1 for � < EF. Also ~ =

As long as the potentialis undercritical,it is simple to isolatethe variouscontributionsto eq. (3.97)and to determinethe chargeof the system.Let us considera superheavynucleussurroundedbyenoughatomic electronsso that the atom is neutral.Then the secondterm of eq. (3.97) includesthedirect term of the Hartree—Fockapproximation (eq. (2.44)) as well as the vacuum polarizationcorrection.The third term includesthe exchangeterm of the Hartree—Fockapproximationas well asthe electromagneticself-energy corrections. The retardation correction, which the function Frepresents,is usuallyneglectedin the Hartree—Fockcalculations.Both the secondandthird termsofeq. (3.97) haveto be renormalizedto recoverthe physicallyobservablequantities.

The self-consistentsolutionof eq. (3.97)is a formidabletask,which hasnot beendone,althoughitsfeasibility has been explored [33]. Fortunately,the smallnessof the fine structure constantneverallows the field correctionsto become very large in the undercritical case and restricting theself-consistentapproachto the electronsin the bound statesis adequate.

In the overcritical case,where vacuum polarization effects becomereal as well as virtual, theself-consistenttreatmentof field effects maybecomeimportantandeq. (3.97) furnishesan acceptablestartingpoint. In section6 below,we discussa self-consistenttreatmentof real vacuumpolarizationthat uses the relativistic Thomas—Fermi(RTF) model to incorporatethe many-body effects. Theself-consistentmethod furnishesonemeansfor justifying the RTF approach,but a simpler derivationwill be presentedbelow.

4. Laboratorytestsof the theory of electronsin overcritical fields

4.1. Overview

The motivation to study the subjectof this paperstems not only from the desire to prove andextendthe region of validity of quantumelectrodynamics,but also from the needto find a new and

282 1. Ratelski et a!., Fermions and bosons interacting with arbitrarily strong external fields

valid descriptionof the electromagneticphenomenain the vicinity of superheavynuclei of chargeZ> 173. Today thereis little hope that suchnucleiactuallywould be stable.Howeverin the wakeofthe associatedinvestigationsit hasbeenrecognizedthat much of the pertinent information can beassembledfrom experimentsinvolving heavy-ionscattering[34].Our interestin theprecedingsectionswas concentratedon potentialsof variable strength;the heavy-ioncollision systemis a mock-up ofthe laboratoryconditions that we desired,assumingfor the moment that the distancebetweentheheavy ions changesinfinitely slowly. We haveherea situationin which the strengthof the externalpotential for the atomic electronschangesfrom the undercritical value Z at large internuclearseparationsto an overcritical strength2Z at smalldistancesbetweenthe colliding nuclei.

Basic to this intuitive picture arethe two assumptionsthat we would like to investigatein detailfurtherbelow:

a) The inner-shellelectronsfeel the electromagneticpotentialof both colliding ions,b) The motion of the heavyions is slow enoughin comparisonto the motion of the electronsthat

the latter adjustsinstantly to the externalfield createdby the colliding nuclei.Both a) and b) can be satisfied when suitable experimentalconditionsare imposed,as is now

apparentfrom experimentsthathavebeendone [83]. They may not be fulfilled if either the relativemotion of theions is too fastas,e.g., in relativistic heavy-ioncollisions with GeV/nucleonlab energiesor when the heavy ions do not penetrateeach other deeply enough in low-energy collisions(keV/nucleonor forwardscattering).

Having avery complexnuclearsystemat hand,we do not wantthenuclei to cometoo closeto eachotherandto penetratethe Coulombbarrier. This latter conditionsetsa limit on the maximumkineticenergythat is allowablein suchexperiments.A lower limit maybe obtainedby noting thataminimumdistanceR~mustbe reachedin the collision, called the critical distance,suchthat the lowestboundelectronstatebecomescritical. We will review the calculationof R~in moredetail further on in thissection. From this calculation we have borrowed the bestavailable values [76] to calculate thekinematicregion of interest.In fig. 4.1 we showthe dashedareaof thekinetic energyof the projectileion as afunction of its chargeZ in asymmetriccollision, in which both conditionsaremet. To obtainthe lower limit the effects of electronscreeningon R~havebeenestimated— the dashedline resultingfrom actualcalculationswith barepoint nucleiapproachingeachother.We recognizefrom fig. 4.1 thatfor collisions such as U on U between3 and6 MeV/nucleonwe fulfill the necessaryconditionsthatallow us to study overcritical phenomena,excluding at the same time most of the complicationsstemmingfrom the nuclearforces. There remains,however, the possible Coulomb excitations ofthe ions which will be discussedat the endof this section.

6 i • I I> Lit/It

77. ~

z 6-~o \C’w / +-J \~ SPONTANEOUS e~ PRODUCTION POSSIBLE

Z41~ -

~ 2 NOe SCREENING ~-_

j POINT NUCLEUSI I • • I.

86 90 94 98 102zFig. 4.1. Shadedthe kinematicregion (kinetic energy(MeV/nucleon)versusthe nuclearchargeZ) in which spontaneousdecayof the neutralvacuumin symmetricheavy-ioncollisions withoutinterferencefrom the nuclear forcesmay occur.

J. Rafelski et at., Fermions and bosons interacting with arbitrarily strong external fields 283

As this review is being written, this kinematic region has been just reachedwith the newacceleratorfacility at the Gesellschaftfür SchwerionenForschung(GSI) in WixhausennearDarm-stadt, Germany[138].It is the hope of the authorsthat by the time this review reachesthe reader,someevidencepertainingto the theoreticalconceptsoutlined in sections2 and 3 will have beencollected.We will summarizein the third part of this section the bestcalculationsrelevantto suchexperiments.

Since the heavy-ioncollisions were introduced as a tool to study the X-ray transitionsand thebehavior of quantum electrodynamicsin strong fields through the creation of superheavyquasi-moleculesby Rafelski,FulcherandGreiner[34,35,83] in 1971,the field hasdevelopedsofast that it isimpossibleto reviewall of the developmentshere.We will thereforeconcentrateexclusively on theaspectsdirectly relevantto the conductof an experimentto prove the predictedchangeof the groundstateof QED from neutral to charged[28,29] and refer to the excellentreviews of Meyerhof [83],Muller [139] and Scheidand Greiner [140]for more extensivediscussionof some other importantquestions.

4.2. Adiabatic approximation in heavy-ion collisions

We beginby discussionof the question:canthe adiabaticcondition be satisfiedin a collision withthe ion kinetic energychosenin the shadedareaof fig. 4.1. We define the collisiontime as thedurationof the penetrationof the inner electronicatomicorbit by theheavyions. This time mustbe comparedwith the period of the orbit of the boundelectron. If the collision time is long enoughso that theelectroncan adjust to the combinedCoulombfield which it experiences,then the adiabaticapproxi-mation will havesomevalidity.

Let usassumein a semiquantitativediscussionthat thediameterof an electronshell is given by theone-electronSchrodingertheory*

d~—n2/(m~Za). (4.1)

Then the orbiting time may be obtained if the ‘velocity’ of the electrons is known. Since therelativistic K-shell electronswe are consideringcan not move faster than the speedof light and arenot muchslowerthan c/2 we find

Torb 2’irn2l(meZa), (4.2)

which for the U-atom Is112 stateis -~ lOx 1021 sec.

Let usdefine the penetrationratio x

x=d/a (4.3)

wherea is the distanceof closestapproachand d is the distance(K-shell diameter)at which we startcounting the collision time. Thenthe collision time may be expressedin terms of d andx as in ref.[141].

* Thisestimateis not thatbad,sincetherelativisticeffectstendto increasethebinding,while themanyelectroneffectsdecreaseit andboth effectsareof the sameorderof magnitudefor Z~ 100.

284 J. Ratelski et a!., Fermions and bosons interacting with arbitrarily strong external fields

3/2

TcoII = 2B1 (~)[(x2— x)”2 + ln(x~2+ (x — 1)1/2)] (4.4)

where— 1/2B — (2ZIZ

2a/~~~d)

and /Lred is the reducedmassof projectile andtargetnucleiof chargeZ1, Z2.Taking d as estimatedby eq. (4.1) we have shown TCOII in fig. 4.2 for severalsymmetricheavy-ion

collisions. We see that the maximal collision time is obtainedfor x= \/3, i.e., for not too largepenetrationsof the K-shell. However we lose only a factorof four when deeperpenetrationsup tox = 90 are considered.We find from fig. 4.2 that the collision times vary between 40 and 400 x1O_21 sec. For the U—U collision at the Coulomb barrier we find TCOII — 40 x lO_2i sec, which issomewhatlonger than the orbit time estimatedabove. This difference should provide a sufficientmargin so that methods based on the adiabatic approximation are valid.

While conditionsfor validity of theadiabaticapproximationseemto be accessible,therearereasonsfor avoiding an extremeadiabaticregime. We need nonadiabaticeffects to excite the K electronbefore the critical distanceis reached.We remind the readerhereof the fact that only emptycriticalstateslead to spontaneouspositron production, as shown in section 3. It may be in place here tomention that for lower Z collisions like Br on Br the adiabacity is much better. No substantialexcitationof innerelectronstateshasbeenseenin experimentsdone in the past.Also, it is apparent,that insteadof arrangingthe experimentto makeoptimumuseof the availableadiabaticy,we maybecompelledto choosemore violent collisions to increasethe numberof electronexcitationsbeforeR~is reached.

From the above consideration,we recognizethat the electron statesthat have been discussedschematicallyin the precedingsectionsshould be obtainedfor the actual experimentalsituationfromthe solution of the Dirac equationwith two Coulomb centersat a fixed distance.Subsequently,time-dependentperturbationtheory allows the calculation of all physical effects in this particularbasis.

4.3. Solution of the two-Coulomb center Dirac equation

Theproblemsinvolved in solvingtherelativistic two-centerequation[61,62]areprobablyan orderofmagnitudelargerthan thosethat areencounteredin similarnonrelativisticcalculations.This is because

o~*82~

‘-K-SHELL RADIUS/DISTANCE OF CLOSEST APPROACH ~X

Fig. 4.2. Scatteringtime asafunction of thepenetrationparameterx for severalheavy-ionsystems.

J. Rafeiski et a!., Fermions and bosons interacting with arbitrarily strong external fields 285

until now no operatorhasbeenfoundfor the relativistic two-centerproblemthat plays the role of thesquareof theangularmomentumJ2,andwhichwould thenallowthe separationof theeigenstateprobleminto radial-like and angular-like one-dimensionaldifferential (or matrix) equationssimilar to theprocedureemployedin nonrelativistictwo-centercalculations.

Let us first discussthe symmetriesand common nomenclatureof the moleculartwo-Coulombcenterproblem[62]. Whereasthe total angularmomentumof the electron orbital is not conserved(since the heavy ions rotate in the electron’sadiabaticframe of reference)its projectionJ. alongtheaxis that joins the nuclei commuteswith the Dirac operator HD since we have maintainedtheazimuthal symmetryaroundthis axis. Therefore,letting the z-coordinatepassthrough the nuclei asshownin fig. 4.3 we havein

(4.5)

an operatorthat commuteswith HD. Every electronic orbital is characterizedthereforeby a uniqueeigenvaluen

= ncit~, (4.6)

andas we will showfurther below, n can assumeonly the values

n = ± l/2,±3/2,±5/2 (4.7)

The solutionswith ±n aredegeneratein energy.The valueof n is commonlydenotedby Greekletterso,iT,ô,...forJ~=±1/2,±3/2,±5/2

Whenthe two-centerpotential V is in addition parity invariant, i.e., if both nucleiarethe same

V(p,z)= V(p,—z), (4.8)

(p is the cylinder coordinateas shownin fig. 4.3) then the eigensolutionshavein addition good parity

I

Z2~>’~/

/__~ /~L2 //ri

Fig. 4.3. Geometryof thetwo-centerproblemwith nucleiZ1, Z2 locatedat z = ± R/2.

286 J. Rafelski et a!., Fermions and bosons interacting with arbitrarily strong external fields

andarereferredto either as “gerade” (even)or as “ungerade”(odd)parity states,often shortenedbylettersg andu.

A completeset of statescorrespondingto the two-center Coulomb potential is defined by thespecification of two additional quantumnumbers,usually chosen to correspondto the quantumnumbersof the asymptoticatomicstatesthat arecontinuouslyreachedin the unitedatom limit, e.g.,afull descriptionof the boundelectronstatesin a heavy-ioncollisionwould be isl/20g, 2P3/3lTg’ etc.

We are now ready to consider the Dirac equationwith a cylindrically symmetric potential. Webegin by separatingthe azimuthal angulardependenceof the wave function in the cylindricalcoordinatesfor the Dirac equation,

(a . p + f3m + V(p, z))çli = �~i, (4.9)

with

a~pS~Sk®o~p, (4.10)

the matrix SlSk as discussedin eq. (2.3) andthe two-centerCoulombpotential V(p, z)

V(P~z)=VI+V2=_[(+R/2)2+2]I/2_[(R/2)2+2]1/2 (4.11)

Suitablemodificationsfor finite size andelectronscreeningcan be addedwhendesired.The nucleiarelocatedat z = ±R12, y = 0, x = 0 as shown in fig. 4.3. We can write the differentialoperatoras

f.8 .8 .18 \a p = exp(—icr34/2)~— 1 -~—~3 —1 o~—1— -~ o~2)exp(io34/2), (4.12)

usingthe cylindrical coordinateswith

x = p cos41, y = p sin 41, (4.13)

which implies for the wave function:

I/J~ = exp[— iIØ (o3/2)41+ in4’(I® I)]~(p, z), . (4.14)

with unit matricesI as introducedin section2 and l/f~ in eq. (4.14) will be single valued if n is halfintegral,cf. eq. (4.7). The explicit form of eq. (4.14) is

/exp(i(n — 1/2)41) 0 0 0 \— ( 0 exp(i(n+ 1/2)41) 0 0 ‘4 14’

— I 0 0 exp(i(n — 1/2)41) 0 J Xfl~

\ 0 0 0 exp(i(n+ l/2)41)/

We thenfind the eigenvalueequationfor the reducedwave functionX~:

6ime+ = �,,,y,,’ (4.15)

with the matrices SSk and~, asdiscussedin eq.(2.3). So far we havebeenunableto maintainexplicitrepresentationindependenceof our formalism. Assumingthe usualrepresentationof the matricesasin section2, eq. (4.15) couldnow be solveddirectly by numericalmethods.The two dimensions(p,z)in which a numericalsolutionwould haveto be carriedout requireratherprohibitive effort. Therefore

J. Rafe!ski et a!., Fermions and bosons interacting with arbitrari!y strong external fields 287

what hasbeendone is to discretiseone or both of the remaininginseparabledimensions.The lattermethodleadsnaturally to a representationof the Hamiltonian associatedwith eq. (4.15) in a certaincompletesetof states.In this approachit is essentialto choosean appropriatecoordinatesystemthatrepresentsadequatelythe symmetryof the solution and thus allows us to minimize the numberofbasis states.

The first of suchapproacheswas that of MUller, RafelskiandGreiner [61]who usedthe coordinatesystemof prolatespheroidalcoordinates~ and i~,

= (r1 + r2)/R, i~ = (r1 — r2)/R, (4.16)

where r1 is the distancefrom the ith nucleus(seefig. 4.3),

r12 = [p2+ (z±R/2)2]~’2. (4.17)

The inverseof eqs. (4.16) and (4.17) are:

p = ~ R[(~2— l)(l — ~2)]i/2 z = ~~ (4.18)

with 1> ~2 >0, ~> 1. We note that for large distances~ approachesr = Vp2 + z2 (in units of R/2),while ~ approachescos0 = z/r. The Coulomb centerscorrespondto the points e = 1; ~ = ± 1. Thetwo-centerCoulombpotentialreadsin thesecoordinates

v 2a(Z1+Z2)~+(Z2—Z1)~~ 419R (~2_~2) . ( . )

An eigensolutionof the two-centerCoulombproblemwas thenfound [61,62]by diagonalizationofthe Dirac Hamiltonianin afinite setof basis functionsof the form

ns —~‘/2 n+S , n+a

XNl(~,~j) = e LN s(~)P1 (‘i)x5~ (4.20)

wherewe introduceda new variable

~‘=(~—l)/a, ~‘>0. (4.21)

The quantity a is to be determinedbelow,and the spin-dependentparameteris:

~ _J—l/2, s=l,3~~+l/2 s=2,4, (4.22)

wheres countsthe componentof the spinorfunction 9!!. The exponentialfactoraccountsfor the factthat we aredealingwith boundstatesolutionswhich as ~‘ -+ ~ mustvanishexponentially.The L~(~’)arethe associatedLaguerrepolynomialsdefinedby

L~’t’1— a F’—N +1~’) 423

— F(a+ l)F(N + 1) ~ , a ‘ ‘

whereF1 denotesthe confluenthypergeometricfunction.

The P~(~)are the associatedLegendrepolynomials. As the internaldistanceR approacheszerotheygo over into P’~(cos0) of the one-centerproblem.The quantitiesx~arethe four-unit spinorswitha one atthe sthplace and all othercomponentszero. Thus the indices aredefinedin the set

{N,l,sJN=O,l,2,3,...;l=n+ô~,n-i-o~-~-1,...;s=l,...,4}. (4.24)

On this set,the functions,eq. (4.20), form acompletebut not orthogonalbasis.The scaling parameter

288 J. Ratelski et a!., Fermions and bosons interacting with arbitrarily strong external fields

a introducedin eq. (4.21) ensuresthatfor ~ —* ~ the basis functionsshow the behavior

9!’~(~, ~) ~ exp(—2r/(aR)), (4.25)

which hasto be comparedwith the asymptoticbehaviorof the boundstatesof the Dirac equationfora certaineigenvalue�

9!’E~~? exp(—(m~---�2)”2r). (4.26)

Thus from a good guessof the expectedeigenenergyof the particular stateof interesta value of amaybe deducedwhich reducesthe numberof basis statesnecessaryto describethe eigensolutionsadequately.In particular,we have

a = (2/R)me[l — (�/m~)2]~2. (4.27)

The detailsof the diagonalizationof the Dirac Hamiltonianare not trivial. In view of the fact that theabovedescribedmethodhasbeenrecentlyabandonedin favor of an integrationof theDirac equationin one dimension[76], we refer for further information to the detailedpaperof MUller and Greiner[62], or MUller’s Ph.D. thesis[142].

Beforegoing into thediscussionof thenewmethodlet usdescribetherelevantresults[61]obtainedbythediagonalizationmethod.In fig. 4.4we showthemostdeeplyboundquasi-molecularstatesof lscr and2PI/2~for thecolliding ion systemsU on U, U on Cf andCf on Cf. The dashedcurvesindicatethe pointnucleusresults,while thefull lines arecalculatedincludingfinite nuclearsize.Theelectronscreeningef-fecthasnot beenincluded.We noticethatas Rbecomessmaller,thebindingenergyincreasesfaster— thewave functiontendsto collapsetothecenterof thecolliding systems— (nottothenuclearcenters).Suchatendencyis arelativisticeffectandis not foundwhenthecombinedchargeof the projectileandtargetissmall comparedwith a~= 137. This is illustrated in the molecularcorrelation diagram for the

S

me — ___________________________________________________________________________

/ POINT NUCLEI

® U-U (Z,~Z2~92)

~ U-Cf (Z1r92,Z2~98

® Cf-Cf(Z~Z~98)

Fig. 4.4. Quasi-molecularcorrelationdiagramsfor severalsupercriticalsystems;both point andfinite sizenuclei areconsidered.From Muller andGreiner 1621.

J. Rafeiski et a!., Fermions and bosons interacting with arbitrarily strong external fields 289

nonrelativisticsystemof Br + Br in fig. 4.5. Hereas thedistancebetweenthenucleidecreaseswithin themolecularorbitals little influenceis seenon the molecularwave functions.This is explainedby the factthat the wave functionsarelocalized well outsideof the centerof the collision.

Returningto the relativistic case(fig. 4.4) we notefurther that for Cf—Cf collisionseventhe 2p1,2~statereachesits critical distancewell aheadof the Coulomb barrier. Furtherwe see that the finitenuclearsize effect is small (of the orderof 2 fm) for the lso statesand is not essentialfor the 2p1,2osolutions.

The precisevalues of R~for homogeneouslychargednuclei with radii i’11 = l.2A”3 fm that follow

from the two-centercalculationsarecontainedin table4.1.At this point we wish to notethat the resultsof figs. 4.4 and4.5 relateto thequestionof the validity

of the adiabaticapproximation.In the caseof the Br—Br collision, thefact that the eigenvalueschangelittle as R changesfrom 500fm to the distanceof closestapproachpromotesthe conditionsunderwhich the adiabaticapproachis useful. Of course,the featuresassociatedwith the rotation of thecoordinateframemustbe also dealtwith. In the caseof the high Z collisions,this featurepromotingthe utility of the adiabaticmethoddoesnot occur.The implication is that relativistic effectsassociatedwith the tendencyof the wave function to collapseenhancethe productionof vacanciesin the lsstate.

Although the abovecalculationswere successfullycarriedout, the diagonalizationmethodhas itslimitations as a techniquefor solving the Dirac equationwith a two-centerCoulombpotential. It hasbeenrecognizedthat any finite basisusedto diagonalizethe Dirac equationwill havesomeflaws, thereasonbeing that the eigenvaluespectrumof the Dirac equationis not boundedbelow. Thus, theeigenstateof interest can never be the lowest eigenstateof the differential equation. There areinfinitely many states of lower energy,and it is thus impossible to orthogonalizethe trial wavefunction with respect to these states. In the particular case of the basis describedabove, the

I I I I ——

35Br —35Br 3p3,~3d

IOU -

10 02 IO~ IO~ tO

5R[fm]

Fig. 4.5. Quasi-molecularcorrelationdiagram for Br on Br collision. From Muller, RafelskiandGreiner[611.

Table 4.1Critical distancein heavyion collisions [621

R~(lso) R~(2p112o)

U-U 34.7fm -U—Cf 47.7fm 16.1 fmCf—Cf 61.lfm 25.4fm

290 1. Rafelski et al., Fermions and bosons interacting with arbitrarily strong external fields

diagonalizationprocessgaveadditional spuriousstatesand the convergenceof the numericalpro-cedurehad to be studiedin greatdetail. Convergenceof the eigenvaluedid not automaticallyimplyconvergenceof the wave function.

In view of this situation,Rafelskiand MUller [76] havedevelopeda new computercodethatfor allintents andpurposesis capableof solvingthe Dirac equationexactlyby a numericalintegration.Toavoid the large numericaleffort associatedwith partial differential equationsin two dimensionsanapproachwhich discretisesoneof the dimensionshasbeenchosen.In particular,a solution 9!’ of eq.(4.9) hasbeenconsideredas expandedin the spinorharmonicsx~

= f~(IrI)x’(f)\K,th ig~(Ir~)xff~(P))

wherex~are as definedin section2, andwe imply the standardrepresentati~nof the Dirac matrices.Similarly let us considerthe potentialV as given by the multipole expansion

V(r) = ~ V~(lrl)Y1m(~). (4.29)

Insertingeqs. (4.28) and (4.29) into the Diraceq. (4.9) andprojectingwith the spinors

P~= (xm, 0), P~= (0,x~), (4.30)

leadsto an infinite setof coupleddifferentialequationsfor the radial functionsf~’,g~.The couplingbetweendifferent functionsis a consequenceof the multipole characterof the potential V. Severalselection rules are immediately apparent,which reflect the particular symmetry of the two-centerpotential.Thereis no in dependenceof the radial wave functionsbecauseof the azimuthalsymmetry;for Z

1 = Z2 only radial functionswith the sameparity coupleto eachother. Thus for the evenparityeigenstatesonly functionswith K = (— 1)~n,n ~ I can couple to eachother,while K = (— )~n,n ~ 1appliesto the odd parity states.

The approachdescribedabove is more efficient than a straightforwardintegrationof the Diracequationand reducesthe necessarycomputationaleffort. The numericalapproachthat has beenchosento solve the eigensystemof the linear first-order differentialequationsreflects the desire toobtaingood eigenfunctions;thedifferential equationsare integratedfrom the origin to alargevalueofr, and it is then requiredthat the wave function falls off exponentiallythere.For further numericaldetailswe referto the work of Rafelskiand MUller [76, 143].

4.4. Thecritical distance

There is one particularaspectof the solutionsof the two-centerproblemwhich hasthe greatestimpact on all calculationsinvolving spontaneouspositron production. We recognizedeasily as thecharacteristicparameterthe critical distancebetweenthe heavy ions at which the binding of theelectronexceedstwice its mass.From dimensionalargumentsalonewe would expectthat the totalpositron productioncross sectionrises as R~.However, the actual functional dependenceis muchsteeper,sinceif the is statejoins the lower continuumearlier,it will alsopenetratemoredeeplyintoit, andmoreof the eligible stateswill decaybecausethe width of the resonanceincreasesas the statedives moredeeplyinto the continuum.

.1. Rafelski a at., Fermions and bosons interacting with arbitrarily strong external fields 291

It is therefore of greatimportance to make an accurateprediction of R~.Conversely,if themolecular K-shell ionization cross section can be determined,the measurementof the positronproductioncrosssectionwill determinelimits on R~.Severalefforts to obtainthe valuesof the criticalradius should be mentioned.The simplestone, but quite accurateconsideringthe crudenessof theapproximationis to usethe monopoleapproximationfor the two-centerpotential V. that is,

f—2Za/r, r>R/2, 431Vo_l_4Za/R, r<R/2, ( . )

which correspondsto a model in which the nuclearchargeis distributedon a shell with the diametercorrespondingto the distancebetweenthe nuclei.The critical distancein thisapproximationis a lowerbound on the critical distance calculated with the full potential. The results of such monopolecalculationsare shown in fig. 4.6 together with the exact solution obtained by the numericalintegrationof the Dirac equationby Rafelskiand MUller [76]. Both curvesdo not includethe nuclearsize or electronscreeningeffects.We seethat the monopoleapproximationis rathergood for smallR~andbecomesworse as the influenceof higher multipolesgrows for largerR~.

Among the manyindirect methodsproposedto calculatethe critical chargeby others,refs. [66—74],thereareseveraloriginal theoreticalideas.Theseweredevelopedto find theparametersof thepotential,given the eigenvalue� = —m of a bound stateof the Dirac equationboth for sphericalandtwo-centerproblems.It is impossibleto reviewthe manydifferentapproaches.We mentiononly thatrecentlynewandunexpectedresultshavebeenobtainedby MarinovandPopov[68,73—75]which arealsoshownin fig.

4.6. Wenoticetheseveredisagreementwith otherresultsherewhichagreewith theearlierwork of Popov[63,64,69,70].Thesenewresultsfor R~aresystematicallyabout15 fm larger; thedifferenceis attributedby MarinovandPopovto thesingularityof theelectronwavefunctionat thenuclearcenters.At presentitseems[76] that the form of the wave function chosenoveremphasizesthe importanceof the nuclearCoulombcentersrenderinga badconvergenceof the numericalalgorithmwhich doesnot seemto havebeencarefully studiedin refs. [68, 73—75].

Whereasthe disputeddifferencein the theoreticalcalculationsis of the orderof 40% of R~(for Uon U collisions),the combinedeffect of the electronscreeningeffects will move R~by approximately5 fm (12%) in the oppositedirection [76].Extensivenumericalcalculationsare necessaryto solve thetwo-centerproblem including all the possible relevant effects. The associatedwave functions areneededfor further investigationof the dynamic effects, someof which will be consideredin thefollowing subsection.

12C ‘III’ I

POINT NUCLEINO SCREENING -

86 90 94 98 102z

Fig. 4.6. Comparisonof the critical distancein heavy-ioncollision with the monopoleapproximationandMarinov, Popovand Stolin [68,73—75]calculations.From Rafeiskiand Muller 176],

292 1. Rafelski et a!., Fermions and bosons interacting with arbitrarily strong external fields

4.5. Inducedandspontaneouspositronproductionin heavy-ioncollisions

When discussingspontaneouspositron productionin section3 we have for the purposeof thecalculationsswitched on the supercriticalinteractionat time t = 0. The decayof the unstableneutralstate was then studied assumingno further change of the external potential. This situation isillustrated in fig. 4.7a.Quite a differentpicture is seenin fig. 4.7b, wherethe potentialchangefor thetwo-centercollision is presentedschematically;at time t 0 the potentialexceedsthe critical valueandafter sometime r~falls againbelow the critical threshold.From the abovediscussionof validityof the adiabaticapproximationfor U—U collisions, it is clear that T~ should be larger than 10 x1021 sec.

We wish to comparesucha time intervalwith the decaytime of the neutralvacuum.This time canbe obtainedfrom the width of the quasimolecularresonance

~ l/F(R). (4.32)

While the position of the resonancein the continuum may be found by integrationof the Diracequation,with some boundarycondition at a large distancefrom the nuclei [76], for � — m, thecalculation of the width of the resonanceinvolves calculation of the scatteringphasesof thetwo-centerproblem and is numerically very difficult. However, a semi-phenomenologicalapproachhelps hereconsiderably.It had beennoticed alreadyby Peitz, MUller, Rafeiski and Greiner [60]thatthe width of the resonanceandresonantenergyare relatedto eachotherin a way that is independentof the preciseform of the potential. In our casethat meansthat [78]

FIR) = (Ep(R))27(Ep)/me, (4.33)

whereF0 is the kinetic energyof the resonance

E0 = — ~r + me (4.34)

andy(E0) is auniversalfunction (within certainlimits) independentof theform of V. i.e., of R. In fig. 4.8

we showy(E~)— all thenumericalcalculationsup todatefall within thedashedregion.We noticethatforE0> m/2, y is essentiallyconstantas predictedby Peitzet al. [60] andthat for E0—*O, y tendsto zeroas noticedoriginally by Popov [50].

Given the relation of eq. (4.33) we can calculatethe width of the lscr resonancein e.g., a U—Ucollision. In the vicinity of the Coulombbarrier R ‘— 15 fm we haveE0~>me/2and thereforewe findfrom eqs.(4.32) and(4.33)the lifetime of the neutralvacuumstate,namely

Tvac ~ l0_20 sec. (4.35)

Thus the lifetime of the neutralvacuumis longer thanthe time for which the supercriticalpotentialiscreatedin the collision.In view of the decreasingvalueof y(E0)for smallE0 we recognizethat only afraction of the K-holeswill spontaneouslydecayby positronemissionduring theheavy-ionencounter.

V-VcJ ~ V~VcJ ~

(a) (b) C

Fig. 4.7. Schematiccomparisonof supercriticalityin a ‘switch-on’ gedankenexperirnentwith theheavy-ioncollision.

J. Rate! ski et a!., Fermions and bosons interacting with arbitrarily strong externa! fields 293

00/01102.5

E~[ml

Fig. 4.8. Systematicsof thereducedwidth y of thesupercriticalvacuumasa function of thekinetic energyE~of the resonance.

In view of the time scalesinvolved a propertreatmentmust include the time dependenceof thesystemin question,i.e., broadeningof the spontaneousdecayassociatedloosely with Heisenberg’suncertaintyrelationmust be considered.In addition thereis a furthercomplicationthat involves theinducedproductionof an electron—positronpair by the time-dependentCoulombpotentialwhich mayoccur beforethe critical distanceis reached.This processhasbeencalled, for obviousreasons,theinducedpositronproduction[78].We will discussthosepoints further in the nextsubsection.

Elaborate calculations have been performed to obtain the associatedcross sections for bothspontaneousandinduceddecays.The crosssectionfor positronproductionis given by

dE0dfl1 = W1(E~)~_‘~ (4.36)

wheredcrR/dfIJ is the Rutherford crosssection for the ions, calculatedin a semi-classicalapproachassumingthe ions follow classicaltrajectories.

The probability W1(E0) representsa folding of the positronproductionprobability W(E~,R) withthe probability L of havingthe supercriticalstateionised.

= f W(E0,R(t))L(E0,R(t))dt, (4.37)

whereit is customaryto replace‘dt’ by ‘dRflvRI’ with IVRI = dR/dt, the radial ion velocity that dependson all kinematicparameters(index I on W) of theheavy-ioncollision andis discussede.g.,by Peitzetal. [60]. The normal choice is to considerthe scatteringangleand the kinetic energyof the collision

IVRI = v~ C_l[(R/R0)2_ C(R/R

0)+~C2cot2(0/2)]”2, (4.38)

whereR0 is somescaleof length suitable for the particularcalculation,

C = ZlZ2a/Ro(~sredv~/2)’, (4.39)

and v0. is the ion velocity in the lab frame at infinite ion separation,relatedto the lab energyin the

294 J. Rafelski et a!., Fermions and bosons interacting with arbitrarily strong external fields

usualfashion,lab I 2E1 = 2 P~redVoo. (4.40)

Neglectingthe changein the ionisationprobability of the supercriticalstate,e.g. setting

L(E0, R(t)) L~ (4.41)

whereL0 is the ionisationat the point of criticality we have

= L0 J W(E0,R)dR/~vRj= L0W1(E0). (4.42)

The essentialdynamicalproblemis to obtain W1(E0). It is generallypermissibleto neglectthe effectconnectedwith the decreasingionisationof the supercriticalstatesas theyarefilled by spontaneouslyproducedelectronsbecausethe lifetime of the overcritical states is large in comparisonwith thecollision time. Further,if all of the other time dependenceis neglected,then [60]

W(E0,R) I’(E0(R)) . la(E0(R))~2, (4.43)

where F’ and a are as discussedabove and in section3. W1(E0) then follows in a straightforward

manner[60].The approximationsusedto obtaineq. (4.43) are not valid,however,sincetime dependenceof the

overcriticalportion of the potentialcan not be neglectedwhen comparedto the neutralstatedecaytime. We refrain here for the time being from the discussionof the necessarytime-dependentperturbationtheory. It will be consideredin section 4.6. In figs. 4.9—4.12 the essentialresultsthatincludethe time dependenceof the collision,as obtainedin refs. [78]and [81] are summarized.Figure4.9 showsa comparisonof theinduced(by the time dependence)andspontaneouscontributionsto the

80- u-u -

812.5 MeVCM - ENERGY

CENTRAL COLLISION

60 - -

COHERENT SUM

040 - PRODUCED DURING-’

CRITICALITY

INDUCED BEFORE20 - AND AFTER- —

CRITICALITY

OO 0.5 l.O 1.5 2.0

E~[MeV]

Fig. 4.9. Comparisonof the positronproductionprobabilitiesfor inducedandspontaneousdecaysof thevacuum in U—U centralcollision at theCoulombbarrier.Smith, private communication.

I. Ratelski et at.. Fermions and bosons interacting with arbitrarily strong external fields 295

positron productionprobability in a central U—U collision of the Coulombbarrier. We note that thedominantcontributionariseswhen the potentialis overcritical, though, the positronproductionmaybeginearlier,the energybeingprovided by the relativenuclearmotion via the time dependenceof thenuclearelectric fields. A significant interferenceeffect betweenthe induced and spontaneouscon-tributions during and before or after criticallity is noticed. We can integratethe curve and discoverthat roughly 5% of the available Iso--holeshavetime to decayin the heavy-ioncollision. The actualcrosssectionsare shown in fig. 4.10 wherethe ionization probability L0= lO_2 hasbeenassumed.Infig. 4.lOa the crosssection in a centralcollision is shown as a function of positron energy.Differentcurves correspondto different distancesof closestapproach.In similar non-central collisions thepositron productioncrosssection is governedby the correspondingdistanceof closestapproach(fig.4.11). In fig. 4.lOb the total cross sectionfor positron productionin U—U collision as a function ofheavy-ionenergyin a center-of-masssystemis shown.With L0 = lO_2 the expectationfor the cross

~ :~:‘ I I51m I :1~ /~/M.

~IO b / //

I6~- CENTRAL 30 :.: / ~.

COLLISION 2.0 300 ‘600 01/

E~[MeVJ E~M[MeV]

a b

Fig. 4.10. (a) Positrondifferential cross sectionsfor U—U centralcollisions at different distancesof closestapproach(E~:M= 815.5. 609.7, 478.5.

406.3, 398.2MeV, respectively).(b) The total integratedpositroncrosssection.From Smith, Peitz, Muller,andGreiner[78].

I I IU-U E~=IMeV

b[fm}

Fig. 4.11. Impactparameterdependenceof I MeV positronproductionat severalheavy-ionenergies.From Smith,Peitz, Muller andGreiner[78].

296 J. Rafe!ski et a!., Fermions and bosons interacting with arbitrarily strong external fields

sectionsis of the order of 0.5 mb. In fig. 4.11 the differential positron crosssection is plotted as afunction of impact parameterb for a fixed positron energyE0 = 1 MeV. The maximum yield isencounteredat b = lOfm, correspondingto aheavy-ioncollision at 900.

Due to the collision dynamicsa vacancyin the2P

1,2o- level in U + U (which does not becomeovercritical)canalso lead to the productionof a positron.It hasbeencalculated[144]that the crosssectionis smaller (by a factorof 20) than the emissionfrom the iso- state,so that the contributionsfrom evenhigher statescanbe expectedto be negligible.

As the amountof supercriticalcharge (Z1 + Z2— Zcr) in the systemincreases,the cross sectionincreasesrapidly for similar heavy-ionpaths.This is seenin fig. 4.12 in the comparisonof the threesystems,U+ U, U + Cf, and U — Cf, for which the moleculardiagramswere shownin fig. 4.4. As thedistanceof closestapproachdecreases,the total crosssectionsfall off rapidly.

4.6. Treatmentof the time-dependenceof heavy-ionscattering

We will now turn back to discussthe essentialdetails of the theoreticalwork that made thecalculationsof the induceddecayratespossible.We will seta broaderframein the first paragraphsinorder to includealsoa generaldescriptionof inducedX-ray transitionsin heavy-ionscattering.

In our previousdescriptionsof solving the two-centerDirac equationdynamicaleffects havebeenleft out that originate in the time dependenceof the heavy-ioncollision. We note that both theseparationbetweenthe ions as well as the orientationof the internuclearaxischangeswith time. Sincein the two-centercalculationsthe electronsmove aroundnuclei assumedto be fixed in space,twoadditionaleffectshaveto be accountedfor:

a) changeof the Coulombpotentialas a function of time;b) rotationof the nuclei in the electron’srestframe.The latter effect can be describedmore convenientlywriting the Dirac equationin the generally

covariantform andreplacingthe covariantderivativewith the suitableconnectionsF~[145],[ya(id~ + iI’~— eAa)— m]ili = 0. (4.44)

-5 ___________ ___________10 I I I I

CfCf (a) ~ - Cf-Cf Ib)

CENTRAL WLN ___t5~ 0.5 1.0 1.5 2.0 ~ ~406O

Ep 1MeV] R~15[fmj

Fig. 4.12. (a) Positronproductioncrosssection in central heavy-ioncollisions with 15 fm distanceof closestapproach.(b) Correspondingtotalcrosssectionsasa function of the distanceof closestapproachof the ions. From Smith. Peitz, MOller and Greiner[78].

J. Rafelski et a!., Fermions and bosons interacting with arbitrarily strong external fields 297

The connectionsI’,, for spinordifferentiationareknown [146]

~ =~ yi(~ii._ {i’K}~) _~Tr(ysyl~)ys. (4.45)

and ~ = ~ EjkImYYYY , where~jkt,,, is the totally antisymmetrictensor.The Christoffel symbol { } is

obtainedfrom the metric tensorg~[147]I J 1 jII f3g11 t

9gkI t3g~~ 4 41~ ~ ~—r+r—~i- . ( . 6)

The Dirac ‘y matricesaredeterminedwith the aid of the metric tensor~ in the rotating frame [146]

describedin Cartesiancoordinates(t, x, y, z),

l—o2(x2+y2) 0 —wx wy

0 —l 0 0g~k= — 0 — 1 0 ‘ (4.47)

wy 0 0 —l

wherew is the angularvelocity. The quantitiesy satisfythe anticommutationrelations

{y~,‘yk} = 2g~, (4.48)

and thus in the rotating frame they takethe form

= {/3(l + (w x r)~a),~a} (4.49)

where{f3, a} are the usualDirac matrices.Equation(4.44) can nowbe written [62]

= [a . (p — eA)+ V+ f3m]I/f — . [r X (p — eA)+ ~ o]~!i+ ~i(0 X r) ad’. (4.50)

The last two termsareeasilyrecognizedto correspondto the usualcentrifugaland Coriolis forces.We wish to find eigenstatesof eq. (4.50) havingatour disposalthe solutionsof the two-centerDirac

equationat a given instantof time,

[a . p + /3m + V(r, R(t))]~/i(r, R(t))= e(R(t))i,li(r, R(t)). (4.51)

Here we haveindicatedexplicitly the time dependenceof the internucleardistanceR. Comparingeq.(4.51) with (4.50) we noticetwo additionalterms,the nonradiativepart

HNR = — w [r X (p — eAex)+ ~ o] — ~i(w X r). a, (4.52)

and the radiative term

HradW [rxeAr]. (4.53)

Here Ar is the transverseradiationfield andwe haveused

A = Aex+ Ar, (4.54)

whereAc,, is the vectorpotentialgeneratedby the colliding ions. The radiativeterm maygive rise toinduced transitions betweenquasi-molecularstates accompaniedby emission of quasi-molecularX-raysandhasthereforebeenconsideredin somedetail lately [148,149]. The HNR term togetherwiththe time dependenceof the two-centerpotential is responsiblefor nonradiativetransitionsbetween

298 J. Ratelski et al., Fermions and bosons interacting with arbitrarily strong external fields

quasi-molecularstates.We will now concentrateon thesetransitionsinvolving negativefrequencycontinuuminducedby the HNR perturbation.

This problemmaybe statedmoregenerally.Given

H(t) = H0(t) + H1(t), (4.55)

we are searchingfor wave functionsx suchthat

(H—i-f ~)~=0. (4.56)

Using the basis generatedby

H0(t)ç1i~(t) = �~(t)~i’,(t), (4.57)

with

ç&~(t)= exp[— i�~(R(t))]i~i~(R(t)), (4.58)

we makethe ansatz

x = ~ a~(t)exp[_~ J ~k(t) dt’]I/Jt(R(t)). (4.59)

which wheninsertedinto eq. (4.56)leadsto a setof equationsthat are equivalentto the original Diracequation,

O~(t)= — ~ a,,,(t) exp(iJ [�~(t’) — �m(t’)] dt’)J d~x[~I1~(R(t))[1~~~Hi(t)]~Iim(R(t))]. (4.60)

Before turning our attention to the solutions of eq. (4.60) we would like to makesome specificcomments.As is obviousfrom the aboveequations,I/Jk(R(t))area completesetof statesfor eachR(t)associatedwith the HamiltonianH0(R(t)).Therefore,we find the term iH1 in the integralof eq. (4.60)which servesto adjustthe basis.Alternatively, if we hadstartedwith the completestatic solutionsofH, we would haveobtainedno H1 term in eq. (4.60).

Let us now turnto the particularcaseof positronproductionin heavy-ioncollisions,which wewilltreat in the undercritical basis. Equations (4.60) can now be written as a set of two equationsdescribingthe decayof the initial state~0’into the setof statesn, that is,

= — a0(t)V~+ ~ a~(t)exp{— iw~0(t)}V~0, (4.61a)

= a0(t) exp{iw~0(t)}V~0—~ a~.(t)exp{iw~~(t)}V~~, (4.6lb)

where

Wnm(t) = J [e~(t’) — �m(t’)] dt’, (4.62)

Va,,, = — V,n Jd3xcfr~(fj+IH1(t))tlim. (4.63)

J. Rafe!ski et a!., Ferm ions and bosons interacting with arbitrarily strong external fields 299

Equations(4.61) areequivalentto our treatmentof the positronproductionprocessin section3 ifthe operator 8/3t is neglectedand H1 consists only of the overcritical part of the potential,H1 = V — Vcr. In eq. (4.61b) the last term describesthe couplingamongthe continuumstatesandmaybe neglectedas in section3. Theneq. (4.6la) becomesupon substitutingfor a~(t)from eq. (4.61b)

= — a~(t)V® — f { v:,~(~)exp{ — iw~0(t)}I [a0(t’)exp{iw~0(t’)}VE’o(t’)] dt~p(�’)d�’, (4.64)

which is an integro-differentialequationfor a0(t). We havereplacedn with the continuousindex �‘

andincludedthe densityof availablestatesp(�’). It is apparentfrom the form of eq. (4.64) that onlywhen w~.0(t) w~0(t’)can we expect large contributions to the integral over �‘. Therefore anapproximatesolutionof eq. (4.64) is

a0(t) = exp(—~F~(t)— V00(t)), (4.65)

where-m

= f d�’p(�’)~fdt’ V~.0(t’)exp{— iw�o(t’)}j, (4.66)

and.13 ±V® = ij d x( V — Vcr)t~histfris. (4.67)

Insertingeq. (4.65) into (4.61b)we find for the positronamplitude

a~(t)= — f dt’{ V~0exp(f U(Er(t”) — Ecr) — ~F0(t”)]dt”) }, (4.68)

which is the essentialformula used in the calculation of the inducedpositron production by theFrankfurtschool [78,79]. We haveabbreviated

= � + V®, (4.69)

which has a similar meaningas in section3 and in the limit of vanishingdynamical effects it isidenticalto the quantity ~r usedthere.

As a final remarkwewould like to stressagainthat the aboveformalismhasbeendevelopedin theundercriticalbasis.In the overcritical basis,the ls stateis not an eigenstateof theDirac Hamiltonianand is composedof manynegativefrequencyeigenstates,

v>v I v>v’~1’is a(~)~’~de. (4.70)

Accordingly, the setof eqs. (4.60) mustbe solvedwith the different initial condition,

ar(t = t0) = a(e). (4.71)

The appearanceof eq. (4.68) as a first-ordersolutionof eq. (4.60) with am = SmO on the right-handsideof eq. (4.60) is deceptive.We notethat the presenceof the nonperturbativetermsin the exponent,in particularthe term V® of eq. (4.69)is essential.At presentno formulationof the induceddecayin

300 .1. Rafe!ski et at., Fermions and bosons interacting with arbitrarily strong external fields

the overcriticalbasishasbeengiven to checktheresultsof JakubassaandKleber [150].Their work isbasedon perturbativesolutionof eq. (4.60) in the overcriticalbasisand differs significantly from thepreviouscalculationsby the Frankfurt group [78].

4.7. Other sourcesofpositronsin heavy-ioncollisions

Although the predictedcrosssectionsfor the positron productiondue to the decayof the neutralvacuumarelarge (total crosssectionin a U—U collision atthe Coulombbarrier is of the orderof onemilibarn [L0 = 10_2]), the complexity of the heavy-ion collision makes it conceivablethat somebackgroundeffects may contribute significantly. Let us begin our discussionby looking at otherpossiblecontributionsfrom quantumelectrodynamic(QED) effects.

Two prototype Feynmangraphs for pair production are shown in fig. 4.13. In (a) a virtualbremsstrahlungphotondecaysinto e~—e’pair and in (b) the pair is made by one of the photonsexchangedbetweenthe ions. It hasnot beenrecognizedthat a fractionof sucheffectsassociatedwiththe longitudinal part of the virtual photon field is already includedin the calculationsof the inducedvacuumdecay. The difference betweenthe Feynmangraph approachand the vacuumdecaycal-culationsis that two different basis sets are used for the electron—positronwave functions.In theinducedvacuumdecaycalculations,the properchoiceof the basis set andthe useof time dependentperturbationtheory allow us to visualize the processas the excitationof the vacuum into the iso-state.In the Feynmangraph approachbecauseof the large effectivecoupling, (Z1 + Z2)a,manymoregraphsthan thoseillustrated in fig. 4.13 are neededto accountfor the sameeffect. Also excitationsinto statesotherthan iso- statearethenautomaticallyincluded,but it is almostimpossibleto accountfor the occupationof deeplybound electronstates.While the inducedvacuumdecayis equivalenttothe usual graphologyof QED, we would like to stressagain that spontaneousdecay can not berepresentedin this fashion andcorrespondsto the fundamentallynew phenomenonof the decayofthe neutralvacuum.It cannot be treatedby meansof anyperturbativeexpansionthat doesnot reflectthe crossingby the Is-stateof the Fermi surface.

Graphssimilar to fig. 4.13,with an internally excitednuclearstateare shownin fig. 4.14. Althoughwe have taken all possible precautionsand stayed below the Coulomb barrier in the heavy-ioncollision, the long rangecharacterof the Coulombforceswill induce excitationsin the nuclei evenbelowthe Coulombbarrierdistance.The lifetime of ahighly excitedstatewith E*> 2m0— JØ~3sec,althoughmuch longer than the collision time (— 10_20sec), it is not long enoughso that the positronproducedfrom nuclear transitions can be distinguishedexperimentallyfrom a positron producedduring the heavy-ioncollision.

It has beenshown [81] that the dominantbackgroundto the effects consideredin the precedingsubsectionresultsfrom the decayof Coulombexcitednuclearlevels by internal conversioninto an

e~ e

~Z1 (a) Z2 Z1 (b)

Fig. 4.13. Feynmanprototypediagramsfor pairproduction.

J. Rafelski et a!.. Fermions and bosons interacting with arbitrari!y strong external fields 301

Z1 (a) Z2 (b)

Fig. 4.14. Feynmandiagramsfor pair productionby an internallyexcitednucleus.

electron—positronpair. For definitenessthe even—evennuclei238U havebeenchosenandtheCoulomb

excitationsdescribedin the rotation vibration (RV) model [151].Sincelittle is known experimentallyaboutthehighly excitedstatesin uranium,two differenttheoreticalmodelshavebeenconsidered:in (I)all statesof the RV-modelbelowthe Coulombbarrierhavebeenincluded(520states)but only magneticsubstateswith M = 0 were considerednumerically which is exactfor 1800 scatteringonly. In (2) theground state and first I~,y-vibrational bands were used up to spin 40k, including all magneticsubstates.This calculationthen yields the Coulomb excitation probability p~ of the initial nuclearlevel. With the branching ratio P’ for the photon transition from state i to state f and thecorrespondingdifferential pair formation coefficient df3/dE

0, the probability for pair creationbecomes[81]

Cb — ‘~~‘ Cb ~ d~1~(E~)W1 (Er) — ~j Pi (O10~)P1~ ,~,, , ( . )

i.f p

wherethe sumruns over all initial and final stateswith sufficiently large energydifferencefor pairproduction.

If the scatterednuclei and the positronsare measuredin coincidencethen the differential paircreationcrosssectionis given by [81]

dE~d1l1df11~W1(E0), (4.73)

wherethe sum includesboth nuclei.The branchingratios PJ for the various nucleartransitionsfollow from the BE2 valuespredicted

by the RV model. The internal pair formation coefficients have been obtained in the PWBAapproximationincluding the necessarycorrection factors [81]. The numericalresults for Coulombe~epair formation are shown as dashedcurves in fig. 4.15 for the U—U systemat the Coulombbarrier which is an upper limit derivedby consideringsmall distancesof closestapproach.Figure4.15a demonstratesthe unsymmetrizedcross sectionswith respectto the ion angle which will bemeasuredin coincidenceexperimentsallowing the distinction betweenprojectile and targetnucleus(e.g.,

234U—238U). In fig. 4.lsb the correspondingsymmetrized results are plotted for 238U—238Ucollisions. In both casesthe dashedcurves indicate the positronsdue to Coulomb and nuclearexcitation(calculatedwith methods(1) and(2) describedabove)while thesolid lines givethosecausedby the induced and spontaneousdecay of the vacuum. Clearly at rather forward angles thevacuum-decaypositronsoutnumberthosefrom the nuclearbackgroundby severalorders of mag-nitude. At O~,= 20°, for instance,the ratio dcrV,,C/do-Cb was found to be -~300 in the unsymmetrizeddistributionand-~ 50 in the symmetrizedcase.For backwardion angles,however,the crosssections

302 J. Rafelski et al., Ferm ions and bosons interacting with arbitrarily strong external fields

III’U-u

(a) ECM~8OOMeV (b)

~Id2fl,./~ ~—SPONTANEOUS+ INDUCED

(PEITZ ef at.)

IO~- - - ~-.~NUCLEAR COULOMB- —

V EXCITATION (OBERACKERetaI.)I li I Ill

0° 40° 80° 120° 60° 0° 40° 80° 20° 600

Fig. 4.15. Comparisonof positron productionfrom vacuum decay— solid curve— and Coulomb excited nucleusdecay— dashedcurves— in asub-CoulombU—U collision as afunction of the ion scatteringangle.From Oberacker,SoftandGreiner[153].(a) Final ion statesdistinguishable.(b) Symmetrizedwith respectto projectileandtargetnuclei.

differ only by a factor of 2. The correspondingtotal pair formation cross sections are 0vac =

5.0 x 1O~b and 0Cb = 1.25x 10~b (method 1) and 2.28X iO~b (method2), respectively.The differentialcross sectionwith respectto the positron kinetic energyE~is shown in fig. 4.16.

Again, a characteristicdifferencebetweenthe pairsoriginating from Coulomband nuclearexcitation

-3I_

0 500 1000 1500 2000

E~[key]

Fig. 4.16. The sameasfig. 4.15,but asa functionof thepositronkinetic energyF,,. From Oberacker,Soff andGreiner[1531.

I. Rafeiski et a!., Fermions and bosons interacting with arbitrarily strong external fields 303

(dashedcurves)and thosefrom vacuumdecay(solid curve) is found. Since the maximum nuclearE2-transitionenergyin 238U is about 1.8MeV, thepositronspectrumterminatesat E

0 -= 800keV whilethe inducedpositronspectrumcontainsmuchhigher energies.

We concludethe discussionof this most importanteffect noting that the internalconversionof yraysfrom Coulomb-excitednuclearlevels can not be neglectedcomparedwith the spontaneousandinducedpositron productionin overcritical electric fields exceptif we succeedin obtaining highervaluesof L0 than the assumedoneof 10_2. Howeverboth processeshavebeenshownto be separableby their differentdistributionswith respectto the ion-angleandthe positronenergy.

All furthereffects studied so far haveturned out to be negligible as comparedto thatdiscussedabove.For furtherreferencethe readermayturn to the work of the Frankfurtschool[152, 153].

4.8. Coulombexcitationof inner electronicshells in heavy-ioncollisions

To obtainthe positroncrosssectionsin subsection4.5 it hasbeenassumedthat it is unlikely thatthe iso- stateis ionizedandthe arbitraryvalueof L0 = l0_2 hasbeenused.In fact, the predictionsforthe ionization degreeL0 of the Iso- statevary betweena few percent[154,155] and less than io~percentper collision [92]. Since this point is crucial to the experiment,all possibleexperimentalandtheoreticalinformationmustbe gatheredpertainingto this point. Meyerhoff et al. [86]havemeasuredsystematicallymolecularexcitation of the iso- statein variousheavy-ionscatteringprocesses.Withthe presentresourcesthey couldnot go beyondacombinedZ of 120; in view of the systematicyieldof the molecularK-X-rays, Meyerhoff [92] suggestedan extrapolationto the supercriticalregion. Noseparationof differentmechanismsinvolved in ionization of the K-shell hasbeenperformedin thisextrapolation.In the ‘nonrelativistic’ (with respectto electrons)experimentsa different mechanismwill contribute dominantly to the ionization of the iso- state than in the ‘relativistic’ casein whichelectronscollapseto the center(asmentionedpreviously).The extrapolationsof Meyerhoff [92]showan ionizationprobability of lessthan l0~%which we interpretto correspondonly to the mechanismspresentin the ‘nonrelativistic’ case.However, further experimentsand calculationsare neededtoclarify the situationhere.We emphasizethat the theoreticalpredictionsare very difficult andneedprecise relativistic molecularwave functions calculatedin the framework of the two-centerDiracequationwhich only now have becomeavailable [77]. We will now give a preliminary report ontheoreticalwork thatis in progressin order to makea valid theoreticalprediction[155].

In calculationsof the quasi-moleculariso- vacancy formation amplitude during and after thecollision, the simple particle basis is provided by the instantaneous,stationarysolutions of thetwo-center Dirac-equation,eqs. (4.9,4.11),as of now without the inclusion of electron screeningeffects. The time-dependentperturbationtheory is then applied to calculate the variation of theoccupationamplitudesa~(t),as definedthrough eq. (4.59). Theseamplitudescan thenbe found fromsolutionsof an(infinite) setof coupledfirst-orderdifferential equations,eq. (4.60).This approachis inprinciple exact and able to describeall aspectsof the collision dynamics.In practicalcalculations,however,onefacestwo limitations:

(1) The restrictionto a finite subsetof Hilbert space.Whetherthis is justified canbe testedfromthe convergencebehaviorby including morestatesin the calculation,which hasbeendone [155]forquasi-molecularboundstatesandcontinuumstatesseparately;

(2) If continuumstatesare includeda~(t)canonly be obtainedwith reasonableeffort up to firstorder,i.e. taking am(t)= const.in eq. (4.60).

Only first-orderperturbationtheoryhasalsobeenusedin calculatingexcitationsinto boundstates.

304 J. Rafelski et a!., Fermions and bosons interacting with arbitrarily St rong external fields

Since the initial occupationprobabilities of the 2so-, 3so-, etc. states are unknown, the presentcalculationsare of qualitative characteronly. The solution of eq. (4.60) in first-order perturbationtheory in the amplitudesa~(t) is given by, for excitationsfrom the iso- state:

a~(t)= — a~J dt’(~~(t’)Ia/at’I~1,,~(t’))exp[j Jdt”(e~— EISU)]. (4.74)

Here a~is a constantamplitude (mostly 0 or 1) that showsto what extentthe transition Is—* n isallowedby the Pauliprinciple.In anactual U + U collision the 2so-and3so-stateswill be filled, so thatin generala~,,= a~’,,,,= 0. Under theseconditionswe find

= ~ a~—0.1, (4.75)

whichis a measureof the relativeerror involved usingeq. (4.74) to obtain a~(t). Setting,e.g.,a~,,,,=1in a fictitious experiment,Betz et a]. [155] find i~P—0.5 which shows that a true coupled-channelcalculationcould becomenecessary.We furthernotethat sinceinstantaneous,stationarysolutionsofthe two-centerDirac equationare usedto calculatethe basis functions,H1(t) hasbeenneglectedintransitionfrom eq. (4.60)to (4.74).

In the co-rotatingcoordinatesystemonecan splitthe matrix elementinto a radialandrotationalpart:

a/at—~E~/~R— iO .1 (4.76)

For symmetric collisions the rotational part couplesto the 3dir stateas lowest state and— mostimportant— the correspondingmatrix elementvanishesfor R —*0. On the otherhand,the radial partcouplesto all nso-, ndo-, etc. stateswith large matrix elements(nscrja/aRIlscr) at close distances.Thereforethe set{n} in eq. (4.74) hasbeenlimited to includeonly iso-, 2so-,3so-, 4so-boundstatesandthe Eso- continuum states.The boundstatesolutionsof the one-electrontwo-centerDirac equationhave beenused in ref. [155] as describedin subsection4.3, eq. (4.28).The calculationof the radialmatrix elementscanbe facilitatedby useof

E(~,,Ia/aRli~i~,)= E(Em — E,,Y’(t/inIaVtc/aRI~/im), (4.77)

here ~ is thetwo centerpotentialgiven in eq. (4.11). It is essentialwhenusingeq.(4.77) to insuretheorthogonalityof the wavefunctions(~/i~,~/,,,,), which mustbe eigenfunctionsof the sameHamiltonian.The matrix elementsbetweenboundstatesas obtainedfrom eq. (4.77) by the Frankfurtschool [155]are shown in fig. (4.17a) (dashedlines) for the U on U collision. Betweenneighboring states,e.g.,ls—2s,2s—3s,3s—4s, etc. they are almostequal.Henceonly the matrix elementswith the iso- stateareshown.They do not vanish for largeseparationsR but approacha constantvalue. This is dueto thefact that the molecularstatest~~(R)arenot asymptoticsolutionsof the scatteringequation,but ratherstates

= fi~(R)exp[— i/2mev . r] (4.78)

should be used [155]. Includingthis “translationfactor” in first order in the projectile velocity v onefinds the correctedmatrix elementsshown in fig. 4.17a by solid lines. They vanishrapidly beyondR = 2000fm, indicating the point wherethe two uraniumK-shells just beginto influenceeachother.

The mostnoteworthyfeatureof the matrix elementsis, however,the steepincreaseatvery smallR. For comparisonthenonrelativisticmatrix elementhasbeenscaledto U + U [dash-dottedline in fig.

I. Rafelskj et a!., Fermions and bosons interacting with arbitrarily strong external fields 305

B lm~c/5l 0) g [mM/fll bI

1(33 2(1) 50310032(33) 500RIfm] RIIm]

Fig. 4.17. (a)Radialcoupling matrix elementsof the Iso- to the2so-,3so-,4so- levels in U + U asafunction of internuclearseparation.Thedashedlines arewithout thetranslationfactor. (b) The radial coupling elementsof the iso- level to thecontinuumstates(energiesin electronmasses)inthemonopoleapproximation.From Betz,Soft, MUller andGreiner [155].

4.i7a]. The differenceis due to the fact that for Z~+ Z2> 137 the relativistic nso- and np112o- wavefunctionsprobethe rapidly changingpotentialbetweenthe ions particularlyfor very small R.

The smooth behavior of the matrix elementssuggestedcalculating them also in the monopoleapproximation,usingthe potential of quasinucleusof radius~R,as given in eq. (4.31) insteadof thetwo centerpotential. The results are in agreementwithin 2% of the exact curves in the range20 fm ~ R ~ 400 fm. This justifies the calculationof the radial matrix elementsto the Eso- continuumstatesin the monopoleapproximation[for both, initial andfinal, statesin orderfor eq. (4.77) to remainvalid] as shownin fig. 4.17b.The total transitionstrengthinto the continuum,

= J dEj(Eso-j3/8RJiso-~I2, (4.79)

is larger than into the 2so- bound state, 1D2,,,,,1

2. For the bound states Betz et a]. [155] findI ~ n4 const,i.e., the summationis rapidly convergent.This behaviorimplies that the energydensityof ns statesis dE/dn n4 for Z = 184.

The complexamplitudesa,,(t) follow in a straightforwardmannerfrom numericalintegration ofeq. (4.74) along classicalCoulombtrajectoriesfor the heavy ions. For excitation into otherthan2so-andEso- statesanumberof channelsaddup coherently(e.g., the one-stepexcitation lso-—3so-andthetwo-stepprocessiso-—2so-—3so-).

The total excitationprobabilitiesja~!2andf dEIaEI2 areshownin fig. 4.18 as a functionof R, undertheassumptionthat the final stateis vacant.The curvesshow oscillationswith a maximumtwice ashigh as the final excitationprobability P(b). The riseof Ia(t)12 beyondthe distanceof closestapproachcan be understood.During the approachamajorportion of the iso- electronicdensity flows inwardfollowing themotion of the nuclei.When this motion is (suddenly)reversedthe electronif left behindand redistributes over other states. This accelerationprocessweakensquickly for high impactparameters.The excitationsinto the continuumvary alsoover the trajectory:After the collision theyare comparableto excitation into the 4s state(which will be practicallyvacantfrom simple velocityarguments),whereasin the regionwherethe iso- stateof U + U hasdived into thepositroncontinuum(R ~ 35 fm) theyaredominant.

306 1. Rafelski et a!., Fermions and bosons interacting with arbitrarily strong external fields

olt/

1S0~nSt~ /CabotexcitohmnU°U /

otE~0~~lb

03M~

020 //0.15

~ ~

JLII 103 100 Wi 3003 R[tn~

Fig.4.i8. Totalprobability for Coulombexcitation of a lso’ electronin U + U intoa vacantnso-level andinto thecontinuum(Eso’)at zeroimpactparameter.Thepoint of closestapproachis indicatedby thedotted line. From Betz,Soft, MUller andGreiner[155].

The projectile energydependenceof the excitation probability is shown in fig. 4.19a, the impactparameterdependencein fig. 4.19b. The final excitation after the collision is denotedby f(n), theaverageexcitationover thediving regionby d(n). To obtaintheoneelectronionizationcrosssections,the probabilitiesmustbe multiplied by a factorof 2 to accountfor the spin multiplicity of the iso-state.For example,at 1600MeV andzero impactparameterL

0 is predictedto be0.08.With this number an essentialpoint has beenprovenby the Frankfurt school [155]. Substantial

Coulomb ionization of the Iso- state in a violent, sub-Coulombbarrier, U on U collision may beachieved.The assumptionof L0 = 10_2 madein subsection4.5 seemsto bevery realisticin view of theresultspresentedin fig. 4.19.

1 Probability Probability

~ ~

0 400 803 120) i~i~o 50 10) 150 20]~E~5lMeV] bltml

a) b)

Fig. 4.19. (a) Energydependenceof the Coulomb ionization of theU + U Iso- level at zero impact parameter.f(n) denotesthe final atomicionizationprobability; d(n) theaverageprobability overtheregionwheretheIso- statehasdived into thenegativeenergycontinuum.(b) Impactparameterdependenceof theCoulomb ionizationof the iso- level at E,~5= 1600 Mev. From Betz, Soff, MUller andGreiner[155].

I. Rafelski et a!., Ferm ions and bosons interacting with arbitrarily strong external fields 307

With decreasingincidention energythe excitationprobabilitiesfall off verysteeply,until at tandemenergies (EIab~ 200 MeV) they are much less than l0_6. The same is true for increasingimpactparameters.

We concludethatthe iso--vacancyproductionin U + U is substantial.The influenceof relativity onthe behaviorof electronsin superheavyquasimoleculesis dominant.

5. Spin-zeroparticlesin strongexternalfields

With section4 wehaveconcludedthatpart of our subjectin which contactwith experiencewill beforthcoming in the reasonablynear future. In the next two sections we shall deal with purelyspeculativeor evenacademicquestions.Thus as seenfrom fig. 2.20, muonsbecomesupercriticalforZ> lOs. Since the most importantdeterminantof this result is the massof the particle, we couldreasonablyassumethat a similar value of Z must be reachedbeforeany interestingbehaviorwouldensuefor pions. (This is verified by explicit solution of the Klein—Gordon equation, as will bediscussedbelow in subsection5.2.) For this reasonand also because,in any event, our discussiondoes not include strong interactions, the developmentin this section is, for the moment,purelyacademic.It is evenmoreacademicthanit needbe becauseof the omissionof weakinteractions.Thislatter failing is, however,easilyrectified as we indicatein subsection5.13.

The contentsof this sectionare then to be consideredas the study of a well-posedproblem inmathematicalphysics.Our view is that it is of someintellectual interestto perceivethe differencebetweenthe electroncaseand that of the pions,thesedifferencesarisingto considerableextentfromthe spin-statisticsconnection.After discussionof the single particlepropertiesof the Klein—Gordonequation,we turn to the many-particleaspectsof our problem[98,156]. This will provideuswith thetrue physical significance of the critical field. We shall first review the conventionalquantizationbelow the critical field and then discuss the minimum changesand additions necessaryto make(mathematical)physical senseabove this point. We deal first with the Coulomb case where weencounterthe onsetof a chargedboson condensationin the nucleusaboveZcr. Then we describeapproximatecalculations to characterizethe propertiesof the system for small (Z — Zcr). Sub-sequentlywe consider the caseof large (Z~Zcr)and discussthe numericalcondensatesolutions.Finally,we turnto the distinctcaseof shortrangepotentialsin which neutralcondensatesare created.Throughoutthis sectionwe rely heavily (without furthermention)on the refs. [98,101, 156].

5.1. Formsof the Klein—GordonequationSeveralforms of the Klein—Gordon (KG) equationareknown, eachparticularlyusefulin different

circumstances.In the conventionalform for the eigenvalueproblem in an externalpotential V(r),takenas the time componentof a four-vector,we write the equation(h = c = 1),

(E— V)2p(r)=(p2+m2)co(r), (5.1)

wherep(r) is acomplexscalarfunction.For the mostpart,we shallunderstandthat V is the potentialenergyof a negativelychargedparticle of mass m in the field of a fixed extendedpositive charge

308 .1. Rafelski et al., Fermions and bosons interacting with arbitrarily strong external fields

distribution,

V(r) = — Ze2f(r) (5.2)

where f(r) cuts off the Coulombsingularity at the origin. We shallalsoreport resultsavailablefor ashort-rangepotential[16,49,51, 94,97—101].Thoughfor specific illustration in the lattercasewe shallhavein mind a square-wellpotential,the resultsquotedremainqualitatively correctfor anypotentialsatisfying the condition [49] lim(r—*co)r2V(r) = 0. Simply for brevity’s sake,we shall refer to thescalarparticle as a pion.

The formal propertiesof the KG equationare exhibitedmoreclearlyby introductionof a formalismof first order in the energy(time derivative)[96]. In termsof the two-componentvector

(5.3)

which is a dimensionalhybrid, eq. (5.1) becomes

= ~‘4, (5.4)

~‘=~(1+r3)(p

2+m2)+~(1—r3)+r1V. (5.5)

Here ~, (i = 1, 2, 3) arethe conventionalPauli spin matrices.Since~Wis Hermitian,we recognizethat

the fundamentalscalarproductis the integralof the densitydab(r) = ~(r)Tl~b(r), (5.6)

providingthen an orthogonalitytheoremfor two solutionsof eq. (5.4) belongingto differentenergies.For the norm of anygiven solution,we compute*

f 4i+r4~_2~fc~*(�V)~’. (5.7)

Since V is everywherenegative,the norm (5.7) is certainlypositivefor � ~ 0. On the otherhand,forthe negativeenergycontinuumsolutions(the systemenclosedin a box for this discussion),the normmustbe negative,by analyticcontinuationfrom the limit V= 0. Thus, if we choosethe original KGscalarfunctionto be normalizedaccordingto the relation

f I~V= [2(� — (V))]’, (5.8)

where

(v)=Jco*v~p/Jc*co, (5.9)

* For concisenesswe omit thevolume element— d3x — whereit is self-explanatory.

J. Rafe!ski et a!., Fermions and bosons interacting with arbitrarily strong external fields 309

we seethat, in general,the solutionsetfor given V divides into two subsets4i,, and4 characterizedas follows:

f 4~TI4,,, = 1, �p(V)p>0,

f 4~r~4,,,—l, �~—(V)~<0. (5.10)

For sufficiently weak potentials,the two setsof solutionsdefinedby (5.10) are cleanly separated.Under thesecircumstances,we havea completenessrelation

~ 4,,,(r)[4,(r’)i-~] — ~ 4,~(r)[4,~(r’)r~]= 16(r — r’), (5.11)

whereI is the unit 2 x 2 matrix.For sufficiently strongpotentials,however,membersof the two sets(5.10)maycoalescein energy.

As we shall describebelow, this coming togethercan occur in two distinct ways dependingon therange of the potentialand/or on the angularmomentumof the orbits. Under any circumstances,itrepresentsthe limit of validity of the single-particledescription.The bulk of this sectionwill then beconcernedwith establishinga suitablemany-particledescription.

A third form, of the KG equation is sometimesuseful becauseof its analogy with the non-relativistic Schrödi’ngerequation.For this form, we write

~eff~= (p2/2m)p+ Veff~, (5.12)

where

~eff= �‘[l + (e’/2m)], Veff = V[1 + (�‘/m)— (V/2m)],e=�’+m. (5.13)

In fig. 5.1, the effectivepotential Veff correspondingto the “nuclear” Coulombpotential

V = (— 3a/2R)[1 — ~(r/R)~], r < R

r>R, (5.13a)

0.8 er-to -

Er-i.)

~

0>~ ~

-0.4- \.~‘—~~~Er-0.8 -

-08 - ~Er_0.9 ar7,Rr5/m_

0 I 2

log [mr}

Fig. 5.1. The curvesrepresenttheeffectivepotential, ~ of eq. (5.13) correspondingto thepotential (5.l3a)for variousvaluesof theparameteredefinedin thesameequation.The valuesof e,,,, are representedby severalhorizontal lines with different associatedstriation.

310 J. Ratelski et a!., Fermions and bosons interacting with arbitrarily strong external fields

of a nucleuswith radiusR = (5/rn) andcouplingstrengtha = 7 is shown for severalvaluesof �. Thecurveshighlight the fact that the relativistic term (— V2/2m) in eq. (5.13) is always attractive.Thehorizontallines representthe correspondingvaluesof �eff.

5.2. Approachto critical potential in a Coulombfield

For a Coulombpotential like (5.2), the deepestboundorbit behavesqualitatively as shownin fig.5.2, i.e., in a mannerqualitatively similar to the spin ~ case.This conclusionwas arguedin detail byPopov[49,95]. We presentherean abbreviatedargumentapplicablewhenthe massis that of thepion.To carry throughthis argument,we utilize a virial theorem(derivedbelow), which appliesto boundstates,

— (V)) = m2 — Za�(f’(r))+ Za(f’(r) V(r)). (5.14)

In the presentdiscussionwe choosefor simplicity the potentialof achargedshell:

f(r)=r/R, r<R

=1, r>R. (5.15)

Equation(5.14) thenyields with V0 = (Za/R)

~2 �(V)= rn2—(�V

0+V~)P(R), (5.16)

where

P(R)= JI~I2/JI~I2. (5.17)

We increaseZ andaskfor the propertiesof the boundorbit � —* — m. We borrow[49]the result that ~remainsintegrable(confined)right to the limit, just as in the electron case.The argumentfor thisresult is similar to that reviewedin section2. This allows us to apply (5.16) in the limit. Furthermoresince R > (h/rnc), which is applicableto the pion, we mayapproximateP(R) by unity, i.e. the pion is

-0.) I —

-0.2-

IC I’S I’9 210

Fig.5.2. Energy,�,of thelowestboundstateof theKlein—Gordonequationasafunctionof nuclearchargefor thepotentialdefinedin eq. (5.13a).Theabscissais actually thestrengthofthepotentialattheorigin. The insetmagnifiestheneighborhoodof thecritical point atwhichereachesthe topof thenegativeenergycontinuum.

J. Rafelski et al., Fermions and bosons interacting with arbitrarily strong external fields 311

always inside the nucleus.If we then set � = — m, we find

V0__(V~rr2m. (5A8)

Thus �—(V)~m >0, andthe limiting boundorbit hasa positivenorm.A few more stepsarerequiredto completethe argumentthat the slope is correctlyrepresentedin

fig. 5.2. The exactconsequenceof (5.16) for � = — rn is

—m(V)=(V~—rnV0)P(R), (5.19)

or, sincethe left handside is positive,

V~>mV0 (�=—m). (5.20)

Let usnow form the expectationvaluefrom (5.1) with V—~AV,which yields the relation

— AV)2) = (p2) + m2. (5.21)

With the help of (5.1) again,the derivativeof this equationis

(�— (V))d�/dA= (�V)— (V2). (5.22)

Under the approximationsconsideredand, in particular,from (5.8) and (5.20) we obtain from (5.22)

(setting(V2) (V)2),d�/dA — 2m <0. (5.23)

In fig. 5.2numericalsolutionsof the KG equationfor I = 0 statesareshown.The Coulombpotentialof a chargedsphereof radius 10m’ hasbeenchosen,andthe lowesteigenvalueis givenas a functionof the potentialdepth V

0 = — V(0). We seethat the discretestatejoins the negativeenergycontinuumat V0= 2.i74m. This correspondsto a nuclearchargeZcr= 1986,with “nuclear” radiusR = 10rn~=14.6fm. The value of Z~.increasesto 3007 with A = 2Z and normalnucleardensity,whereA is thenumberof nucleons.The insert in fig. 5.2 is an enlargementof the neighborhoodof the critical point.The dashedline (an anti-resonancein the continuumas we shallseefurtherbelow) was obtainedfromactual numericalintegrationof the KG equation.The dotted line is the approximatebehaviorof thesolutionafter it is modified to include many-bodyeffects as discussedin section5.6.

As hasbeenemphasizedin the cited ref. [49], the normalizability of the solution at � = — m is aconsequenceof the fact that Veff, eq. (5.13) contains, in the energyrange considered,an effectivecentrifugal barrier (shown in fig. 5.1), here of relativistic origin and falling off like r~.For thepurposesof confiningthe wave function the r

2 behaviorprovidedby the usual centrifugalbarrier (tobe addedto (5.13) whenthe orbital angularmomentumis non-vanishing)is sufficient to accomplishthesame purpose.Thus for I > 0, the distinction between long and short-rangepotential becomesimmaterial, the behavior of the bound states for increasing Z being covered under the presentdiscussion.Only the caseof ashort-rangepotentialfor I = 0 requiresa separateconsideration,givenbelow.

Before turning to that discussion,we considerbriefly the derivationof (5.14). This requires theidentity

([r . p, {(�— V)2 — p2 — m2}]) = 0, (5.24)

or(p2) = ((r . V V)(�— V)). (5.25)

If we work this out for V given by (5.2) and insert in (5.21), the result is (5.14).

312 J. Rafe!ski et a!., Fermions and bosons interacting with arbitrarily strong externa! fields

5.3. Approachto criticality for short rangepotential

The behaviorwe wish to describenow, as illustrated in fig. 5.3, is relevantonly for the s-waveof ashort rangepotential.For the specialcaseof a squarewell we have

V(r) = A9(R — r). (5.26)

For anypotential (proportionalto A), it follows from (5.1) that

~(r;�, A) = ~‘(r,— �, — A). (5.27)

This expressionof chargeconjugationin fact assuresus that if the “negatively chargedparticle”follows the energycurvesmarked�_ in fig. 5.3, thenthe “positively chargedparticle” possessestheorbits indicatedby the curveslabeled�.~

We continue the discussionby consideringthe effective Schrödingerequationfor the positiveparticlemoving under(5.26)with A > 0. If we “sit” at �‘ = 0 in (5.12) and(5A3), we have

Veff(E’ = 0) = A[1 — (A/2rn)]. (5.28)

For IA I > 2m and increasing,this becomesan ordinary attractivesquarewell of increasingstrengthandthuswe musteventuallyhaveaboundstatebranchappearas in fig. 5.3, branch�~. Coupledwith(5.27)anda standard,non-relativisticdiscussionfor smallA, themain featuresof fig. 5.3follow almostimmediately.For a squarewell in the s-statethe limiting value � = 0 is not a memberof the pointspectrumas the wave function delocalizes.Consequentlyneither is the endpoint �‘ = — 2m of curvee.Thus (eq.(5.9))(V)_= 0 atthis point and

[�_(V)]e._m2m <0. (5.29)

It follows by continuity that the solid anddashedportions of the curve �.. canmeetonly at a pointat which � — (V) = 0. At suchapoint

�(V)—~V2)=—(v2~-i-(v~2�o. (5.30)

Thus from (5.22)de/dA = 0, andthe occurrenceof the vertical tangentis establishedfor this case(andthis caseonly).

———~~

-m

Fig. 5.3. Schematicrepresentationof the lowest bound-statebranchfor a negative pion in theattractiveshort rangefield. The branchfor thepositivepionrelatedby chargeconjugationis also shown.

J. Rate/ski et al., Fermions and bosons interacting with arbitrarily strong external fields 313

5.4. Limit of the singleparticle description

As long as the potentialstrengthkeepsus below the critical points Zcr andAcr markedon figs. 5.2and5.3,the theoriesstudiedin subsections5.2 and5.3 provideuswith a perfectlysensibleone-particlequantummechanicsprovided that we associatethe indefinite metric with chargeratherthan proba-bility. For a long-rangepotential,only negativelychargedparticlesareboundby a field generatedbypositivecharge,as expected.For the short-rangeinteraction,andconfining attentionto an s-state,wefind a “surprise” in that for A0 ~ A ~ Acr, a positivepion can alsobebound.This is a purely relativisticeffect which can be understoodwhenwe recall that the chargedensity is not positivedefinite, but fora ~ is in fact negativenearthe forcecenterallowing an averageattractiveinteraction.

Beginning in the next subsection,we shall learn how the critical point representsthe limit ofphysical meaning for any many-body or second-quantizedtheory, based on the single-particleHamiltonian (5.5). Here we shall convinceourselvesthat this point also representsa limit for anysensibleone-particlequantummechanics,first for the Coulomb caseand then for the short rangeinteraction.

The description of the supercritical KG equation in the Coulomb potential is similar to thetreatmentof the electronproblemand utilizes a reducedHamiltonian.Supposethat V in eq. (5.5) issupercritical.We write

V= Vcr+ V’, (5.31)

where Vcr is just critical. The basisprovidedby ~, where

= ~cr + r, V’, (5.32)

is complete.We shall study two types of statesbelonging to ~C,bound statesnearthe top of thenegativeenergycontinuumand negativeenergycontinuumstates,both underthe approximation thatthesecan be representedas superpositionsof 4,(_), the boundstateat the top of the negativeenergycontinuumandof the negativeenergycontinuumstates4,~.

Thus we seeksolutionsof

= (5.33)

by meansof an approximateexpansion

= a(e)4,~+ f b~.(�)4,~.de’. (5.34)

With the definitions ~ = — m)

= �~ +f ~ V’4,(), (5.35)

V~= f 4,T1 V’4,(), (5.36)

ignoring continuum—continuummatrix elements,and rememberingthe signsof the norms,we find theequations

— �0)a(�)= f br’(�)V de’, (5.37)

314 1. Rate/ski et a!., Fermions and bosons interacting with arbitrarily strong external fields

(e — e’)b~(e)= — a(e)V~. (5.38)

Looking first for boundstatesolutions,we solve (5.38) in the form

b~.(�)= — ~ (5.39)

Insertedinto (5.37), we obtainan eigenvalueconditionwhich canbe written (e = ~R + ie1),

— — d ‘ IV�I [CR—C] 54ER — Co j ~ (ER — &)2 + �~

and if e~� 0

1= Jd�’ Iv�I: 2 (5.41)(ER—C) +EI

It is straightforwardto see that (5.40) and (5.41) havea pair of conjugatesolutions� = �R± Iei forwhich ER< — m and such that the correspondingstatesc1+ have vanishingnorm. By an argumentwhich we shalldevelopbelow, this implies that beyondthe critical point ~‘ is no longer seif-adjointandthus thereis no one-particlequantumtheory.

But first, let usnotethat (5.37) and(5.38) possessscatteringsolutions,in which (5.39) is replacedby

b~(e)= 5(e — e’) —�~:~~ (5.42)

~ > 0, andwhere(5.37) now yields

a(�)= v~/{e+ iô — �~+f ~ , d�’}. (5.43)

Becausee,~< — m, it is easyto seethat the denominatorof (5.43) is of the resonanceform. The minussign in front of the secondterm of (5.42) implies, however,that this is an anti-resonance,the phaseshift falling ratherthanincreasingby ~ It alsofollows directly that thesescatteringsolutionshavetheexpectednorm

J t~i-1$~ = — ô(� — �‘). (5.44)

We finally considerthe questionof completeness.We show that the continuumsolutions4~justconsideredcannot (as we expected)be complete.By completeness,of course,we mean that theycannotspantheundercriticalsubspaceformedby 4,(_) and4,~,i.e. wewish to showa contradictioninthe assumption

— ~ I(r)[~~(r’).r~]= 4,(_)(r)[4,~_)(r’)r~]— ~ 4,~(r)[4,(r’)r~]. (5.45)

But from (5.45),we concludeimmediately

— ~ I(4,~_)TI,~�)I2 = 1, (5.46)

J. Rate/ski et a!., Fennions and bosons interacting with arbitrarily strong external fields 315

which is impossible.Thus the normalizeablesolutionswith complexenergymust play a role in thecompletenessrelation. We shall not, for lack of physical interest,pursue that role in the presentcontext.

For the short-rangepotentialand the particlesin an s-state,the majorconclusionis the same,butthe details are ratherdifferent. Here for supercriticalfields we look for bound states which aremixturesof the two slightly undercriticalboundstatesnearthe vertical tangentandfind two complexconjugateenergiesandnorm zero.

We turn briefly to the detailsof this calculation.Let 4,~,,be the single positiveand negativenormstatesthatwe include in the calculation.We measureenergyfrom the positionof the vertical tangent;then 4k,, hasenergy6,, and 4~energy(— O~).We look for solutionsof

= wr,ttO, (5.47)

where

(5.48)

where~r0 is subcritical (seediscussionbelow), but ~Wis supercritical,andwe write for c1,

‘1’ = a~4~+ a~4i~. (5.49)

From(5.47)—(5.49)and associatedremarks,we obtainthe equations

wa,, = 6~a~+ ~ +

— wa~= O~a~+ ~ + ~ (5.50)

To study eqs. (5.50), let us recall that from the known movement of the two unperturbedeigenvaluestoward a commonpoint, V,,~,and ~ are oppositein sign. As o~,and O~—~0,we shallsupposethat theyapproacha commonabsolutevalue,

— V,,,,~V~~=V (5.51)

(also set ~ = U). Undertheseconditions,the secularequationimplied by (5.50)yields the solutions

w+—V±iU. (5.52)

Thesesolutionsfurthermoreimply that

= ± ia,,, (5.53)

which yields that for eithersolution

I ~tq,0 (5.54)

An interestingdifferencebetweenthis caseand the Coulombcaseis that herealreadyat the criticalpoint we lose completeness.

We considerefforts to constructquantumtheoriesusingthe complexsolutionsfoundaboveto beill-advised, to saythe least.The sensibleway to proceedforms the context of the remainderof thissection.

316 J. Rafelski et a!., Fermions and bosons interacting with arbitrarily strong external fields

5.5. Conventionalquantization

This is achievedmostelegantlyin an externalfield by useof the first-orderformalism. We adoptthe Lagrangian

1(t) = I d~r[i~t(r,t)r,q~r,t) ~ (5.55)

This yields the canonicalmomentum

1~(r,t) = i~~(r,t)’r,. (5.56)

Thus the requiredcommutationrelationsare

[~(r,t), t~t(rIt)] = r,3(r — r’). (5.57)

The Hamiltonianwhich follows from (5.42) and (5.43) is

(5.58)

which we considertogetherwith the chargeoperator,

Q=—IeIJ~ri~, (5.59)

and both areto be takenin normal form with respectto the vacuumdefinedbelow.As long as we are in the subcriticalregime, the completenessrelation (5.11) will guaranteea

satisfactoryquantumtheorywhenusedin conjunctionwith the expansion

~(r)=~d,,4i,,(r)+~I~&(r), (5.60)

where,as will be momentarilyevident, the a,, are destructionoperatorsfor negativemesons,the L~creationoperatorsfor positivemesons.The assumptionof the non-vanishingcanonicalcommutators

[a,,,d,,t]=5,,,,(5.61)

[ba,b~]= On,,

othersvanishing,in conjunctionwith (5.11) satisfies(5.57). If furthermorethe vacuum is definedbythe statements,

d,,~vac)= 1’,, vac) 0, (5.62)

we find from (5.58)and (5.59),

= ~ + ~ �,,Ib~b,,, (5.63)

~=—IeI(~ a~d~—~l~6~). (5.64)

For Z<Z~, the standardquantizationgiven above is completely satisfactory,i.e., our modelsystemis in principle apossiblephysicalsystem.Hereonemustbackup, at leastbriefly, to the valueof Z = Z

0< Z~,.when the smallest bound stateeigenvalue�,, becomesnegative.Becausethe boson

I. Ratelski et a!., Fermions and bosons interacting with arbitrarily strong external fields 317

numberoperatorsareunboundedabove,it would thenappearthat H is unboundedbelow.If we couldignore weak interactionsthis would be no sourceof worry becausethe stateIvac) would still be theground stateof the sectorof zerocharge.Until onereachesthe valueZ = Zcr, therewould still beanenergygap againstthe productionof a boundnegativepion and a free positive pion. On the otherhand,for Z slightly greaterthanZ0, the vacuumbecomesunstableagainstthe reaction

vac—~(1TibOUfld+ e~+ p.

This processwould be strongly suppressedbecauseof the Coulombbarrier facedby thee~.It mustberemembered,however,that in reality, at Z = Z0 the nucleuswill contain both e andp~clouds, theformerbecausewe are so supercriticalwith respectto electrons,the latter becausewhenthe lowestboundorbit of a ~ in theeffectivesupercriticalCoulombfield crosseszeroenergywe can inversethe~z

decay,

e’-~x+v+.

In principle then both e and p. can serveas sourcesof ir. The physics of this situationwill bediscussedin section6.3.

For the present,we shalldevelopthe theory that is appropriateto asituationthat ignoresthe weakas well as the stronginteraction.In section5.14, we shall indicatewhat modificationsarerequiredinthe treatmentthat follows whenweakinteractionsarepermitted.

In this case,as Z-~Z,however,we approachan inherentinstability, in that at the limit we nolonger haveauniquegroundstate.Forexamplethestateof chargezerowith a ir in thelowestboundorbit and a ir~at restat the bottomof the continuumis degeneratewith the Ivac).

To remedythis difficulty, as hasbeenshown[98,156], we include in H the Coulombinteractionofthe chargedpion field. Beforepresentinganyformalismlet usconsiderthe qualitativeoutlinesof theassociatedphysics. Suppose the system is just subcritical and Z is suddenly increasedto asupercriticalvalue. Whatensuesis somewhat(thoughnot completely)reminiscentof the correspond-ing electroncase.The systemwill liberatepositivemesonsandbind negativepionsto itself. This willcontinueuntil the emissionof ir~ is energeticallyforbidden.Roughly speakingthis occurswhen thescreeningeffect of the bound ir is sufficient, in conjunctionwith the nucleus,to producean averagefield which is effectively subcritical. (The idea of anaveragefield becomesincreasinglyaccurateas Zincreasesandwill form the basis of the considerationof section5.9.)

We remarkon the fundamentaldifferencebetweenthe bosonandfermioncasesin that in the latterthe maximum numberof fermionsemitted is the spin degeneracyof the “diving” orbit andthat divingcannotbe prevented.

In accordancewith the discussionabove, we study the Hamiltonian supplementedwith theCoulombenergyof the mesonfield,

= I ~ + ~e2I Pr)P(~), (5.65)

~(r) = ç~t(r)r,~(r), (5.66)

definedup to ordering problems,i.e., up to additional constantsand single-particleoperatorswhichwill not enterour discussiondirectly in the approximationswhich will be considered.The operatorQof eq. (5.59) continuesto be a constantof the motion.

318 1. Rate/ski et a!., Fermions and bosons interacting with arbitrarily strong external fields

5.6. Thegroundstatefor small (Z— Z~j,long rangepotential

We shallbaseourstudyon an approximateform of eq. (5.65), obtainedas follows. We expandtheoperator4, in terms of the completeset of orbitals generatedby ~cr, the just critical single particleHamiltoniandefined in eq. (5.32). We then omit from this expansionthe positive energycontinuummodesandall boundstatesexceptthe onenearthe top of the negativeenergycontinuum.We thusput(with a slight changeof notation 4,~-~

~(r) d04,0+ fd�~~�(r). (5.67)

Furthermorein the resultingHamiltonianwekeeponly termswhich arediagonalin modenumbersorcreateandannihilatea single pair of irk. This yields

= �0â~a~+U0(d~â0)2+f de[V~~�d

0+V~á~—Jd�e~�

+~J d�u~Ea~a06~a0+~â0â~â0]+~~ (5.68)

Herethe matrix elements�,,, V~areas definedin eqs. (5.35)and(5.36). Similarly

U0 = ~e~I d3rd3r’ [4,~(r)r,4,~(r)]. [~~(r’)i~,4,

0(r’)] (5.69)J

= ~e~f d3r d3r’ [4,(r)T

14,0(r)][4,~(r’)r14,0(r’)] (5.70)Ir—rI

In thedevelopmentwhichfollows, it is well to rememberthat the matrixelementsU0, U~arefixed,whereasthe elements(�~+ m) and V~areproportionalto (Z — Z~~)andwill be varied in the discussion.It is also well to rememberthat the parameter~ hasbeendefinedwith referenceto the single particleHamiltonian ~. The actualHamiltonian— andalso (5.68) — will not “go supercritical”until Z somewhatexceedsZcr. We shall seethis as a specialcaseof a calculationto which we now turn.

Supposethat Z is large enoughto yield the stableground statewith pion chargeq. For q not toolarge the structureof (5.68) suggeststhe approximaterepresentationof the statevector

IG(q)) A(q!Y”2(á~Ivac)+ J d�B~[(q+ 1)!]2~(á~)~jvac). (5.71)

From the Schrödingerequation

H~G(q))= W~iG(q)) (5.72)

andfrom (5.68),we derivethe equations

W~A= (q�0+ q

2U0)A + (q + 1)1/2 J ~ (5.73)

J. Rate/ski et a!., Fermions and bosons interacting with arbitrarily strong external fields 319

w~B~= [(q+ 1)�~+ (q + 1)2U0— e]B~+ (q + 1)1/2~ (5.74)

where

V~+ (2q + 1) (J~. (5.75)

The inherentlimitation of this perturbativemethod to small q is clear from the rapid growth of thecoupling termsin (5.73)and (5.74)with q.

Fromtheseequations,by eliminatingB~in the usualway appropriateto aboundstatesolution,weobtainthe following eigenvalueequationfor W

t”~,

-m,i

w~—w +( +1) ~ 576)q q j W~°’~—W

0—/10+E’

where

Wq = qe0+ q2U

0, (5.77)

/.Lq = E0+(2q + 1)U0. (5.78)

It is not difficult to see that for eq. (5.76) to yield a real solution w~°~as a powerseriesin (Z — Z~~)andthereforeapossiblestablestate,wemusthavethattheparameter~sq> (— m).Forfixed (Z — Zcr) > 0,the chargeof thegroundstateis theminimumvalueof q for whichthis conditionis satisfied.Fromtheassociatedcondition~sqi <— m, we thendeducethe importantcondition

Wq < — qm. (5.79)

Since furthermore W~< Wq from (5.76), we concludethat the energyper particle is below (— m).This suggeststhe possibility of resonantscatteringof positivepionsas treatedin the nextsubsection.

We remark finally that our equationsmayhave a solution for q = 0, if (Z — Zcr) is small enough.Thus— as is hardly surprising— the inclusionof the pion—pion interactiontemporarilyretardsthe onsetof the bosoncondensation.

5.7. Scatteringresonances

We shall illustratethe possibility of resonantscatteringof positive pions,notedjust above,in thesimplestapproximationwhich exhibits the phenomenon.We takethe casethat the groundstatehaschargeunity andconsiderthe scatteringof the positivepion, asymptoticenergy(— �)(> 0). We write

B~.(�)~a~vac)+ A(�)Ivac). (5.80)

This correspondsto an approximationin which the groundstate is representedonly by its simplestcomponenta~Ivac).We then allow only for potential scatteringand for virtual annihilation into theunstableunchargedvacuum,but do not allow for transitionsinto stateswith additionalpion pairs. Inthis approximationwe areto solve

HI�~)= (W1 — ~ (5.81)

320 J. Rate/ski et a!.. Fermions and bosons interacting with arbitrarily strong externalfields

with W1 given by (5.77) for q = I. From (5.41) we obtainthe equations

� + E’)B~,(�)= V~°~A(�), (5.82)

C + W1)A(C)= I d�’V~B~.(�). (5.83)

Equationsof this form havebeensolvedinnumerabletimes includingsection3 of this review.Herewe find

1 V<°)*V~°~

B~(�)= ô(E — �‘) + — � + iO + C’ — C + iO + W, — F(— C + iO) (5.84)

V,01

A(�)=+iO+w~p(+.o)~ (5.85)

where— In

IVt0i2F(y) = I dC ~ (5.86)

PreciselybecauseW, <— m, (5.84) implies resonantbehaviorof the scatteringanalogousto the caseof resonantscatteringof positrons from supercriticallybound electroniccharge.The presentdis-cussion can be extendedto the caseof arbitraryground statechargeand to include an improveddescriptionof that state.

The position and width of the resonancepredictedby (5.84)—(5.86) is in preciseaccordwith thevaluesobtainedin discussionof the transientphenomenonto which we next turn and will be givenmore explicitly below.

5.8. Decayof the unstablevacuum

Following the previous considerations,it is natural to turn next to the study of a transientphenomenonin the usual idealization.We assumethat for time t ~ 0, the systemis in the (stable)unchargedgroundstate.At t = 0, we suddenlyturn up the externalfield to a valuefor which the stablechargeis unity. We thenseekthe statevectorof thesystemutilizing oncemoretheHamiltonian(5.68)anda representationof the statevector of the form

It) = A(t)I vac)+ I dCB�(t)I.~â~Ivac). (5.87)

The amplitudesA(t), B~(t)(cf. (5.82),(5.83)) aredeterminedfrom the equations

iô,A(t) = I dCV~°~B~(t), (5.88)

iD5B~(t) = (W, — C)B~(t)+ V~°)*A(t). (5.89)

The solutionof theseequationssubjectto the initial conditions

A(0)=1, B~(0)=0 (5.90)

is againaclassicproblem,seesubsection3.4. Thesolution in thenarrowresonanceapproximation,valid

J. Rate/ski et a!., Fermions and bosons interacting with arbitrarily strong external fields 321

for t ~ F~,whereF is the width characterizedbelow, is

A(t) = exp(— iWt) (5.91)

B~(t) = B(�){exp[— i( W, — C)t] — exp(—i Wt)}, (5.92)

where

B(e) = — V~°)*[W_W, + �]~‘. (5.93)

If we write

W= W~—~iF, (5.94)

thenapproximatelythe resonanceenergyis given as the solution of the equation

~(O) 2

WRPIdEWW+ WI—ER, (5.95)

and

I’ 2irI V~~I2. (5.96)

For the readerinterestedin repeatingthe exercise,(5.93) is derivedfrom (5.89), whereas (5.95) and(5.96) follow from (5.88) for t ~ F~in the narrowresonanceapproximation.Equation(5.96) itself maybe derivedalsofrom the normalizationcondition

IA(t)12 + I dCIB�(t)I2= 1 (5.97)

under the same restrictions,and expressesthe final disposition of probability according to theequation

fd�~B~(o~)j2=1. (5.98)

The connectionbetweenthe considerationsof this subsectionand thoseof the previousone arewell known: The parameters(5.95) and (5.96) are precisely thosewhich characterizethe resonancescattering.

5.9. Self-consistentcharged condensateequations

We now describeanapproachvalid for large (Z — Zcr). We shall usethe meanfield methodwhichfor largeq is sufficient to yield qualitatively correctphysics.

Let (Z — Z~~)~‘ 1. We assumethat the ground statehaschargeq and that it can be describedingood approximationby the statevector

q) (q !)~h/2(~t)n I vac), (5.99)

where

= I ~t(~~)ni4,o(,~, (5.100)

dlvac)=0, (5.101)

322 1. Ratelski et a!., Fermions and bosons interacting with arbitrarily strong external fields

and

I 4,~(r)r~4,~(r)= 1. (5.102)

We next calculatethe expectationvalue of I~,eq. (5.65) with respectto (5.99) as a trial function.The result containsasemi-classicalpart which is extractedby simply replacing4, —~á4,0in (5.65) andcalculatingthe resulting expectationvalue. There are also quantumfluctuations to which we shallreturn in the last part of this section.We thusfind

W(q)= (qI4Iq) qJ4,~x4,

0+~q2e2 J po(r)po(r) (5.103)

Po(T) = 4~(r)r~4i~(r). (5.104)

We shall determine4,0(r) and q from the variationalprinciple

= O[ W(q)— ~sqJp0(r)] = 0, (5.105)

wherejsqis the Lagrangemultiplier for the condition(5.102). Varying with respectto 4~atfixed q, weobtainthe condensateequation

1sr1cI~(r)= ~C$(r)+ T1 Vcondensate(r)~t~(r)~‘~~(r), (5.106)

where

(5.107)

— 2 f d3r’p(r’)

V condensate~1J— e j — , , . 0and

p(r) = tI~tTItI~. (5.109)

Herethe physicalsignificanceof the parameterjx is establishedby computing(q + I IHIq + 1) in thesameapproximationas led to (5.103). This yields

(q+lIHIq+1) W(q)+,a. (5.110)

We thus seethat jx is theenergyof the last pion added.Thus we mustseeksolutionsof (5.106)underthe condition~srealand

~—m. (5.111)

In principle weshould proceedas follows: Fix ~x at — m andfind the “correct” solution(seebelow) to

(5.106). In generalthen

1~~tr1~t’=q (5.112)

will not be an integer. Now increase~ slightly until q becomesthe next larger integer. For large(Z — Zcr) this stepis of no practicalimportanceandneednot be carriedout.

It is obviousphysically that in addition to the correct(stable)solutiontherewill existsolutionswith

I. Rafelski et a!., Fermions and bosons interacting with arbitrarily strong external fields 323

smaller valuesof q. Thesewill describethe lowest lying memberof the point spectrumwith realeigenvalueof a non-self-adjoint ~ whichalso hastwo complexconjugatemembers(or more!)of thetype describedin section5.4.

At the stationarypoint, we have,using(5.106),

W(q) 1mm W0= p.q—~e2q2 f Ir—r’I (5.113)

In the limit q—~~ (which requiresZ -+ ~), this may be viewed as a phasetransition to a chargedsuperfluidstatewith macroscopicoccupationof the mode 4

0(r), sinceanotherdefinition of 40(r),equivalentto the one given, is

(q~ci(r)Iq+ 1) = \[q4,0(r) tF(r). (5.114)

An independentderivation of (5.106) can be basedon the definition (5.114) and the Heisenbergequationfor 4,(r), but we shall not developthis remarkin detail.

Still anotherdescriptionvalid for large q replacesthe chargeeigenstate(5.99) by a coherentstate

lao) = exp[a0dt]I vac), (5.115)

wherea0 = q”

2 At this point we note that q may be considereda continuousfunction of Z with ~sfixed at — m.

5.10. Solutionsof the condensateequations

In this section,we solve the condensateequations(5.106),(5.108).It is clearfrom the onsetthat thecomplicatedintegro-differentialequationsmust be approachednumerically. To this end, it is con-venientto convertthem back into the secondorder form, eq. (5.1). Introducingthe total potential,

Vtotai = VCoulomb + Vcondensale (5.116)V~+qV

0,

we obtainfrom eqs.(5.106) and(5.4)

(/5 — V101,,i)2çoo= (p2+ m2)~

0. (5.117)

Herethe wave function~ is normalizedin the usual way

f d3rp

0=l, (5.118)

where

Po =

2po(/5 — Vtotai)4p0.

In the expression(5.116) for the total potential,we have

Vo=e2fd3rl ~ (5.119)

lr — r I

For the restof this discussion,the Coulombpotentialwill be that of theuniformly chargedsphere

324 J. Rafelski et a!., Fermions and bosons interacting with arbitrarily strong externalfields

of radiusR,3ZaIf I/r\2]

r<R,V~= Z (5.120)

r>R.

Motivated by the (hypothetical)example of isospin zeronuclear matter (f = (ZIA) = ~) at normal

density,we choose

R=bA”3, (5.121)

where

b = [(1.2 fm m,rc2)/hc](pn°/pn)”3, (5.122)

andp°,, is the normal nucleardensity.We areinterestedin solutionsof the equations(5.116)—(5.1I9)inthe limit Z> Zcr = 3006.5 (with p,, = p~,I =

In solving theseequationsan interativeprocedurewas utilized: Given V~(in the nth iteration)wecansolveeq. (5.117)

(m+ V~+ ~ V~)2~~= (p2 + ~ (5.123)

with /5 fixed at — m and q~servingas eigenvalue.(Even if Z < Zcr, solutionsexist for q~< 0.) Asexplainedin subsection5.9, we are interestedin the smallestq(fl) that yields a solution. Given theeigenvalueq~,we subsequentlyobtain

— ~j ~ (n)1,r(n)

~totaI ~ q ~ ,

and(n) (n) (n) (n)

Po = ~ ( m — Vtotai)co . (5.125)The iteration loop is closedby computationof

r (n)V~’’= e~I d

3r’ Po (5.126))

The aboveprocedureis then repeateduntil the norm of the differenceof successiveapproximationsissufficientlysmall, i.e.,

II ~ — v~n±DII< 5, (5.127)

wherewe havechosenS = i0~.

In practice,the choice= Z V,~, (5.128)

provedsufficient for our purposes.The Klein—Gordon equation(5.123) was solved by integratingfrom r = 0 outwardson a logarith-

mically distributedmeshof points. The eigenvalueq is found in the usualfashion.Assumingq(m) inthe mth integration of the Klein—Gordon equation,with fixed potential V~O””, the integratedwavefunction ~~m) develops as r—*~,an irregular (exploding) part. Comparingwith the behavior forq(m)+ Oq, an improvedq(m~~)is found from the methodof steepestdescent.

J. Rate/ski et a!., Fermions and bosons interacting with arbitrarily strong external fields 325

We turn then to discussionof the numerical results.The free parameterin our solutionsis the“nuclearcharge” Z, and all quantitieswill be given in units of m. We havechosen,arbitrarily, valuesof Z = 3025,3500 and 3750 to encompasstheessentialfeaturesof our calculations.While the smallestof thesevalues is still closeto the critical value3006.5,the largestoneis significantly supercritical.Infigs. 5.4—5.6 we presentthe wave functionsw~the chargedensityPo’ and the radial chargedensityr2p

0. The significantfeatureto note is the delocalizationof the wave functions,which can, to a largeextent, be associatedwith increasing width of the total potential, shown in fig. 5.7. The mostremarkablefeaturehere is the fact that with growing supercriticality the potentialapproachesmoreand more closely to the value — (2 + �)m,where � — (ir/(r)). This result is in agreementwith ourexpectationsthat the potentialmustbe deeperthan — 2m to generatea boundstateat js = — m. Wenote that the potential shows a local minimum for higher Z, an effect that is associatedwith therelatively high localizationof the condensatechargedistributionas comparedto the assumednuclearcharge.Also the relative similarity of the total potentialsat the origin for higherZ can be understoodin termsof theincreasedinfluenceof the condensatepotentialon the form of the total potentialat theorigin. This is illustratedin fig. 5.8 which exhibitsthe condensatepotentials.For Z = 3750 the strengthof the condensatepotential at the origin is alreadysignificant. This is seenmore clearly in fig. 5.9wherefor Z = 3750 the threepotentials(Coulomb,condensateand total) are compared.Finally fig.

o.ie I I I I I I I I I 5C I I I I I I

o2~ ~ ~ ‘1215

r [m1} r [m’]

Fig. 5.4. Solution g~of eq. (5.117), the Klein-Gordon equation Fig. 5.5. The chargedensityassociatedwith thesolutionsshowninwith self-consistentpotential for various values of Z, the nuclear fig. 5.4.charge.

0.25 I I I 1 I I I I I.4 I I 1 1 I I I 1

~ ~ ~ ~ :~8~z50:r [ml] r [m~]

Fig. 5.6. The chargedensitytimes r2 for thesolutionsshownin fig. Fig. 5.7. The total potential, V,01,~,for the threecasesconsidered

5.4. in thepreviousfigures of this section.

326 1. Rate/ski et a!., Fermions and bosons interacting with arbitrarily strong external fields

0.5 , I I

045 ~ ~Z~375O I I : ~d.

0.05 I I~ -2.So 0 20 300.000 3 6 9 2 5 r [mH]

[m~] Fig. 5.9. Comparisonof the total potential, V,,,~,I, thecondensate

Fig. 5.8. The condensatepotential. V~O,,d.for thethreecasesunder potential. V~0,,j,and the Coulomb potential, VCOOIO,,,b, for Z=

considerationin thetext. 3750.

5.10 exhibits q, the chargeof the condensate(and thereforeof the groundstateor “vacuum”) as afunctionof Z. As expected,q growsmonotonicallywith Z’ = (Z — ~ atfirst linearly, but soon(Z’/q)beginsto decreasemonotonicallywith increasingZ.

5.11. Quasi-particleexcitationsand the stability of the condensate

In this subsection,we showthat the (charged)groundstateis stablewith respectto fluctuationsofzero charge. This emphasizesthe role played by the subsidiary condition in eq. (5.105). Theapproximate,coherentgroundstate(5.115) is in fact an approximateeigenstateof the operator

(5.129)

where

(5.130)

The coherentstateis furthermorecharacterizedby the condition

(a~I~(r)Ia~)= 1(r), (5.131)

which is equivalentto (5.114).

q 50

3000 3200 3400 3600 3800

zFig. 5.10. The condensatecharge,q, asafunctionof the nuclearcharge,Z.

.1. Ratelski et al., Fermions and bosons interacting with arbitrarily strong external fields 327

Furtherdiscussionwill thenbe tied to thedescriptionassociatedwith (5.l29)—(5.130).We shall view

I a0) as an approximatevacuumstatein the supercriticalfield andindicatethat this vacuumhasstableexcitations.In this discussionan essentialrole is played by the completenessof the solutionsof theoperator(~Ceff— /5r~).Besidesthe eigenfunctiontfr~~with positive norm (and eigenvaluezero), it haspositive eigenvaluescorrespondingto positive norm andnegativeeigenvalueswith negativenorm.

According to the standard lore, the quasi-particleexcitations will be the “small oscillations”associatedwith a new operator,~‘(r),where

~(r) = tI~(r)+~(r), (5.132)

andthus

(aoI~(r)Ia~)= 0. (5.133)

Substituting(5.132) in (5.129),usingthe defining equationfor ‘1(r), we find

= F0 +I ~eff — /5ni)X

+~e~I [~(r) + ~~(r)][i~(r’) + ~~(r’))d3r d3r’

Ir — r I

+ e2{f ~ d3rd3r’ + b.c.Ir—rI

+ ~ e2~ [~t(r)T1,~(r)][,f(r’)T1~(r’)Id

3r d3r’, (5.134)Ir — r I

where

= ~~(r)r,~(r). (5.135)

Here F0= W0—jsq (compareeq. (5.113)) is, approximately,the condensationenergyassociatedwith

H’. In the remaining discussion,we shall treat only the terms quadraticin ,~, assumingthat theremainingtermsmaybe treatedby perturbationtheory.

We thusstudy the operator

= I ~~(~eff- ,sr~)x+ e2I ~ + ~‘). (5.136)

We seeka realization of the operator~jr) which brings (5.136) to diagonalform and satisfies thecommutationrelations

[,~(r),~t(r’)] = ‘r

18(r— r’). (5.137)

[~(r),,~(r’)]=0. (5.138)

We showultimately thata solution to this problemcanbe found by studyingthe amplitudes

~a~I~(r)Iv)= f~(r), (5.139)

(aoI~t(r)Iv)= g~(r), (5.140)

328 J. Rafelski et a!., Fermions and bosons interacting with arbitrarily strong external fields

wherethe statesarequasiparticlestates,eigenstatesof ~ For example

(aoI[ri~(r), hl’2]11’) cr,f~(r)

= (~eff— /5r1)f~(r)+ e2J Ir — r’I~[~j~,(r’)+ ~~(r’)]ri~(r), (5.141)

and similarly

— C~g(r)r1= g~(r)(~~ff— /5r1) + e

2~t(r)TIJIr — r’I~[~,(r’)+ ~~(r’)], (5.142)

with

=4~t()f() (5.143)

For an acceptablesolution the physical ç should, from their definition in (5.141) and (5.142), bepositive.

We attempt to demonstratehow (5.151) and(5.142) may meet this requirement.By forming theobviousscalarproducts,we derivefrom (5.141),(5.142),

Fit it~LJf~r1f~—jg~r1g~

= f ~(~ett /5T1)~+ Jg~(~ff—j.tr1)g~+ e2 J~ + Ir-r’I + ~ (5.144)

The third term of (5.144) is positive (unlessthe integralvanishes,see below). Thefirst two terms arenecessarilynonnegative,as we conclude by expandingf~and g~in terms of the solutions of[~‘eff — /5r

1]. This suggeststhat all �~underconsiderationwill fall into two classes.In the first

(fr, f~)= f f~r1f~>0 and (f~,f~)>(g~,g~). (5.145)

By ignoring the coupling terms in (5.141) and (5.142), we see that thesesolutions arise from thepositive energysolutionsof (~t’eff— /5r1). The secondset ariseundersimilar circumstancesfrom thenegativeenergysolutionsof [~C~ff— /5r~]andarecharacterizedby the conditions

(g~,g~)<0 and I(g~’g~)l>I(f~f~)I. (5.146)

There is in addition to the aboveasolutionwith C~= 0, andf,. = — g,. = ~. As is well known fromthe theoriesof superconductivityand superfluidity, this solution does not correspondto a physicalexcitation of the system— it is a spurioussolutionassociatedwith our failure to conservechargeinlowestapproximation.We iteratethat the physicalexcitationscarry positiveenergy.

We short circuit further detaileddiscussion,quoting only the results neededto round out ourpicture:The net effect of a more detailedconsiderationis thatwe maywrite

~(r) = ~ [a~f~(r)+ ~~r)], (5.147)

where

Iv) = ~i~Iao), ~Iao)= 0. (5.148)

.1. Ratelski et al., Ferm ions and bosons interacting with arbitrarily strong external fields 329

It follows that

112 = ~ E~4,,+ constant, (5.149)

wherethe constant(zero point energy)will not be discussed.We havethusachievedour initial aimwhich was to show that(properlyredefined)H2 is positive definite.

A standarddiscussion,usingmuch the same tools as were indicatedabove,can be carriedout toshowthat the vacuumstate lao) is locally stableundertheseconditions.As statedat the onset,it isthenassumedthat the cubicand quarticterms of H’ can be treatedby perturbationtheory.

5.12. Quantizationbeyondthe critical pointA~.for short range potentials

In this subsectionwe return to the discussionof the situation (characterizedby a short-rangeinteraction) pictured in fig. 5.3. For A <Acr, the formal steps of the quantizationprocedureare thesameas given in subsection5.2a. Oneessentialdifferenceis to be noted,however,in the compositionof the expansion(5.60), for the Coulomb case,the secondsum over orbitals with negative normcomprisedonly scatteringstates,since the boundstateswereof positive norm. Here for A > A0 butA <Acr, we encounteraboundstateof negativenorm,which upon quantizationis to be pictured as aboundorbital for a positivemesonat a positiveenergy (cf. (5.63)).

With this interpretation,the quantumfield theoreticalsignificanceof the point A = Acr is again thatit representsa limit of physicalsensefor the quadraticHamiltonian. For at this point the vacuumloses its significance as the unique state of lowest energyand becomesinfinitely degeneratewithstates containing any number of chargedpairs ira, i.e., we have the onset of an overall neutralcondensationprocess.To havea possiblephysical system,we mustagainadda stabilizinginteractionto the system.Thoughanarbitraryrepulsivequarticinteractionwould do, we shall againmakeuseofthe self-Coulombinteraction,i.e., of the Hamiltonian (5.65), to illustratethe ideaswhich appearto beadequateto treatthe supercriticalregion.

When V in ~Wis supercritical,we introduceagainthe decomposition

(5.150)

where 7C0 is subcritical.This is necessaryin the presentinstancebecause~Cno longer generatesacomplete set when A = Acr, a point which we shall addressmore carefully below. The special“dangerous”orbitals consideredin subsection5.4 will now be called 4,±(r).We also define

= ~~04,’ (5.151)

o =~(�±—�..)>0, (5.152)

(5.153)

wheretheseall refer to the nearcritical eigenvalues.

To establishthe stability of the Hamiltonian (5.65) in the presentinstance,we write

~(r) = â4,±(r)+b~tçtk.(r)+~(r). (5.154)

For the remainderof the presentsubsectionwe drop the terms arising from ~, since thesemakereferencesto the non-dangerouslevels: a full discussionincluding theseterms will be considered

330 1. Rate/ski et a!., Fermions and bosons interacting with arbitrarily strong external fields

subsequently.We alsorewrite the first termsof (5.154) by meansof the operators

4 = (a +

= i(dt— 1)/2, (5.155)

yielding

~(r) = 4tfre(r) + ij5t4,0(r), (5.156)

where4,~and4,,, arethe combinations,

= (1/V2)(4±+ 4—), (5.157)

= (l/\/~)(4.~.— 4,...). (5.158)

With the aid of definitions (5.150)—(5.153)andthe furtherdefinitions

Vab = J 4~T~V’41b, (5.159)

Uab.cd = e2f (4,~rI4,b)(4,~rI4,~)’/~r— r’I, (5.160)

L = btb — ata = : i(qtpt — pq):, (5.161)

we obtainby straightforwardtranscription(where Ueeee= Ue, etc.):

Hred(5+ V,,0)j3

t~5+(8+Vee)4t4+(/5 V,jL+~Ue(4t4)2+~U~0{4

t4,j3tfl}+~U0(15

t13)

2

+ ~{4~4,I~}Ue oe — ~{~5t~51~}Uooe + ~I~Uoe oe . (5.162)

The subscripton 1~remindsus thatwe aredescribingonly two degreesof freedom.For A Acr~1, the basis defined by (5.150) and (5.151) should be very satisfactory for the

diagonalizationof the Hamiltonian(5.162).The unperturbedvacuum(we considerthe subspaceL = 0)will mix with stateswith atmost afew pion pairs to yield a new stablevacuum.This is guaranteedbythe positivedefinite characterof thequartictermsof H. In contrastto the Coulombcase,we shallnotpresentany specific approximatecalculations.

For A — A~. A’ ~ 1, the mixing becomeslarge and the treatmentby meansof the basis definedabove cumbersome.In this casethe stability of the ground state maybe inferred from a classicalapproximationfor which

((414)2) (4t4)(4t4) (q2)2, (5.163)

~ q2p2, (5.164)

etc.With

8+V,,,,=—B, O+Vee=~A, (5.165)

we have(1=0)

W(q2,P2) = (~red) = — Aq2— Bp2+ ~ U~(q2)2+ Ueoq2p2 + ~ U

0(p

2)2. (5.166)

As will be seenbelow, we can, without loss of generality,choosep2 = 0. The variation of W with

J. Rafe!ski et a!., Fermions and bosons interacting with arbitrarily strong external fields 331

respectto q2 thenyields the minimumat

q2A/Ue, (5.167)

andW~A2IUe, (5.168)

which is proportionalto (A’)2.

5.13. Mean field approximationfor the neutral condensate

Further progressand substantiationof the above simplification dependson recognizingthat thevariational expression(5.166) may be derived as the expectationvalue of Hred with respectto acoherenttrial function

a’, b’, 0) = exp(dtal+ ~tbl e~°)Ivac). (5.169)

(This is true insofar as la’l2~1, lb’I2~’1, and requiresrememberingthat (5.169) is not normalized.)

With the identifications

q2 = 1q0I

2, p2 = 1p0l

2, (5.170)

we have(immediatelydroppingthe subscriptzero)

\/2q = a’+ b’ e’°, (5.171)

\/2p =i(a’—b’e’°). (5.172)

Average chargeneutrality is assuredby choosinga’ = b’. The choice 0 = 0 for the arbitraryphasefurthermoreguaranteesp = 0. Both hereandbelow,however,it is necessaryto retainthis phaseuntilthe final stagesof the formulation.

The observationsjust madesuggestthat the previousclassicalapproximationcanbe improved byalso varying the operatorsat,b~,i.e. by varying the functions &(r) and Ø,,(r) in (5.156). Theexpectationvalueof Hred with respectto a generaltrial state(5.169)againhasthe form (5.166)withthe moreconcisedefinitions of A and B,

A = —I 4,~4,e, (5.173)

B =—I Ø~’4,O. (5.174)

It is importantforthe sequelto noticethat(5.166) is afunctionalof thequantitiesq4,~andp4,0,(andtheir

hermitian conjugates).The variation is to be carriedout subjectto the constraintsthat 4,...given by

4,±~(2Y112(4,~±4,,,) (5.175)

have positive and negative norm respectively.From the variational expression(5.166), we thussubtract

- C±lahj2I 4,~r~4,+- �4b’12I 4,~r~4,, (5.176)

332 J. Ratelski et a!., Fermions and bosons interacting with arbitrarily strong external fields

the coefficientsof the normsrepresentinga definition of Lagrangemultipliers. This is transformedbymeansof (5.171), (5.172),(5.175),into the expression

— ~ [I~I~+ I~121 [14,~r~4,~+J4,~r~4,,,]

— 8[IqI2 + p12] [f4,~r~4,,,+ J~~T14?e], (5.177)

wherewe havealso assumedchargeneutrality andusedthe definitions (5.152)and (5.153).Variationof the sum of (5.166)and(5.177)with respectto Iql2~andlpl24,~yields the equations

~‘eff4,e= /.LT

14,e + &r14,0, (5.178)

~‘eff4,,,_ /5r~4,,,+Or14,~, (5.179)

or equivalently

~‘eff4,±= e+r

14,±, (5.180)

where

~eff = .~C+Vefi, (5.181)

veff(r) = e~1f pegfr’)Ir — r’I’, (5.182)

and

Pefffr) = IqI2&~(r)r

1&(r)+ pI24,~(r)r

14,,,(r)= IqI

2pe+IpI2po. (5.183)

An understandingof the structureof the non-linearproblem posedby (5.178)—(5.183)requiresfurtherdiscussion.For instance,whathappenswhenwe set p~2= 0? Sincethis correspondsto settingthe phase0 = 0, we areled to study the equivalenttransformation

4i~.= e~°4,., 4,.~.= 4,±, (5.184)

or in termsof 4,~and4,~

— ~ (1 + e’°)4,~+ ~ (1 — e’°)4,0 (5185)

= ~ (1 — e’°)4,~+ ~ (1 + e’°)4,,,. (5.186)

With the helpof (5.171)and(5.172), we canfurthermorewrite

IqI2’xn, 1pI2—(l—x)n, (5.187)

where

n = Ia’12+ lb’I2, x = ~ (1 + cos0). (5.188)

By meansof trivial algebrawe now find as expectedthat (5.178),(5.179)or (5.180) is invariant in form

underthe transformation(5.185,6)but becauseP~ XPe~U X)P,, (5.189)

J. Ratelski et a!.. Fermions and bosons interacting with arbitrarily strong external fields 333

the definition (5.183) is simplified to

Pefffr) np~(r)rnp(r). (5.190)

Thus we haveshownthat the solution for p12 � 0 can be transformedinto one with p12 = 0 withoutaffecting any physical results.But to obtain a generalformulationof the problem, this must be doneafter the variation hasbeencarriedout.

If we apply (5.185,6) to the energy(5.166) (or equivalentlyset IpI2 = 0), we find

W(n2,0)=—A’n+~U~n2, (5.191)

which is a minimum for (cf. (5.167))

flAIUe, (5.192)

wherewe henceforthdropthe primes.Rememberingthe definitions (5.160),(5.165),we seethat p~ff(r),(5.190) is independentof the scaleof 4,~

The last observationis the essentialoneto understandthe limit o—*0. In this limit Pett will remainfinite, but of the separatefactors,n = q2-+0 andp~—~x.It is strongly suggestedthat we rescalethefunctions~e’ ~ We thereforedefine

4, = q4,e, i/i = 4,,,/q, (5.193)

thusretainingthe scalarproductbetweenthem (eq. (5.199)below). With

(5.194)

equations(5.178,9) become

(~‘eff/5TI)4, = q4ATI~c, (5.195)

(~‘eff /5r1)~fI= Ar14,, (5.196)

where

(5.197)

v = er1 I p(r’)!r — r’l~, (5.198)

I if,4,=1, jiin~t=J~~=0~ (5.199)

= ~ etc., (5.200)

p(r)=çb4,. (5.201)

In this new form the structureof the self-consistencyproblem is completelyclear. The set(5.195)—(5.201)will havea self-consistentsolution for any (reasonable)q ~ 0. (At self-consistencyq

2will, in fact, be given by (5.192).) The limit q = 0 may be takenwith impunity. In this limit, theself-consistencyproblembecomesindependentof ~f~’andof its eq. (5.196). This meansthat we havelost one solution of our self-consistentK.G. equation.Equation (5.196) now definesa non-vanishingfunctionsuitably orthogonalto all solutionsof theK.G. equationandprovidinga missingfunction tomakeup a completeset.

334 1. Rafelski et al., Fermions and bosons interacting with arbitrarily strong external fields

Let 4’a be all solutionsof

~‘eff4,a = EaTi 4,a, (5.202)

other than 4, itself. As describedin subsection5.2a,thesewill divide neatlyinto two sets4,~,4,~,withpositive andnegativenorm.For an arbitraryvector

f(r) = (~‘~), (5.203)

we have(including the limit S = 0, but not confinedto it) an expansiontheorem

f = f~4’+f~t/i+~ f~4i~+~ f,,4,,,, (5.204)

where

foJiiif’ f~J4uf~

~ f,,J4,,,f. (5.205)

The discussionjust completedprovidesuswith the basicmachineryneededto carry out a completequantizationin the supercriticaldomain A > Acr. If we usethe aboveresultsin the limit 0 —~0, we usethe operatordecomposition

4,(r) = 4,(r)+ ~(r), (5.206)

but thishasthe sameform as (5.132),andthereforethe quantizationcarriedout in subsection5.11 canbe takenover verbatimand will not be repeated.

5.14. Modifications necessaryfor weakinteractions

We shall finally study the modificationsneededin the considerationsof the previoussectionsinorder to makethem relevantwhen the effects of weak interaction are turned on. For the presentstudy, it suffices to considerthe weak interactionas an externalsourceof ir, turnedon in order tokeepthe valueof the chemicalpotential,introducedin eq. (5.105),fixed at /5 = — me,whereme is theelectronmass.The self-consistentfield treatmentof subsection5.9 carriesthrough preciselyas giventhere,andit is only necessaryto supplementthe previousdiscussionwith numericalresultsrelevantto the new valueof the chemicalpotential.Theseareexhibitedin figs. 5.11—5.17which areto be set inoneto onecorrespondencewith figs. 5.4—5.10(exceptfor selectedchangesof scalewhich are noted).For the considerationsof subsection6.3, the most important result is that shown in fig. 5.14 whichimplies a limiting potentialof depth-= — m,~.

We next consider the relevanceof the remainder of the many body theory presentedin theprevioussections.Again, the prime observationis that the combinationof electromagneticand weakeffects implies the creation,if anything,of a chargedpion condensate,so that within this approxima-tion the theory of the neutralcondensatedoesnot enter.Evenfor the chargedcondensate,the resultsof subsections5.7 and 5.8 cannotbe takenover for (Z — Z

0) small. In placeof resonantscatteringof

J. Rafe!ski et a!., Fermions and bosons interacting with arbitrarily strong external fields 335

0.16 I I I I I I I I

0.14 - I Ii3.0 -8000.12 - 2.50.10

0.08

0.06 ~ I I E 2.0 - Z 1600~. 1.5 -

0.04 -

0.02 -

C0 3 6 9 12 15 ~ I I

3 6 9 12 15

r [md] r [md]

Fig. 5.11. Solution g, of eq. (5.117), the Klein-Gordonequation Fig. 5.12. The chargedensity associatedwith the solutionsshownwith self-consistentpotential for various values of Z, thenucleon in fig. 5.11 normalizedto thevalue I pd

3x= q.charge,when the chemicalpotential is chosenas the negativeoftheelectron’s rest energy.

55 —0.5 I I I I I I

~Zi8

_50 -0.6 -454035 -0.7 - Z~l600 Z 1230

~ 25 2

I ~Z~l200: ~I Ii ~ -0.8 -Z 600’ ~ -0.9 - ,~-Zel80020 15 ~i~i /5.

0 3 6 9 12 IS ~2o 3 6 9 12 15

r [m’] r [m1]Fig. 5.13. The chargedensitytimes r2 for thesolutionsof fig. 5.11 Fig. 5.14. The total potential, Votai, for thethreecasesconsiderednormalizedasin fig. 5.12. in the previousthreefigures.

0.50 I I I I I __________________0.6 I I I I

0.450.4

0.40

Z 1800 ________________0.35

-0.2 - 7Vcofld0.30 ~.~ 0.25 -0.4 ~Z~I80O ~totaI

-0.6>~ ~ I ______________> -0.8-

0.10 -

0.05 —i~~I 200 - - I .4I I

3 6 9 2 15 1•6o 10 20

r [m~] r [rn’]

Fig. 5.15. The condensatepotential, V.O~d,for the three cases Fig. 5.16. Comparisonof ~ V.Ofld, and the Coulomb potentialunderconsiderationin thetext. V~

0~10,,,5for Z= l800.

336 J. Rafelski et a!., Ferm ions and bosons interacting with arbitrarily strong external fields

35011! 1111111 I III

~°~IIIIIIlI1200 300 400 I 500 1600 1700 1800

zFig. 5.17. Condensatecharge,q, asafunction of the nuclearcharge,Z.

~ through the electromagneticinteraction, there is, in principle resonantscatteringof positrons

through the weak interactions.(We shall not develop this idea in detail.) Finally subsection5.11concerningsmall oscillationsaboutequilibrium shouldessentiallyapply without change.

6. Applicationsandextensionsof the theory

We haveseenthat in the vicinity of a sufficiently largeassemblyof chargedparticles (held togetherin one place by e.g. the strong interaction)first the electron-positronfield and then the mesonfieldbecomesovercritical. In the first part of this section,we describethe responseof the overcriticalelectronfield to an everincreasingstrengthof the externalpotential[102].In particular,we focusontwo gedankenexperiments.In the first one, we considera spherically symmetricexternal potential,while in the secondwe return to the discussionof Klein’s paradox.Then the results obtainedforovercritical electron,muonandpionfields arecombinedto studythe neutralizationof chargednuclearmatter.

Subsequently,we turn our attention to questionsof current interest that are related to thegravitationalinteraction.The singular gravitationalpotential of a black hole will give rise to someeffectssimilar to thosediscussedin the previoussubsections.An importantdifferenceis that in ourpre-viousdiscussion,we were able to abandonthe overcritical singular 1/ r potential in favor of a physi-cally smoothedpotentialat the origin, but suchan approachis not possiblefor the gravitationalfield.We haveto go to the limit of the singularblackholepotential.The existenceof a true eventhorizonisonly consistentwith singularfield solutions,which in turn makethe interpretationof the behaviorofparticlescloseto the singularity difficult. In this section,we indicatethe possibilityof a new approachin which the elementaryfields gravitate,and,at the sametime,experiencethe gravitationalforce. WediscussBosecondensationin thegravitationalfield. The original investigation[112]was undertakentopursue the possibility that a flow of energy from the gravitational field to the scalar field wouldweaken the singular structure of the metric. However, it has been shown from rather generalargumentsthat the formationof aneventhorizoncannotbe avoidedwhensufficientexternalmatterispresentalthoughsignificantmodificationsof the metric mayoccur.

In the final applicationof the methodsdiscussedin this paper [157], a model field theory isdiscussedin which the external field is replacedby a self-consistentpotential. It is shownthat the

I. Rate/ski et a!.. Fermions and bosons interacting with arbitrarily strong external fields 337

neutralvacuumremainsstableas the lowestboundstatejoins the lower continuum.This stability is aconsequenceof the fact thatno externalenergysourceis present.

6.1. The chargedvacuum in the ThomasFermi approximation

The point at which the Is wave function joins the continuumsolutionsof negativefrequencyhasbeendetermined(section2) to be aboutZ = 172, under“realistic” assumptionsandextrapolationsofthe knownpropertiesof nuclearandelectromagneticinteractions.Similarly, the nextcritical point, forwhichthe 2P112 stateis expectedto join the continuumis aboutZ = 183.At this point the chargeof thevacuumbecomes4. Soon,as we increasethe nuclearcharge,higher angularmomentumstateswillalso join the lower continuum,and the chargeof the vacuumwill rise even faster.Thereafter,theself-interactionof the vacuumchargewill becomean importantaspect.In particular,we canalreadyforeseethat the..vacuumchargemay screena substantialpart of the nuclearchargeandpreventmorestatesfrom joining the lower continuum,or, from anotherviewpoint, the repulsiveinteractionof anelectronwith the surroundingvacuumchargemaybecomecomparablewith the attractiveforce of thenuclearcharge.

It hasbeenproposed[102]to treatthat complexsituationby useof the relativistic ThomasFermimethod(RTF). The densityof electronsis as usualrelatedto the Fermi momentumkF by

e ~ (6.1)

The effect of the spin degeneracyis included.The relativistic relation betweenthe Fermi energyEFand Fermimomentumis

k~’[(E~—V)2—m2]0(EF—V—rn), (6.2)

where V is the electrostaticpotential.The stepfunctionensuresthat k~is a positive quantity.

Fromeqs.(6.1) and (6.2) we now obtainfor the chargedensityof the groundstate(OIp(x)I0)= (eI31T2)[(EF— V)2—rn2]3120(E~—V—rn). (6.3)

At this point, we makecontactwith the formalism of section3. The total chargeof the vacuumisequalto the chargecarriedby all of the statesthat have joined the lower continuum,as indicatedschematicallyin fig. 6.1. In the Thomas—Fermimodel,this sum over all of thesestatesis representedby an integralover all the momentumstateswithin the Fermi sphereof radiuskF. IntroducingthetotalchargedensityPT, which is composedof the external“nuclear” part pN andthe electronicpart,

PTPN+P, (6.4)

andusingCoulomb’slaw,

i~V(r) = —epT(r), (6.5)

we find

A V(r) = —epN(r)— (e2/3ir2)[(E~— V)2 — rn2]3120(EF— V— m). (6.6)

For neutral atomic systemsone must take EF = m which gives (for —2mV~>I V2J) the usualThomas—Fermimodel. Equation(6.6) is very useful andmay beapplied to describemanyinterestingproblemsin atomic physics [158]. In particular,when supplementedwith the relativistic exchange

338 . J. Rafelski et a!., Fermions and bosons interacting with arbitrarily strong external fields

\-~.;-- -~-,i_____ I-ET1 w340 524 m346 524 lSBT\ V-rn

Fig. 6.1. Energyand momentumof thesupercriticalstatesin a potential V on whichtheThomasFermiapproximationis based.

(Dirac) term, it maybe usedto describethe electronicdensityin heavy-ioncollisions,as discussedinsection4.

The essentialstep forward in ref. [102] was to take E~= —m. This meansthat only the statesaccessibleto spontaneousdecayarefilled. We will showbelowthat the Hamiltonianassociatedwitheq. (6.6) showsa minimumas a functionof EF. InsertingEF = —rn into eq. (6.6) yields

AV(r) = —epN(r)— (e2/3ir2)(2mV+ V2)3120(—V — 2m). (6.6’)

Equation (6.6’) may be derivedfrom a Hamiltonian and thus viewed as determiningthe chargedensitywhich minimizesthe total energy,that is,

oHIap=0. (6.7)

This Hamiltonianis

H[p] = 2J d3xJ ~ 0(k~— k2)[(k2 + m2)~2+ m] + i-_f d3xJd3y ~ + p(x)][PNCY) + i’(.v)]

(6.8)

which maybe consideredthe basisof the discussion.The first term of eq. (6.8) is associatedwith theenergyrequired to make electron—positronpairs; the positron escapesto infinity with zero kineticenergy.The lastterm describesthe Coulombenergyof the nuclearandelectronicchargedistributions.It is interestingto note that the choiceEF = —m minimizesH whenconsideredas a functionof EF,that is,

oH 2 i ~= ~ j d xkF(x)(EF— V(x))(EF+ m). (6.9)

(/L~F ~1TJ j

This expressionvanishesif and only if EF= —m. Taking anotherderivative with respect to EFestablishesthat thisis a minimum.The groundstateis only stableas long as the nuclearsystemPN isisolatedfrom externalsourcesof electrons.If the latter condition is relaxedand an inexhaustiblesupplyof electronsis available,we mustaccountonly for the kinetic energyof theseelectronsandmodify the term involving m in eqs.(6.8) and(6.9) to read—m insteadof +m. ThenEF = m is the onlystablesolution,correspondingto neutralatoms.

I. Ratelski et al., Fermions and bosons interacting with arbitrarily strong external fields 339

An observantreaderwill havenoticed that the term involving +m in eq. (6.8) correspondsto theLagrangianmultiplier, which shouldread—EF thatis associatedwith chargeconservation.We haveset itequalto +m sincethis is the energynecessaryto createa free positron.Thereforeeq. (6.8) properlyexpressesthe physicalsideconditionsthat for kF—~O.the energyrequiredto createa particlepair is 2mand not ni — EF.

The nuclearchargedistribution on which the numericalresults presentedbelow are basedis theFermi distributionof eq. (2.37), normalizedto give the nuclearchargeZ. The radius R is determinedby the numberof nucleons,that is,

R = r0A~3, (6.10)

andr0 is relatedto the nucleardensityby

0 1/3

r0= I.2fm(pN/pN) . (6.11)Herep~is the densityof normalnuclei (0.17 nucleons/fm

3).The surfacethicknesst of eq. (2.37) hasbeenchosenas before to be 2.5fm. The resultspresentedbelowdependonly slightly on the form ofthe chargedistribution,provided that the r.m.s. radiusis chosenthe same.The systemis specifiedcompletelywhenthe ratio of protonto nucleonnumbers

f=Z/A (6.12)

is given.We now proceedto discussthe solutionof eq. (6.6’). Sincethe chargedensityof the vacuummust

be confined to the vicinity of the externalcharge,we requirea solutionsuch that

V(r) —~ —ya/r, (6.13)

(a is the fine structureconstant).For everychoiceof Z, y is determinedby the boundaryconditiononthe electrostaticpotentialat the origin

aV/or Ir=O = 0. (6.14)

Equations(6.6’) and(6.14) arethereforeeigenvalueequationsfor y, the unscreenedpart of the nuclearcharge,andZ — ‘It gives the chargeof the vacuum.

Integrationof eq. (6.6) is straightforward.MUller and Rafelski [102] have assumedf = ~ (equalnumberof protonsandneutrons)andnormalnucleardensity(PN = p~).Their resultsfor y areplottedin fig. 6.2. Fromthe figure, one can seethat y increasesmonotonicallywith Z andthat y/Z decreasesas Z increases.In fact, MUller and Rafelskishowedthat

y/Z-~0 as Z—*cx. (6.15)

The single-particleresultsare denotedby crossesin the figure and agreereasonablywell with anextrapolationof the Thomas—Fermiresultsinto the realm of small valuesof Z — y.

In accord with our expectations,the chargegeneratedby successivelevels joining the lowercontinuumis sufficientto screenmostof the barenuclearchargeas it increaseswithout bound.This isthe significanceof the limit y/Z—*0 as Z—*co. In fig. 6.3 we considerthe approachto infinite nuclearmatter. As shownthere,both the vacuumchargedensity and the potentialapproacha limit as Zincreasesfrom l0~to l0~.The radial chargedensity,calculatedfrom the right handsideof eq. (6.6), isshownin fig. 6.4. The resultsare scaledwith y so thateachcurve is normalizedto unity. We seethat

340 J. Rate/ski et a!., Fermions and bosons interacting with arbitrarily strong external fields

I I I0

////

///7

I ~

///

UJIO2. /

Ca /~ ./I z-r0

10

I I I02 i03 l0~ I0~

zFig. 6.2. The unscreenedchargey andthetotal chargeof thevacuum(Z — ~)asa function of Z.The crossesdenotepoints from single particlecalculations.The dashedline denotesthe nuclearchargeZ. From Muller andRafelski[102].

~IOO~ ~ _250 (a) -0.8 - 7 (b)

-300———- I L -IC I II 0 100 000 0 00 1000

fm fm

Fig. 6.3. The solutionsof therelativisticThomas—Fermieq. (6.6) for selectedvaluesof thenuclearchargeasafunction of r. Curve I, Z = 600:curve2. 1000; curve 3, 2000; curve 4, 5000; curve 5, 10 000; curve 6, 10~curve 7, 10”. (a) The self-consistentpotential;(b) thecorrespondingchargedistributionof the vacuum.From Muller and Rafelski[102].

5.0 I I I

4.0- 7

r 3.0

1~ ~I lO 100 I000

fmFig. 6.4. The total chargedensities,scaledwith y. Sameselectedvaluesof nuclearchargeas in fig. 6.2. From Muller andRafelski[102].

J. Rafelski et a!., Fermions and bosons interacting with arbitrarily strong external fields 341

the chargedensitiesbecomemore and more nearly thoseof a surfacedipole with clearly definedregionsof positiveandnegativecharge.

Before closing this subsection,we note that manyof the resultsobtainedhere could be obtainedmoreformally with the Hartree—Fockapproachof subsection3.7.

6.2. Resolutionof Klein‘s paradox

As we havejust seen,a large bare nucleuswill spontaneouslyproducepositronsuntil all of thenegativeenergystatesbelow � = —rn arefilled. This stablesituationis illustrated schematicallyin fig.6.5, wherea profile of the nuclearpotentialand the chargedvacuumin its vicinity is shown.As weemphasizedbefore,the potential includes, in a self-consistentmanner,the effects of the chargedvacuumas well as the nuclearcharge.At the nuclearsurfacethe overcriticalpotential is reminiscentof the potentialstepusedto illustrateKlein’s paradoxin subsection2.1. We believethat this similarityis the key to a refined treatmentof Klein’s paradox.

Before proceedingwith the discussionof Klein’s gedankenexperiment,we notethat in the presentcasethe sign of the potentialwell is oppositethat of our discussionin subsection2.1 (fig. 2.2). Apictorial representationof the situationis shown in fig. 6.6, whereall of the statesbelow � = —m areoccupied,in contrastwith fig. 2.4.Thecalculationof thereflectionandtransmissioncoefficientsof awaveincidenton the potentialwell shouldbe basedon thegroundstateof fig. 6.6. Thenthe resultswill differmarkedlyfrom eq. (2.13). In particularwedo not expecttheabsolutevalueof thetransmissioncoefficientto be anomalouslylarge for deeppotentialwells. Thus the behaviorof eq. (2.13) for large V0 can beunderstoodas aconsequenceof an incorrectchoiceof the stateof reference.

If our conjectureas to the correctchoiceof the stateof referenceis valid, thenit is important toexamine the influence of theseoccupiedstateson other states.Since the vacuumstatemust notchangewith time, the appropriatesolutionsfor x<0 are standingwaves,in contrastto the planewavesof subsection2.1. Anotherwayof recognizingthe standingwavesas the propersolutionsinsidethe potential well involves the limit of a large nucleus:The boundstatesthat joined the negativefrequencycontinuumbecomestandingwavesas ‘one side’ of the nucleusis takento infinity.

We now show that standingwave solutionsinside the well imply standingwavesoutsidethe well.

V

_________________________ ______________________ E

m

____________________0 ‘I-~ ,— z _______________________

/ -m- rn / Er .~A/VVW

_______ ~V+mV1. V

Fig. 6.5. Schematicpicture of a cut along z-axis of a self-con- Fig. 6.6. Semi-infinite potential step with supercritical states in-sistentpotential well with many supercriticalstatesindicated. dicatedto the left.

342 J. Rate!ski et al., Fermions and bosons interacting with arbitrarily strong external fields

Inside the well we have

cos(p’x + q’)

~<~= N{~ii~_V+m)}sin(p’x+~’ (6.16)

whereas before,p’ = V’(� — V)2— in2. Outsidethe well the solution is an in- or out-goingplanewave,

~>0_ Aetox(0 m))+B e~x( /(+ rn)} (6.17)

Fromcontinuity of the wavefunctionwe find that

NcosQ=A+B, (6.18)

{ip’/(e — V+ m)}N sin~ = (A — B)p/(�+ m). (6.19)

We maychooseN to bereal. Thenwe have

ImA = —1mB, (6.20a)

ReA = ReB. (6.20b)

A consequenceof eqs. (6.20) is the vanishingof the current for x > 0. A further implication of eqs.(6.20) is that we maywrite

2A = Nij eia (6.21a)

2B= N’q e’~. (6.21b)

Substitutinginto eqs.(6.18) and(6.19) wefind

p’/(E — V+ m)tan~ = p/(~+ rn)tan0. (6.22a)

cos~ = i~cos0. (6.22b)

The wavefunction outsidecan now be written,

cos(px+ 0)

— N~ip/(�+ rn)} sin(px+ 0)). (6.23)

Thus we find that both the outsideandinside solutionsarestandingwaves.In our presentdiscussion Klein’s gedankenexperimentconsists in scatteringof an (incoming)

positron of momentump = — in2 with —m > �> V + rn. The energyof the positron is E~=

We havejust shown abovethat all of the incomingcurrent is reflected,leading in consequenceto astandingwave outside of the potential well. Thus no matter is transmittedinto the region of thepotential.

We completeourexerciseby consideringbriefly the caseof an arbitrarily orientedwave.If p is not

I. Ratelski et a!., Fermions and bosons interacting with arbitrarily strong externalfields 343

orthogonalto the surfaceof the potentialwell, we obtain an eigenfunctionin a similar way. Writing

p = p~+ p,~, (6.24)

we find the standingwave,in the directionnormal to the surfaceof the potential,to be

t/’(p, s) = N exp(ip±. x1)X

(6.25)

Jf cos(p~x+ O)x~ O( ) + cos(p~x+ ‘p)x~ O(—x)UJu p/(�+ m) sin(p~x+ 0)x~J X ~io~p’/(�— V+ m) sin(p~x+ ~)x~

Since the lowest componentof the spinor of eq. (6.25) doesnot vanish, the vectorp’ is determinedcompletelyto be

p’p/pIV(�— V)2—rn2. (6.26)

The relations amongthe phases~ and 3 are the same as those of eq. (6.22), if I~Iand p’I aresubstitutedfor p andp’ there.

As a final step leading to the descriptionof the theory in the presenceof an arbitrarily strongpotential step,we nowgeneralizethe RTF model of theprevioussubsectionto includethesemi-infinitecase.The situation that we deal with is shownin fig. 6.6. A potential step showntherewill be aconsequenceof a constantexternalchargedensityPN. Our aim is to describethe systemwith a specialemphasison thoseaspectsconnectedwith a selfconsistenttreatmentof the vacuumcharge.When theexternalchargeis localizedto the left of fig. 6.6,

Pw(X) = O(—x)p~, (6.27)

thenit is apparentthata solution which minimizesthe Hamiltonian of eq. (6.8) musthavea vacuumchargeconfinedalsoto the left. Again the requisitevalueof the Fermi energyis

= —m. (6.28)

Thereforeall stateswith

V+rn<�<—rn, (6.29)

within the rangeof the potentialwell are occupiedwith electrons.The associatedelectronchargedensity,(eq.(6.16)), becomesin this case:

ep = —O(—x)~-~(2mV+V2)3120(—V—2rn). (6.30)

Neglectingatfirst the inhomogeneityof the solution,we find that V is determinedfrom thecondition,

P~PN°, (6.31)

andis givenby

V0 = {rn — [rn

2+ (3ir2ep~/e2)213]”2}O(—x). (6.32)

Thus wehaveshownthat thepotentialstepresultsfromthecombinationof theeffectsof thebackgroundchargeand the vacuumcharge.

We now return briefly to discussthe inhomogeneityof the solutionsof semi-infiniteTF equation:

—A V(x)= epN+ ep(V(x)). (6.33)

344 J. Rafelski et al., Ferm ions and bosons interacting with arbitrarily strong external fields

The boundaryconditionsfor the solutionsare V = 0, x>0 and that the solution (6.32) applies forx —~ —~.The latterconditionandCoulombforce requiresthe solutionto oscillatearoundthevalue V0,as illustratedin fig. 6.7. Fromthe formerrequirementweobtainthe approximatesolution:

V(X)~rn_~~~X2O(_X), VJ42m. (6.34)

In conclusionof this paragraphwe wish to remind the readeragainthat the abruptpotentialchangeofKlein’s paradox is creatednot only by an external charge,but also by the vacuumcharge. Wepostponethe treatmentof the Klein’s paradoxto a future work andproceedwith the discussionof thephysical implications of the neutralizationprocessesin the next subsectionusing the formalism ofsection3.

Before doing so we makea commentconcerningthe validity of our assertionsfor macroscopicchargedistributions. In the microscopicchargedistributions, like those associatedwith nuclei, anovercritical value of the total charge must be first exceededin order for spontaneouspositronproductionto begin. In infinite chargedistributionsanyarbitrarily weakPN ~Spermissible,sinceoverinfinite volume sufficient strengthfor overcriticality will be attained.It is thereforenecessarytoconsidermore carefully the effects. The barrier separatingthe particle and antiparticle statesmaybecomemuch too wide to allow the spontaneousdecayof the neutralvacuum. The most commonsituationof this natureoccurswith the Van de Graaff generator.Therethe potentialchangeof severalMeV is distributedover macroscopicdistances.In consequence,the decaytime of the neutralvacuumis so largethat ‘sparking’ of the vacuumshould neverbeobservedin the vicinity of suchgenerators.

6.3. Neutralizationof nuclearmatter

In the two precedingsubsectionswe havediscussedthe neutralizationof the background(nucleon)chargeby the vacuum chargedensity associatedwith the fermion field. We wish to combinethoseresultsnow with thoseobtainedin section5 concerningthe neutralizationthrough a pion conden-sation.

As illustratedin fig. 6.3,a constantlimiting value Vjjm of the potentialV is reachedas the chargeofthe nucleusincreases,providedthat the nuclearchargedensityis keptata constantvalue.Thatmeansthat asidefrom surfaceeffects, exactneutralizationof the chargesmustoccur in finite, large chargedistributions,c.f. eq. (6.31).This leadsto the valueof Vijm identicalto the right handside of eq. (6.32),whenthe pion condensationeffects areomitted.

Sincem 4 I V11I forsupercriticalnuclearmatter,wehaveincludingtheeffectof theelectronfield only,c.f. eq. (6.32):

V~m= —(31r2ePN1e2)”3. (6.35)

With

PN = Ze/(~irR3), (6.36)

~ (PN

-‘~~\J0 X

Fig. 6.7. Semi-infinite backgroundand inducedelectronvacuumcharges.Qualitativepicture.

J. Ratelski et a!., Fermions and bosons interacting with arbitrarily strong external fields 345

and R as given by eq. (6.10) we find, (heref = Z/A),

V~m= — 315[MeV]. (f~p~/p~)W. (6.37)

A remarkablecoincidenceis that

3l5[MeV] = 2.25m,~> 2m,~. (6.38)

For f = ~ andPN = p~we find V~m= —250MeV from eq. (6.37) in agreementwith resultspresentedinfig. 6.3. In view of this valueof Viim we mustconsiderthe pion condensationeffect in more detail, inorder to obtainthe properphysicaldescriptionof the neutralizationprocess.

We notethat the channelfor the muonic vacuumchargeis alsoopen.We briefly give the essentialdetails in order to have a complete description.The chargedensity associatedwith the muonicspontaneousdecayis:

ep~= — [(E~ — V)2 — rn~j312. (6.39a)

An importantdifference [159] from the electronvacuumchargeconsistsin the properchoiceof E~.Inaddition to the spontaneousdecayof the neutralvacuum,the muonic stateswith — me> E~> — rn,~

arefed by the weak decayfrom the electronchannelas shownin fig. 6.8. Thus we find:

Er”’ = E = — me. (6.39b)

We note in passingthat sinceall fermionsare ‘connected’in sucha way by weakdecays,all Fermisurfaceslie at Ef= — me. Thus in principle,we have

0 = epN + ~ [(— rn~— Vi~m)2— rn ~]3/29(_ Vi~m— rn, — me) (6.40)

as the neutralizationconditionfor V~m,includingthe effectsof the Fermionfields only. We haveused:

(i) — f — (negativelychargedfermions) (6 41)

sg — 1. + (positively charged fermions).

Nuut~:

Fig. 6.8. Supercriticalelectronandmuon statesin apotentialwell. The weakdecayof supercriticalelectronsguaranteesthat theFermi surfaceisthesamefor both electronsandmuons.

346 1. Ratelski et a!., Ferm ions and bosons interacting with arbitrarily strong external fields

Sincewe anticipateI V~,I< m~,we needonly to considerthe muonic channelin additionto the lightestfermion, the electron.

As long as V~mI< m,~,a solutionof eq. (6.40) is the actuallimit on V that maynot be exceeded.Letus now discuss the case that I V~mI>m,~.The behavior of the numerical condensatesolutionspresentedin section5 suggestanotherphysicallimit to the electrostaticpotential.The strengthof thepotentialseenby the pionscannotgreatly exceedthe value —m~in view of the subsequentformationof the neutralizing chargedcondensate.If the nuclear charge is increased‘suddenly’ beyond thecritical limit, then V reachesthe limit —2m,~.Subsequentslowerweakdecayof thevacuumallowsthelower limit, —m,~to the electrostaticpotential,as suggestedby the resultsof subsection5.14.We wishto note thatthe inclusion of the nuclearinteractionmayalter the thresholdm,~.We will proceedwithour discussionneglectingthis effect at normalnucleardensities.Thus the considerationof leptonandpion fields simultaneouslyleadto the more restrictivelimit for the electrostaticpotential:

I ViimI = min(m,~,I V~m~). (6.42)

This limit implies that the nuclearchargeis completelyneutralized,with somebalancesurfacechargey=Z—(qe+q~+q,~)suchthat:

V11m —ay/R, (6.43)

where R is the radiusof the dipole surfacechargeand coincideswith the nuclearradius. Since asZ—~oa,R -~~ but Vijm remainsconstant,we expectthat for largeZ the surfacechargey—s’Z”

3from eq. (6.43).

Another importantconsequenceis the cancellationof the Z2 dependentCoulombenergyterm fromthe total energy.For the Coulombenergyof the remainingsurfacechargewe find

= = ~IV1~~~yZ”

3. (6.44)

The Coulombenergymusthoweverbe supplementedwith the term associatedwith the kinetic energyof the vacuum,(i.e. of fermionsandpionsthatneutralizethe nuclearmatter)which we now consider.To wit, we use the Hamiltonian, eq. (6.8) with PN replaced by PN + ~ and the kinetic termsupplementedto include muonsandpionsas well:

H=Jd3x{2 J ~~4[(k2+m~)h1’2+me]+2 J 4[(k2+rn~)h/2+rne]}

IkI<k~ IkI<kf

+Jd3x(IiTI2+IVcI2+mI2)+~Jd3xd3y[pN+p,r+pe+p,~](x)4JlI[pN+p~+pe+p,J(y),

(6.45)

where

k~e.wt=(me V)2—m~,,~.). (6.46)

The first term in eq. (6.45), the contributionof the (highly relativistic) electronsis:

~ = Jd3x 2J ~ IkI = J~kp~d3x = ~ V11~~qe, (6.47)

1k 1<k f~

J. Ratelski et al., Ferm ions and bosons interacting with arbitrarily strong external fields 347

wherewe haveusedthe relativistic limit of eq. (6.2):

k = I V11mI (6.48a)

and

q~= f d3xpe. (6.48b)

A similar result is obtainedfor the muonic kinetic energy,thoughherewe mustbe more carefulnowaboutthe relativistic effects, since I V

11,,,~ m,~.We findL’w _~ii- ru 1

-‘-‘kin — 4 ~‘ urn cL ~ urn

1 . q,~

wherethe coefficient c is given by the expression

c = (i +!.~~)_rn~~ln [+ (P~+1)1/2] (6.50)

Pf.w I VIIm I~~ m~, m,.

and

p~ (V~mm~j~2. (6.51)

We find, inserting the value Vjjm = —m,~above,that c = 1.21, while for Vtjm = —2m,~we find c = 1.06(and the relativistic limit appliesagain). Finally, the pion kinetic energyin eq. (6.45) is castinto theform:

d3x(e,~— Viim)p,~ ~ (6.52)

Thus the energyper nucleon residing in the chargedfields, exceptfor the binding energyof thenucleons,is:

EtA = m,~q,,/A + ~IVuimI(cq,JA+ q~/A). (6.53a)

Neglectingthe surfacecharge(aswe haveneglectedthe surfaceenergy)we maywrite,

q,~= Z— q~.— q~, (6.53b)

and we find,

E/A = m~f+~(~V11m m~)+~”(~c.V1jm~mr). (6.54)

The quantitiesqe/A,q,JA arewell definedin the limit of theinfinite nuclearmatterwhich we now may

take. With constantchargedensitieswe find usingeqs.(6.10), (6.11)qe/A ~ =~—~.~!!211 (e.~)(0.85)~, (6.55a)

qJA = ~— [( Vu~mIm~)— (m,~/rn~)J312(p~IpN)(0.85)3. (6.55b)

We notethat for V11,,. = —m,~,the numericalvaluefollowing from eq. (6.55a)is q~/A 0.087. An even

smallernumberfollows for the muonic charge.Thus approximately80% of the neutralizingchargeisprovided by the pion condensateat normalnucleardensities.Thereforewe mayin first order neglect

348 J. Rate/ski et a!., Fermions and bosons interacting with arbitrarily strong external fields

leptonic termsin eq. (6.54) to obtainthe energythat replacesthe Coulombenergyin the neutralizednuclearmatter(1 =

EtA (m,,/2) = 70 MeV. (6.54’)

More precisefiguresobtainedfrom eq. (6.54) aregiven in table 6.1 below.The consequencesof the considerationspresentedaboveare:

(1) Thelimit to the electrostaticpotentialdoesnot exceedm,,.. Dependingon thedensityof theexternalchargedistribution— whichis forourpurposesindependentof electromagneticinteractions— only afinitenumber of speciesof chargedelementaryparticlescan createa chargedvacuumor form a Bosecondensate.(2) Due to the essentiallycompleteneutralizationof the chargednuclearmatter,the Coulombenergyproportionalto Z2IA”3 becomes,at most, kinetic energyof the neutralizingfields proportionalto Z.

This latter fact hassomeimportanttheoreticalconsequencesfor theoriesinvolving strongly boundnuclear matter. The Coulomb energywas usually thought to provide the stabilizing influence inspeculationsinvolving strongly bound,collapsednuclearmatter.The detailsof suchtheories,requiredisospin zero (f = ~) matter. Then the positive definite, conventionalA513 Coulomb energy wouldoutgrow any large binding energyfor sufficiently large A.

Becauseof the compensationof the repulsiveelectric forces inside the collapsednucleusby theBose condensateand the polarizationof the vacuum,the Coulombinteractionwill not, most likely,stopthe growth of the abnormalnuclearstate. In other words,if an abnormalnucleusis createdin ahot environment(like a star), it will continueto absorbfastneutronsandprotonsin the absenceof therepulsiveCoulombenergywhich will be neutralizedby spontaneouspositronemission.The existence

Table 6.1Total energy per nucleon. E/A, of chargedneutralizednuclearmatter, excluding nuclearbinding energy.f is the fractionof protons inthe matter, PN/P~ is the nucleardensity inunits of theequilibrium value, Vijm is the con-

stant electromagnetic potential maintainedwithin the nuclearmatterdistribution.

Viim PNIP~ f E/A[MeV)

1 0.5 66—m,, 4 0.5 69

8 0.5 69

1 0.5 164—2m,, 4 0.5 93

8 0.5 81

—m,, 1 0.1 II

—2rn,, t 0.1 lOS

—m,, 8 0.1 13.5

—2m,, 8 0.1 26

J. Ratelski et a!., Fermions and bosons interacting with arbitrarily strong external fields 349

of our solar system,now for several billion years,is thereforea strongargumentagainstextrapola-tions concerningabnormal(collapsed)nuclei with the binding energyper nucleuslarger than theaverageenergythat remainsin the systemafter neutralization.The binding of collapsedmatter muststay below the limit setby eq. (6.55), table6.1, otherwisethe universemaycollapse.

In the Lee andWick [160,161] theory of abnormalstrongly boundnuclearstatesthereis a significantgain in the energy(ca. 150—300MeV) whenevera baryon is absorbed.Sincethe limit setby eq. (6.55)hasbeenexceeded,the Lee—Wick nuclearmatteris unstableconsideringall knowninteractions.

Finally let us restateagainour result in a morecolloquial language:Whenthe potential I VI exceedsthe value m,,, the vacuum breaksdown and is stabilized at I VI = m,,. The energyof the Bosecondensateand of the electronsand muon that fill all of the phasespaceavailableis then the onlysignificant, positivedefinite, componentin the energyandleadsto the limit setby eq. (6.55) (all theseremarksare valid for systemslargerthan the characteristiclengthsof the particlesinvolved).

6.4. Gravitational field, gravitational collapse

Work in the areato be treatedin this subsectionhas beenstimulatedto a large extentby somequalitatively importanteffects describedrecently by Hawking [105—108].He hasshown that a starofmassM collapsingto form a blackhole emitsparticleswith an energyspectrumcharacterizedby atemperature

T (hc3/8ITGM)= (0.6)107°K(M0/M), (6.56)

associatedwith the thermalradiationlaws of bosonsor fermions.This radiationcontinuesthrough thecollapseandis capableof carryingawayan infinite amountof energyif the reactionon the collapsingstar (reducedmass) is further neglected.The approachof Hawking in which radiationtunnels outthroughthe eventhorizon,is similar to our considerationsin previoussections.

The work of Hawking has raised a numberof questionstreatedin subsequentpapers[109,110],which have discussedspontaneousand inducedparticleproduction in the vicinity of black holes.There is still a difficulty in understandingthe relation of this work on quantizationof fields in thevacuum Schwarzschildand Kerr [Ill] solutions and the original work of Hawking on particleproductionin gravitationalcollapse.Hawking predictsthat ablackholeformedby collapsecontinuesto emit particleslong after the actual collapseprocess.Thereseemto be two possibleinterpretationsof this result.One is that this emissionis not a directconsequenceof thecollapseandshouldshowupin a correct quantumfield theory in the Schwarzschildspacetime.Such theory has not yet beenformulatedsuccessfully.The other is that the collapseis essentialfor the productionof the particles;the emissionappearsto an observerat infinity to continuefor a long time only becauseit takesthislong for the particlesformedin the collapseto reachhim. At the presentit is an openquestionas towhich interpretationis correct. It doesseemto be possibleto devisequantizationswhich do yieldparticle productionin the vacuumSchwarzschildgeometry.As will be discussedfurtherbelow, whatremainsto be seenis whethera compelling physicalcriterion can be found in favor of a particularformulationof the theory.

The treatmentof the gravitationalfield as an externalfield, the only problemwe shall consider,isvery much like that of any other external field. Any time-independent,spherically-symmetricmass

350 J. Rate/ski et al., Fermions and bosons interacting with arbitrarily strong external fields

distributionhasa metric tensorg,~,,of the particularform [162]

(ec _eA

—r2 (6.57)_r2sin2OJ

wherethe two functionse” ande5 are determinedfrom solutionsof Einstein’sequationsof motion.Given g,,~(to be understoodas the externalgravitationalfield), we can find the coupling of the

gravitationalfield to any otherfield, for both bosonsand fermions by writing the Lagrangianin ageneralcovariantform. The simplestcaseis that of the zerospin Bosefield. With the invariantaction

S= ~Jd4xV—g(—d~g~D”~+ ~2~2) (6.58)

whereg is the determinentof g~,,,we obtainexplicitly for eigenstatesof good angularmomentum

Q(r, t) = ~,(r, t)Yim(1~), (6.59)

the action (with I = 0 for consistencywith the ansatz(6.57)),

S= ~Jdt drr2 exp(‘~~)[—e’~2 + e~(~’)2+ ~2tp2] (6.60)

andthe associatedKlein—Gordonequation:

~ = w2G1~, (6.61)

with eigenvaluew2, where

G1 = exp~ (6.62a)

G2 = exp~ (6.62b)

We note also at this point that for the particular caseof a Schwarzschildsolution in empty spaceii = —A, i.e.,

G2=1, (6.63a)

and

e” = I — 21c1/r= G1. (6.63b)

Here J~= (KM/C2) where K is the gravitationalconstant(K = 6.67 x l0~dyne-cm2/g2)and M the

gravitatingmass.Inclusionof a vectorpotentialin eq. (6.61)proceedsthroughsubstitutionw —~w — V.Equation(6.61) is the Klein—Gordon equationin interaction with the externalgravitational field.

Thereare no discretesolutionsof this equationin the Schwarzschildmetric. It is shownin ref. [112]that the externalfield is of supercriticalstrength.This leadsto a singularbehaviordifferent from anyencounteredin section5, — for the particularcaseof gravitationalinteraction— all boundstatesmeetatw = 0 as the functions G

1, G2, approachthe Schwarzschildlimit (which the Schwarzschildvacuumblack-holesolutionwill be calledhenceforth),in a suitableparametrization.The parametrizationthat

I. Rafe/ski et a!., Fermions and bosons interacting with arbitrarily strong external fields 35 1

has been employed can be associatedwith the metric of an incompressiblefluid model for theconstantmassdensitystar, suitably augmentedto avoid infinite pressure.

The spectrumof eq. (6.61) is shown in fig. 6.9 as a function of the massradiusR (which hasbeenchosenunrealistically small only for numericalconvenience,M = lip,, p, is the mesonmass).Wenotice the collapseof the discretesolutionsas the Schwarzschildmetric solution is approached.Thisbehaviorseemsto demanda formulationsimilar to that given in section5 — a mesoncondensationisexpectedwhich will gravitate and could conceivably cancel a part of the external gravitationalforce— reminiscent of the behavior of the electromagneticinteraction given before. However, nostabilizing mechanismhasbeenfound yet, a point that is understoodby noting that all gravitationalinteractionsare attractive,unlike the casewith the electromagneticforce. This meansthat a Bosecondensatestrengthensthegravitationalforceratherthancancellingit. Thisqualitativeargumentmaybeaugmentedeasily with more detail: The massof thegravitatingobject, found to be

M = 4~fr2 drp0+ r

2 dr[w2 e~~2+ e~’2+ p,2~2], (6.64)

(whereR is the radiusof the massdensityPo) increasesas the Bose condensateis formed below theSchwarzschildlimit. Since(e”, e~”)may be negativedefinite within the blackhole,a decreaseof Mmaybe involved if an eventhorizonis formed.However,existenceof an eventhorizonis necessaryfor this to happen.The only possibleeffect of the Bosecondensatemaybe a softeningof the originalsingularity of the metric at the Schwarzschildradius 2M. These argumentsare basedon a time-independentapproach.Introduction of time dependencemay allow for oscillatory solutions withqualitatively new features.

A similar procedurefor obtaining the field equationsapplies to the Dirac field with somewell-known complications associatedwith the proper treatment of the spin degree of freedom

[145—147].The equationsof motion become

(e~2~ + -~)— ~)f(r) + (e~2�— m)g(r)= 0,(6.65)

(e~2(~_+ —) +~)g(r) + (—e~2�~—m)f(r) = 0,

F

4s : , R

~2M

Fig. 6.9. Spectrumof theKlein—Gordonequationwith externalgravitationalfield. The eigenstatescollapseto e= 0 as theSchwarzschildmetric is

approached.From Muller, Greiner,and Rafelski[112].

352 1. Rate/ski et al., Fermions and bosons interacting with arbitrarily strong external fields

which should be comparedwith eq. (2.18). Inclusion of the vector potential is straightforward,involving the replacementof E by � — V.

It is apparentfrom the form of the Schwarzschildsolutionthat thefermion field will be exposedtosupercriticalsingular potentialsas well. in fact a more detailedstudy [163] shows that the effectsdiscussedin this section arealwayspresent,despitethe ‘weakness’of the coupling constant2M. Inparticularonly resonancesandno discreteboundstateshavebeenfound. The theoreticaldescriptionis more involved andthe difficulties encounteredare reminiscentof the problemsassociatedwith thepure(1/r) potentialin the Coulombpotentialproblemwhich havebeendiscussedin section2.

The work outlined in this subsectionis thus indecisive, again raising more questionsthan areresolved.This subjectremainsopenfor muchfuture investigation.

6.5. The role of supercritical fields in field theory

Quantumelectrodynamicshasbeena pacemakerin the developmentof modernfield theory. It isconceivable,that methodsemployedin the discussionof supercriticalfields will be of relevancein thenonperturbativetreatmentof theoriesinvolving strongly boundstates.

A simple andpossibly relevantexamplearisesconsideringstrongly boundstatesin the (classical)field theory of (attractive)massivevectormesonsinteractingwith fermion fields. The situationhereissomewhatmore involved than in the discussionof section2. Let us assume,for illustration, that thefermionsmovein a self-consistentshell potential.Thenthe total energyof the boundstateis given bythe sum of the Dirac eigenenergiesand of the energyassociatedwith the shell potential,sometimesalso called ‘correlationenergy’,

E~= £�~+ Emeson. (6.66)

The condition for the transitionto a stateequivalentto the chargedvacuumis now

E~<0 (6.67)

andnot EDirac < in.

Unlike the casewith externalfields, it must be expectedthat the critical potentialstrengthcan beexceededby the self-consistentshell potentialanda particlestatebe embeddedin the negativeenergycontinuum.

To illustrate this behavior we considera fermion field coupledto a vector mesonfield with theHamiltonian[Ia] (with A . A A2 = A~—A . A)

H = ~ d3x~[~(ap + f3m - g~(a. A - A°flç~](6.68)

-~Jd3x[~+(VA~)2+W(A . A)- W(0)]+h.c.}~

where W(A2) is an arbitrarypotentialenergyof the vectorfield Aw, which leadsto the equationsofmotion

(a . p + 13m — g~(a. A — A°Thti= ia~/i/at, (6.69)

— AA,.~+ A~aW/aA2= +~g~[u/i,y~]. (6.70)

I. Rate/ski et a!., Ferm ions and bosons interacting with arbitrarily strong external fields 353

We alsohavethe auxiliary condition

(pslawAwlps)= 0 (6.71)

for anyphysicalstate!ps).For stronginteractionsit is mostconvenientto chooseas the basis for the expansionof the Fermi

field ~(t the completesetof solutionsof the Dirac equation(p standsfor both discreteandcontinuousindices),

(a . (p — g~A~)+ ~m+ g~A~°)u/i~= ç,ufr~,, (6.72)

wherethe c-numberfield A~’is the meanfield A~’= (ps~A~Ips).The field operatoris expandedin the

quasi-particleFock spaceas

~(x,0) = fb~(x)de + ~ b~~~(x)+ J d~~(x)d�, - (6.73)rn >�p>~m

m -~

wheneverthe field A~is undercritical.A different expansionis appropriatewhen a resonanceof eq.(6.72) is embeddedin the negativeenergycontinuum.Thenwe have

~(x,0) = J ~ d�+ ~ b0~0+ b0 J a(�)~~d�+ J d [JhE(E)~de] d�’, (6.74)m>ep>—m

m —~ -~ —~

where a(�) and h~(�)are suitable functions that describe the resonancein a negative energycontinuum(see sections2, 3). The extractionof the resonancefrom the negativeenergycontinuumwavefunctionshasbeendiscussedin detail in sections2, 3. If morethan oneresonanceis embeddedin the continuum,a generalizationof eq. (6.74) is in order.

It is importantto stressthe differencebetweenthe self-consistentfield problemand the caseof anexternalfield. The neutralvacuum,characterizedby b~I0)= d~l0)= 0, may be a stable, zero-energystateevenwhen supercriticalbinding describedby (6.74) should occur, sincethe total energyof theexcited state b~l0)may be positive upon considerationof the mesonenergyincluded in (6.66), incontradistinctionto the situationwith external potentials.

Our choiceof the meanfield A~was directedby our desireto eliminatefrom thegroundstatemostcontributions from the virtual quasi-particleexcitations. Therefore A~’ is determinedfrom theequation

—AA~’+ A~(ôW/0A2)A~= g~(ps~~[~, y~]Ips). (6.75)

We considerthe stateb~J0)= p) as a trial state.Then eq. (6.75) becomes(neglectingvirtual vacuumpolarization,c.f. section3)

—AA~’+ A~’(l9WIOA2)A~= g~’0y~’0. (6.76)

Here ~ is the coefficientof b0 in eq. (6.74)or the lowestenergyparticlewave function in eq. (6.73).At this point, the interactionhasbeenmodified [157]to allow bound fermionstatesbound only by

“electric” forces.Therefore the sign on the right-handside of eq. (6.77) hasbeen changedand theresultingfield hasbeencalledattractive:

—AA~’+ A~(DW/8A2)Al= ~ (6.77)

354 J. Ratelski et al., Fermions and bosons interacting with arbitrarily strong external fields

This changewas motivatedby considerationof possibleinternal (SU(3)) symmetryof the fields ofinterestandthe sign maybe understoodin analogyto the differentpossiblesignsof the z-componentof the isospin.This changealsomeansthat the energyof the boundstatebecomes

= EDirac + Em~son= f d3xçli~(a. p + /3m — g~(a. A —

(6.78)

+ ~f d3x[(VA~)2+ W(AC . A~)— W(0)] = Jd3x~,

where

EDirac = �o. (6.79)

In view of the arbitrary(but motivated)changesinvolved in writing (6.78), what follows should beunderstoodonly as illustrating a typeof behaviorin a coupledfield theorydistinct from that found inthe externalfield problem.

Before turning to numericalsolutions, a virial theorem for classical interacting fields may beconsideredto gatherinsight into thequalitativefeaturesof this model.Taking the expectationvalueofthe commutator[x p, a p + /3m + v] betweenany boundDirac wave functiononefinds that

(a~p~=(x~Vv~ (6.80)

where v may be any momentum-independentmatrix potential. In our model v is just the vectorpotential,g~y

0y. A~.From eqs. (6.80)and (6.77) it follows that

(p — g~A~)+g~A~)Jd3x(x . VA~— A~)~(—AA~+ A~(8W/8A2)). (6.81)

Equation(6.79) can berewritten with someeffort to givean expressionfor the energy,eq. (6.78):

E~ + J d3x[( W(A~)— W(0)) — A~(oW/aA~)]+ mf d3x~fi0t~i0. (6.82)

Even if q~rhad beenentirely eliminatedfrom the energy(as is possiblefor a scalarinteraction),eq.(6.82) could not be usedas a basis for a variational principle since the constrainton u/i0 to be thelowest particle solution of the Dirac equation is not implemented. When supercritical fields areencountered,eq. (6.82) is only an approximation,sinceu/’~ is thenasuperpositionof continuumwavesand the virial theorem,eq. (6.80), doesnot apply. The error madeis small for the rangeof couplingstrengthsdiscussedin the numericalinvestigationsdescribedbelow.

A remarkableresult is thedisappearanceof the kinetic energyterm in eq. (6.82). Further,if W(A2)

is a polynomial in A2,E~(A2)doesnot containanyA4 term.Exceptfor thecaseW — A2, an extremumof E~doesnot coincidewith the minimum of W. The effective shift is dueto the interactionwith thefermion field.

Let usdevotethe restof our discussionto the specialcase

W = p,2A2, (6.83)

where js is the vector mesonmass.Then eq. (6.82) becomes:

E~ Jd3x[p,2A~+ mu/i0u/i0]. (6.84)

I. Rafelski et a!., Fermions and bosons interacting with arbitrarily strong external fields 355

The energyE~is positivedefinite if the scalarintegral fd3xu/i0u/s0is positive.This is a sufficient but

not necessarycondition. No lower bound for the scalar integral is known when vector fields arepresent,but in all examplesstudiedit hasbeenfound to be positive definite.

Equations(6.72) and (6.77) havebeensolvednumericallywith W given in eq. (6.83) for the lowestenergy particle state such that the Dirac wave functions (either eigenstatesor resonances)arenormalizedto one. In fig. 6.10 we showthe energyof the boundfermion,EDirac, (eq. (6.72)) andthetotal energyE~,as a functionof the couplingconstantg,. for vector mesonmassesp, = 0.2 and0.4m.The energyis measuredin units of the barefermion massm. In this examplewe see that the totalenergyis always positive and very small for coupling constantsg~>4, while the Dirac eigenvaluesEDirac take largenegativevalues.The scalarintegral is a negligible contributionto E~,eq. (6.84) forstrongcoupling. Exact agreementof numericalresultswith thoseexpectedfrom the virial theorem,eq. (6.84), as long as EDirac>—m hasbeenfound. This is alsoa good checkof the computercodesinvolved in solvingthe nonlinearequations(6.72), (6.77). For the rangeof couplingconstantsshowninfig. 6.10 with EDirac< —m the errormadeusingthe virial theorem,eq. (6.84) to calculateE~with givenA~was found to be lessthan 1%.

The mainconclusionthat canbedrawnfrom thisexerciseis the recognitionthat thetotal energyofthe strongly boundstatelies above thatof that of the neutralvacuum,evenif the shell potential issupercritical.Thereforethe neutralvacuumseemsto be a stablegroundstateof the theoryat leastinthe example considerednumerically. Spontaneousproductionof bound state pairs by the neutralvacuumis forbiddenin the strongcoupling limit, althoughthe energyof eachboundfermion,EDirac, ~5smaller than —m. The localizedboundstate is the lowest energystateof the chargedsectorof theHilbert space.However,sinceE~canbe smallerthanp,, no free mesonsmayexist.

Furtherwe note that essentialin obtaininga propersolution of eqs. (6.72) and (6.77) is a propertreatmentof resonantstatesin theantiparticlecontinuumwhentheyoccur. As describedin section3,

Dirac \Ui \

[m]________— /~‘0.4[mJ

2g2/47r

Fig. 6.10. The Dirac eigenvalueandtotal energiesof interactingDirac vectormesonfields.

356 J. Rate/ski et al., Fermions and bosons interacting with arbitrarily strong external fields

the properdescriptionof the systeminvolves thenthe statement

a(E)~. (6.85)

To obtaina(E)astudyof the phaseshifts may be madeas discussedin section2.Without the properunderstandingof the supercriticalspectruma solution of eqs.(6.72) and(6.77)

would seemdiscontinuousat the critical points,sincewe would takethe lowestdiscretestate,e.g.a 2sstate,in placeof the embeddedIs-stateresonance.

Another way of seeingthe differencebetweenthe presentlyconsideredcaseandthe externalfieldproblemstudiedin the remainderof this paperinvolves the Green’sfunction formulationconsideredin subsection3.5. Therewe learnedthat the pole associatedwith the critical statecannotdrag theintegrationpathoff the real axis,as the potentialattainsovercritical strength.However,we havejustshown that the opposite is true for interacting fields studiedin this subsection.The integrationpathC’ of fig. 3.5 shouldnow be takento obtainthe Green’sfunction, insteadof the previouslyemployedpath D.

References

[1] W. Greiner, panel discussionin: Proc. Intern. Conf. on the Propertiesof Nuclear States,Montreal, 1969 and panel discussionin TheTransuraniumElements,TheMendeleevCentennial,Houston,1969.

[2] PAM. Dirac,Proc.Roy. Soc. (London) Al17 (1928) 610.[3] W. Gordon,Z. f. Physik40(1926)117.[4] 0. Klein, Z. f. Physik41(1927)407.

[5]CD. Anderson,Phys.Rev. 43(1933)491.[6] PAM. Dirac, in Rap.d. 7 con. Solvay d. Phys.,Str. e. Prop. d. Noy. Atom. (1934) 203.[7] PAM. Dirac, Proc.Roy. Soc. (London) A126 (1930) 360.[8] 0. Klein, z. 1. Physik53(1929)157.[9] J.D. Bjorkenand S.D. Drell. Relativistic QuantumMechanics,Vol. 1 (McGraw-Hill. New York. 1964).

[10] F. Sauter,Z. f. Physik69 (1931)742.[11] F. Sauter,Z. f. Physik73(1931)547.[12] S. Szczeniowski,Z. f. Physik73(1931)553.[13] F. Hund,Z. f. Physik 117 (1941) 1.

[14] MS. Plesset,Phys. Rev.41(1932)278.[15] F. Beck, H. SteinwedelandG. SUssmann,Z. f. Physik171 (1963)189.[16] LI. Schiff. H. Snyderand J.weinberg,Phys. Rev. 57 (1940)315.[17] K. Nikolsky, Z. f. Physik62(1929)677.[18] I. PomeranchukandJ. Smorodinsky,J. Phys. USSR9 (1945) 97.[19] Al. Akheizerand yB. Berestetskii,QuantumElectrodynamics(IntersciencePublishers,New York, 1965).[20] P.A M. Dirac, Proc. Roy. Soc. (London) A118 (1928) 341.[21] PAM. Dirac, Principles of QuantumMechanics(Oxford, London, 1958).[22] K.N. Case,Phys.Rev. 80 (1950)797.[23] W.M. Frank,D.J. Land andR.M. Spector,Rev.Mod. Phys.43(1971)36.[24] F.G. WernerandJ.A. Wheeler,Phys. Rev.109 (1958) 126.[25]D. Rein, Diplomarbeit,University of Frankfurt.[26] D. Rein, Z. f. Physik221 (1969)423.[27] W. PieperandW. Greiner,Z.f. Physik218 (1969)327.[28] J. Rafelski,B. Muller andW. Greiner,Nucl. Phys. B38 (1974) 585.[29] L. FulcherandA. Klein, Phys. Rev.D8 (1973)2455.

[30] L. FulcherandA. Klein, Ann. Phys.84 (1974)335.[31] PG. Reinhard,Lettre NuovoCimento 3(1970)313.

J. Rafelski et al., Fermions and bosons interacting with arbitrarily strong external fields 357

[32] PG.Reinhard,W. GreinerandH. Arenhövel,NucI. Phys.A166 (1971)173.

[331L.P. FulcherandW. Greiner,Lettre NuovoCimento 2 (1971) 279.[34] J. Rafelski,L.P. FulcherandW. Greiner,Phys.Rev.Letters27 (1971) 958.[35] J. Rafelski,W. GreinerandL.P. Fulcher,NuovoCimento 13B (1973)135.[36a] B. MUller, Diplomarbeit,Universityof Frankfurt (1972).[36b]J. Waldeck,Diplomarbeit, Universityof Frankfurt (1974).[37] MS. Freedman,FT. PorterandJ.B. Mann,Phys.Rev. Letters28(1972)711.[38] B. Fricke, J.P.DesclauxandJ.T. Waber,Phys. Rev. Letters28 (1972)714.[39] G. Soff, J. RafelskiandW. Greiner,Phys. Rev. A7 (1973).

[40a] GA. RinkerandL. Wilets, Phys.Rev.Letters31(1973)1559.[40b] GA. Rinker and L. Wilets, Phys. Rev.Al2 (1975) 748.[4la] M. Gyulassy,Phys.Rev. Letters 32 (1974) 1393.[41b] M. Gyulassy,Phys.Rev.Letters33(1974)921.[42] E.H. Wichmannand N.M. KrolI, Phys.Rev. 101 (1956) 843.[43a] P. Mohr, Phys. Rev. Letters34 (1973) 1050.[43b] K-T. ChengandW.R. Johnson,Phys.Rev. A14(1976)1943.[44] G.W. Erickson,Phys. Rev.Letters27(1971)780.[45]G. Soff, B. MUller andJ. Rafelski,Z. f. Naturforschung29 (1974)1267.[46] W. Dittrich, Phys. Rev.Dl (1970)3345.[47] S.S.GershteinandYB. Zeldovich,Lettre Nuovo Cimento 1(1969)835.[48] S.S.GershteinandYB. Zeldovich,SovietPhys. JETP30 (1970) 358.[49] VS. Popov, SovietJour. NucI. Phys. 12(1971)235.[50] VS.Popov,SovietPhys. JETP32(1971)526.[51] YB. Zeldovichand VS. Popov,SovietPhys.Uspekhi 14 (1972) 673.[52] B. MUller, H.Peitz, J. Rafelskiand W. Greiner,Phys.Rev. Letters 28(1972)1235.[53] B. MUller, J. Rafeiskiand W. Greiner,z.f. Physik257 (1972) 62.[54] U. Fano,Phys.Rev. 124 (1961)1866.[55]B. MUller, J. RafelskiandW. Greiner,NuovoCimento 18A (1973)551.[561S.J.Brodsky,Comm.Atomic MolecularPhysics4 (1973)109.[57] WE. Meyerhof, McGraw-Hill ScienceEncyclopediaYearbook(1975).[581J.H. Hamilton and IA. Sellin. PhysicsToday26 (1973) 42.[59] B. MUller, J. Rafelskiand W. Greiner,Z. f. Physik257 (1972)183.[60] H. Peitz, B. MUller, J.RafelskiandW. Greiner,Lettre NuovoCimento 8 (1973)37.

[61]B. MUller, J. Rafelskiand W. Greiner,Phys.Letters47B (1973) 5.[62] B. MUller andW. Greiner,Z. f. Naturforschung31A (1976)1.[63] VS. Popov andTI. Rozhdestvenskaya,SovietPhys.JETPLetters 14 (1971) 177.[64] VS. Popov, SovietPhys.JETP Letters16 (1972)251.[65] S.S.GershteinandVS. Popov,Lettre Nuovo Cimento 6 (1973) 593.[66] VS. Popov,SovietPhys. JETPLetters 18 (1973) 53.[67] V.5. Popov, SovietJour.NucI. Phys. 19 (1974) 81.[68] MS. Marinov and VS. Popov, SovietJour. NucI. Phys. 20 (1975)641.[69] V.5. Popov,SovietJour.NucI. Phys. 14 (1972)257.[70] V.S. Popov,SovietJour.NucI. Phys. 15 (1972) 595.[71] MS. Marinov and VS. Popov, SovietPhys.JETP 38 (1974) 1069.[72] MS. Marinov and V.S Popov,SovietPhys.JETP40 (1975)621.

[73] MS. Marinov and VS. Popov, SovietPhys.JETP41(1975)205.[74] MS. Marinov, VS. Popovand V.L. Stolin, JETPletters 19 (1974) 49.[75] M.S. Marinov, VS. Popovand V.L. Stolin, Jour.Comp.Phys. 19 (1975)241.[76]J. RafelskiandB. Muller, Phys. Letters65B (1976) 205.[77]J. RafelskiandB. MUller, Phys.Rev. Letters36 (1976) 517.[78] K. Smith, H. Peitz, B. Muller andW. Greiner,Phys. Rev. Letters 32(1974)554.[79] K. Smith, B. Muller andW. Greiner,J. Phys. B8 (1975) 75.[80] J. Rafelskiand A. Klein, in: ReactionsbetweenComplexNuclei (North-Holland,Amsterdam,1974).[SI] V. Oberacker,G. Soff andW. Greiner,Phys.Rev. Letters 36 (1976)1024.[82] J. Reinhardt,G. Soff and W. Greiner,z. f. PhysikA276 (1976)285.

[83] WE. Meyerhof,Science193 (1976) 839.[841PH. Mokler, H.J.Stein andP. Armbruster.Phys.Rev. Letters29 (1972) 827.[85] F.W. Saris, W.F. van der Weg, H. TawaraandR. Laubert,Phys. Rev.Letters28 (1972)717.

358 J. Ratelski et al., Fermions and bosons interacting with arbitrarily strong external fields

[86] WE. Meyerhof, T.K. Saylor,SM. Lazarus,WA. Little, B.B. Triplett and FL. ChaseJr.,Phys. Rev.Letters30 (1973)1279.[87] C.K. Davis andJ.S. Greenberg,Phys.Rev.Letters32 (1974)1215.[88] B. Muller, R.K. Smith and W. Greiner,Phys.Letters49B (1974) 219.[89] B. MUller andW. Greiner,Phys.Rev. Letters33(1974)469.[90]J.S. Greenberg,C.K. Davis andP. Vincent, Phys.Rev.Letters 33(1974)473.[91] G. Kraft, PH. MoklerandH.J.Stein,Phys.Rev. Letters33(1974)476.[92] W.E. Meyerhof, Phys.Rev.AlO (1974)1005.

[93a]I. Tserruya,H. Schmidt-Bocking,R. Schulé,K. Bethge,R. SchuhandH.J. Specht,Phys. Rev. Letters36 (1976)1451.[93b]W. Wölfi, Ch. Stoller,0. Bonani, M. Stöckli, M. SuterandW. Dappen,Phys.Rev. Letters36 (1976)309.[93c]W. Frank,P. Grippner,K.H. Kaun, P. Manfrassand Yu. P. Tretyakov, Z. f. Phys. A277 (1976)333.[94] H. SnyderandJ. Weinberg,Phys. Rev.57 (1940)307.[95] M. Bavin andJ.P. Lavine,Phys. Rev.D12(1975)1192.[96] H. FeshbachandF. Villars,Rev.Mod. Phys.30 (1958) 24.[97] A.B. Migdal, Soy. Phys. JETP34 (1972)1189;Zh. Eksp. Theor.Fiz. 61(1971)2209.[98] A. Klein andJ. Rafeiski, Phys.Rev. DII (1975)300.[99] A. Klein andJ. Rafelski,in: FundamentalTheoriesin Physics,eds.S.L. Mintz, L. Mittag and SM. Widmayer (PlenumPubI. Corp., New

York, 1974).[100] A. Klein andJ. Rafelski,in: ParticlesandFields, 1974, ed. Carl E. Carlson(AlP, New York, 1975).[101] A. Klein andJ. Rafelski, Phys.Rev. D12 (1975) 1194.

[102] B. MUller andJ. Rafelski,Phys. Rev.Letters 34 (1975) 349.[103] J. RafelskiandB. MUller, to bepublished.[104]w.G. Unruhe,Phys. Rev. D14(1976) 870.[105] SW. Hawking,Nature248 (1974)30.[106] SW. Hawking, Comm.Math. Phys.43(1975)199.[107] SW. Hawking, Phys. Rev.D13 (1976) 191.[108] B.S. DeWitt, Phys.Reports19C (1975) 295.[109]D.G. Boulware,Phys.Rev. D13 (1976)2169 andreferencestherein.[110] S.A. Fulling, Phys. Rev. D14 (1976) 1939.

[Ill] L.M. Ford, Phys.Rev.Dl2 (1975)2963;D14 (1976)658andreferencestherein.[112] B. MUller, W. GreinerandJ. Rafelski,to bepublished.[113] HG. Dosch,J.D.H.JensenandV.F. MUller, PhysicaNorvegica5 (1971) 151.

[114] M.E. Rose,Relativistic ElectronTheory (Wiley, New York, 1961).[115] M. Abramowitzand IA. Stegun,Handbookof MathematicalFunctions(U.S.GovernmentPrintingOffice,Washington,D.C., 1965).[116]E.T. Whittaker andG.N. Watson,A Courseof ModernAnalysis (Cambridge,London, 1963).[117] N.N. Lebedev,SpecialFunctionsandTheir Applications(Prentice-Hall,EnglewoodCliffs, NJ, 1965).[118] B. Simon,QuantumMechanicsfor HamiltoniansDefinedasQuadraticForms(PrincetonUniv. Press,Princeton,NJ, 1971) p. 204.[119]J.M. EisenbergandW. Greiner,ExcitationMechanismsof the Nucleus,Vol. 1 (North-Holland,Amsterdam,1970).[120] B. Fricke and W. Greiner,Phys. Letters30B (1969)317.[121] B. Fricke, W. GreinerandJ.T. Waber,Theo. Chim. Acta (Ben.) 21(1971) 235.[122] B. Fricke and G. Soff,GSI-BerichtTI (1974).[123]DR. Hartree,The Calculationof Atomic Structures(Wiley, New York, 1957).[124] FT. Porterand MS. Freedman,Phys.Rev. Letters27 (1971)293.[125] G. Soff, B. Muller, J. Rafelskiand W. Greiner,Z. f. Naturforschung28a (1973)1389.

[1261J. Vasconcelos,RevistaBrazilierade Fisica 1(1971)441.[127] L.S. RodbergandR.M. Thaler,Introductionto the QuantumTheory of Scattering(AcademicPress,New York, 1967).[128] ML. GoldbergerandKM. Watson,Collision Theory (Wiley, New York, 1964).[129]W. Pauli, Phys.Rev.58 (1940) 716.[130] S.S. Schweber,Introductionto Relativistic QuantumField Theory (Row, PetersonandCo., Evanston,III., 1961).[131]J.D.BjorkenandS.D. Drell, Relativistic QuantumFields,Vol. II (McGraw-Hill, New York, 1965).[132]0. Källen, QuantumElectrodynamics(Springer,New York 1972).[133] J. Schwinger,Phys.Rev. 82 (1951) 664.[134] B. MUller, unpublishednotes.[135]J. Blomquist,NucI. Phys.B48 (1972) 95. -[136] G.W. Pratt, Rev. Mod. Phys. 35 (1963)502.[137] L. GomberoffandV. Tolmachev,Phys. Rev.D3 (1971)17%.

[138] Search& Discovery,PhysicsToday29 (1976)17.[139]B. Muller, Ann. Rev. NucI. Sci. 26 (1976)351.[140]W. Scheidand W. Greiner,prepnint,to be published.

J. Rate/ski et al., Fermions and bosons interacting with arbitrarily strong external fields 359

[141]J. Rafelski,B. MUller and W. Greiner,Lett. al NuovoCimento4 (1972) 469.[142] B. MUller, Ph.D. Thesis,University of Frankfurt, 1974.[143]J. Rafelskiand B. MUller, to be published.[144]H. Peitz, private communication.[145]DR. Bnill andJ.M. Cohen,J. Math. Phys. 7(1966) 238andreferencestherein.[146]We particularlyrefer to Section III andappendixA of ref. [62].[147]R. Adler, M. Bazin andM. Schiffer,GeneralRelativity (New York, 1965)pp. 62—71.[148]R.K. Smith and W. Greiner,SpontaneousandInducedRadiationfrom Intermediate(Superheavy)Molecules,prepnint,Frankfurt.[149]T.C.Y. Chen.T. Ishiharaand KM. Watson.Phys. Rev. Letters 35(1975)1574.[ISO]D.M. Jakubassaand M. Kleber, Z. f. Physik A277 (1976) 41.[151]H. HoIm andW. Greiner,NucI. Phys. A195 (1972) 333.[152]J. Reinhard,G. Soff andW. Greiner, Z. f. PhysikA276 (1976) 285.[153]V. Oberacker,G. Soff and W. Greiner,NucI. Phys.A259 (1976) 324.[154] D. Bunch, W.P. Ingalls,M. Wiemannand R. Vandenbosch,Phys. Rev. AlO (1974) 1245.[155] W. Betz,0. Soff, B. Muller and W. Greiner,haveobtainedrecentlyresultssuggestingthe validity of thecomments;Phys. Rev. Letters 37

(1976) 1046.[156] A. Klein andJ. Rafelski. Z. f. Physik A284 (1978) 71.[157]J. Rafelski and B. MUller. Phys. Rev.Dl4 (1976) 3532.[158] N.H. March,Self-ConsistentField in Atoms(New York, 1975).[159]J. Rafelski andB. MUller. to be published.[160]A. Bodmer.Phys. Rev.D4 (1971) 1601.[161]T.D. Leeand G.C. Wick. Phys. Rev.D9 (1974) 2291.[162]Ref. [147]p. 166.[163]M. Soffel. B. MUller and W. Greiner,J. Phys. AlO (1977) 551.

Note addedin proof

Sincethis articlewas completed(February1977)severalimportantnewdevelopmentshaveoccurred.Certainly,themostexcitingis themeasurementsof thepositronproductioncrosssectionsjustcompletedatGSI in Darmstadt[NI—N4].Therehavealsobeentwo importanttheoreticaldevelopments.Thefirst ofthese[N5]is thattheinducedpositronproductionprocessconsideredin section4 isnot theonly importantproductionprocessin whichenergyis transferredto theelectron—positronfield from thenuclearelectricfields. In section 4 only thoseinduced processeswere con:ideredin which the electronchargewasproducedin one of the lower bound states(2p or Is before the critical distanceis reached)or in theovercritical state.(Seefigs. 4.4 and4.9.)

~[pb] s~(x-~1) .[~b[ p,~2(x-.1)

Za-25fm 2a.Z5fm /

/ nduc~is / induc~{2p 2~

si~keoff(x s~keoft(x 1)sum sum

0_i z ______________ z~0 170 180 190 160 170 180 19J

Fig. NI. The Z dependenceof the shakeoff and induced positron production.

360 1. Rate/ski et al., Ferm ions and bosons interacting with arbitrarily strong external fields

trr [pb] Total Positr~iCross Section10 Pb-Pb .~. Pb-U - UU

I . \estinote

1& ~ \.

\ \\ + Experiment lGSt)

— total positrnnyietd ‘ only Epi,2o.Zpi,2o\.

\ —— QEOlstOmoft.in~.iced)nuclear b~kgrxund

1 ‘.1 I I I I

15 20 25 20 25 20 25 2a[fm]

ao 5.5 5.0 4.5 4.0 3.5 5.5 5.0 45 4.0 35 ~ ~ 4.0 3.6E/A 1MeV/am]

Fig. N2. Total positronproductioncross-sectionsasa function of ion energy.

Approximately equalin importanceis the processwhereinthe electronis producedin a (positive-energy)continuumstate[N5]. Thisprocessis verysimilarto theusualpair productionprocessin thefieldof aheavynucleusby achargedparticle.However,in thepresentcaseperturbationtheoryis inapplicable.In fact, themostimportantnewfeature,acoherentactionof alargenumberof virtual photons,wouldnotbeuncoveredwith perturbationtheory.This processhasbeencalled “vacuumpolarizationshakeoff”,whichemphasizesthatsomeof thevirtual positronssurroundingeachof thetwo centersof chargehavebecomephysicalwith theinputof energyfrom thenuclearelectric’fields.The Z dependencesof inducedandshakeoffpositronproductionare shownin fig. Ni. Thereit is shownthat the two processesareofcomparableimportanceandthat the size of the inducedcontributionincreasesmorerapidly with Z.

Thefirstquantitativecomparisonof theoryandexperimentisshownin [N3,N4],figs. N2 andN3. In fig.N2 thetotal positronproductionrateis depictedas afunctionof ion energyfor Pb—Pb,Pb—U andU—Ucollisions.Thetrendsof thedataandcalculationsareencouraging.In particular,theslopesseemtoagree.For Pb—PbandPb—U collisions the magnitudesseemto disagreeby a factorof about2. This is not toodisturbingsincethedatapointsmaycontainanormalizationerror,andseveraltheoreticaleffectsthatareprobablyimportanthavenot yet beenincluded.For example,ionization of the is electronsinto higherboundstateshasnot yet beenincluded.

The calculatedcurvesindicatethat for bothPb—PbandPb—Ucollisionsthereis an energyregion inwhich QED positronsareconsiderablymorenumerousthanthoseresultingfrom nuclearbackgroundeffects.In neithercasewouldwe expectovercriticalphenomenato occursincethe critical chargeis notexceeded.In U—U collisions the low-lying nuclearstatesare more numerousand thus the nuclearbackgroundincreases.Fortunately,theQED(molecular)positronproductionratesgrowevenfaster[N3,N4]. All of the importantQEDeffectshavenot beenincludedandhencewedo not knowto whatextentthe datasupportthe calculations,althoughthe initial estimateis very encouraging.Further,wedo notknow if overcriticalQED effectscan be separatedfrom undercriticalQED effects.

There are also recentexperimentaldata [N3] from projectile coincidenceexperimentsin U—Pbcollisions atGSI. In theseexperimentsthepositronsin anarrowenergywindow (here500±55keV)are

I. Rafelski et at., Fermions and bosons interacting with arbitrarily strong external fields 361

I -I--

- Eu-u ___________

‘~ -m me5MeV)/MeV

I;”,’ ___Fig. N3. Differential productionprobability for positronsof 500± Fig. N4. New featuresin inelasticcollisions.55 keV energyasafunctionof scatteredparticleangle.Thetwo uppercurvesrefer to the molecularprobabilities of the U—U and U—Pbsystems,respectively,aftersubtractionof thenuclearpartfrom thetotal measuredprobabilities.The nuclearpantis alsoshownin theU—Pb data connectedby the(free-hand)lower curve. The curveconnectingthemolecularU—Pb datais calculatedby theFrankfurtgroup[N4].

detectedin coincidencewith the ion angle&m~The dataareshownin fig. N3 in comparisonwith [N4]

QEDpredictionof positronsfrom inducedvacuumdecayandvac-pol.shakeoff.Theangulardistributionis describedvery well.

At presentdisentanglingtheovercriticalpositronsfrom othereffectsinelasticcollisionsdoesnotseemto be aneasytask.Thus,Rafelski,MUller andGreiner [N6] haveproposedexaminingthe spectrumofpositrons in deeply inelastic collisions. Therethe formation of a nuclear compositemay enhanceovercriticalphenomenaandproduceadistinctivecharacteristicfortheidentificationof these,asshowninfig. N4. The effects of the nuclear magneticfield split the ls levels so that one might expect apositronproductionspectrumwith two peaks.

[NI] H. Backe,J.S. Greenberg,E. Kankeleit et al., GSI preprint.[N2] P. Kienle,C. Kozhuharov,J.S.Greenberg,E. Kankeleit et al, OS! preprint.[N3] H. Backe, E. Berdermann,H. Bokemeyer,iS. Greenberg,L. Handschug,F. Hessberger,E. Kankeleit, P. Kienle, Ch. Kozhuharov,Y.

Nakayama,L. Richter,H.Stettmeier,P.Vincent, F. Weik andR. Willwater,GSI preprint,toappearin: Proc.Intern.Conf. on NuclearStructure,Tokyo (September1977).

[N4] B. Muller,V. Oberacker,J. Reinhardt,0. Soff, W. GreinerandJ. Rafelski,Frankfurtpreprint. to appearin: Proc.Intern. Conf. on NuclearStructure,Tokyo (September1977).

ENS] 0. Soff,J. Reinhardt,B. MUller and W. Greiner,Phys. Rev.Letters 38 (1977) 592.[N6] J. Rafeiski,B. MUller and W. Greiner,Frankfurt preprint, Z. f. Physik, in print.