Electrodynamic Absorber Theory

186 Pages·2010·2.44 MB·German

Electrodynamic Absorber Theory
A Mathematical Study
-
Dirk - Andre Deckert

DoctoralThesis
Electrodynamic Absorber Theory
A Mathematical Study
Dirk-Andre´ Deckert
DissertationanderFakulta¨tfu¨rMathematik,InformatikundStatistikder
Ludwig-Maximilians-Universita¨tMu¨nchen
Theresienstraße39,80333Mu¨nchen,Germany.
Submission: October26,2009
Rigorosum: January7,2010
SupervisorandFirstReviewer:
Prof. Dr. DetlefDu¨rr
MathematischesInstitutderLudwig-Maximilians-Universita¨tMu¨nchen
Theresienstraße39,80333Mu¨nchen,Germany.
SecondReviewer:
Prof. Dr. HerbertSpohn
ZentrumMathematik,TechnischeUniversita¨tMu¨nchen
Boltzmannstr. 3,85747Garching,Germany.
ExternalReviewer:
Prof. Dr. GernotBauer
FachbereichElektrotechnikundInformatikderFachhochschuleMu¨nster
Bismarckstraße11,48565Steinfurt,Germany.

Abstract
This work treats mathematical questions which arise in classical and quantum electrodynamics
whendescribingthephenomenaofradiationreactionandpaircreation. Itconsistsoftwomajor
parts:
Inthefirstpartclassicaldynamicsarestudiedwhichallowforradiationreaction: Forclassical,
non-rotating, rigid charges we give explicit representation formulas of strong solutions to the
Maxwell equations, prove global existence and uniqueness of strong solutions to the Maxwell-
Lorentz equations in a general setting in which we also allow for a class of infinite energy
solutions, negative masses of the charges, and for both cases, Maxwell-Lorentz equations of
motionwithself-interaction(ML+SI)andwithout(ML-SI).Weprovethatpossiblesolutionsto
theWheeler-Feynmanequationswithboundedaccelerationsandmomentaarealsosolutionsto
theML-SIequations. ThisgivesrisetoauniquecharacterizationofWheeler-Feynmansolutions
by position, momenta and their Wheeler-Feynman fields at one instant of time. Moreover, we
give a reformulation of Wheeler-Feynman electrodynamics for rigid and non-rotating charges
in terms of an initial value problem for Newtonian Cauchy data. With it we prove existence
ofWheeler-FeynmansolutionsonfinitetimeintervalscorrespondingtotheNewtonianCauchy
data. We discuss how this method could yield global existence of solution to the Wheeler-
Feynmanequations.
In the second part quantum dynamics are studied for systems of an infinite number of Dirac
electrons which interact only with a prescribed external field that allows for pair creation: We
construct the time-evolution for the second quantized Dirac equation subject to a smooth and
compactly supported, time-dependent electrodynamic four-vector field. Earlier works on this
(Ruijsenaars) observed the Shale-Stinespring condition and showed that the one-particle time-
evolutioncanbeliftedtoFockspaceifandonlyiftheexternalfieldhaszeromagneticcompo-
nents. ThebasicobstacleintheconstructionisthatthereisneitheradistinguishedDiracsea,i.e.
Fockspacevacuum,noradistinguishedpolarization. Therefore,thekeyidea(suggestedalready
byFierzandScharf)istoimplementthistime-evolutionbetweentime-varyingFockspaces. We
show that this implementation is unique up to a phase. All induced transition amplitudes are
uniqueandfinite.
In a last part we give a brief outlook on our perspective of a divergence free, electrodynamical
theory for point-like charges which accounts for both phenomena, radiation reaction as well as
paircreation. ItisbasedontheideathattheDiracsearepresentstheabsorbermediumproposed
by Wheeler and Feynman. The presented mathematical results can be considered as first steps
towardsit.
Keywords: Absorber Electrodynamics, Radiation Reaction, Pair Creation, Wheeler-Feynman
Solutions,MaxwellSolutions,Lie´nard-WiechertFields,Maxwell-LorentzSolutions,Functional
DifferentialEquations,InfiniteWedgeSpaces,Second-QuantizedDiracTime-Evolution,Quan-
tumElectrodynamics,QuantumWheeler-FeynmanInteraction
Zusammenfassung (Translation of the Abstract)
Diese Arbeit behandelt mathematische Fragen, die im Zusammenhang mit der Strahlungsru¨ck-
wirkungundderPaarerzeugunginderklassischenFeldtheoriesowieinderQuantenfeldtheorie
stehen.SiebestehtauszweiHauptteilen:
Im ersten Teil werden klassische Dynamiken studiert, die es ermo¨glichen den Effekt der Strah-
lungsru¨ckwirkungzubeschreiben:Fu¨rklassische,nicht-rotierende,starreKo¨rpergebenwirex-
pliziteDarstellungsformelnfu¨rstarkeLo¨sungenderMaxwellGleichungenanundbeweisendie
globale Existenz und Eindeutigkeit von starken Lo¨sungen der Maxwell-Lorentz Gleichungen
in einem allgemeinen Rahmen. Dieser erlaubt es eine Klasse von Lo¨sungen unendlicher Ener-
gien, negativen Massen der Ladungen, sowie beide Fa¨lle, Maxwell-Lorentz Gleichungen mit
Selbstwechselwirkung (ML+SI) und ohne (ML-SI) zu behandeln. Wir beweisen weiter, dass
mo¨glicheWheeler-FeynmanLo¨sungenmitbeschra¨nktenBeschleunigungenundImpulsenauch
Lo¨sungenderML-SIGleichungendarstellen.Diesermo¨glichteineeindeutigeCharakterisierung
der Wheeler-Feynman Lo¨sungen anhand von Ort, Impuls, und ihrer Wheeler-Feynman Felder
zueinembestimmtenZeitpunkt.ZudemgebenwireineUmformulierungderWheeler-Feynman
Elektrodynamikfu¨rstarreundnicht-rotierendeLadungenineinAnfangswertproblemfu¨rNew-
tonsche Cauchy Daten an und beweisen damit die Existenz von Wheeler-Feynman Lo¨sungen
aufendlichenZeitintervallenentsprechendNewtonscherCauchyDaten.Wirdiskutierenwiemit
HilfedieserVorgehensweisedieglobaleExistenzvonLo¨sungenzudenWheeler-FeynmanGlei-
chungengezeigtwerdenko¨nnte.
ImzweitenTeilstudierenwirQuantendynamikenfu¨rSystememitunendlichvielenDiracElek-
tronen, die nur mit einem vorgeschriebenen a¨ußeren Feld wechselwirken, welches Paarerzeu-
gung ermo¨glicht: Wir konstruieren die Zeitentwicklung fu¨r die zweitquantisierte Dirac Glei-
chunginAbha¨ngigkeitvoneinemglattenundkompakten,zeitabha¨ngigen,elektrodynamischen
Viervektorfeld. In fru¨heren Arbeiten auf diesem Gebiet (Ruijsenaars) wurde die Shale-
Stinespring Bedingung beru¨cksichtigt, und es wurde gezeigt, dass die Einteilchenzeitentwick-
lung genau dann auf den Fock Raum gehoben werden kann, wenn die magnetischen Kompo-
nentendesa¨ußerenFeldesnullsind.DaswesentlicheHindernisbeidieserKonstruktionist,dass
eswedereinenausgezeichnetenDiracSee,d.h.FockRaumVakuum,nocheineausgezeichnete
Polarisationgibt.AusdiesemGrundistdieSchlu¨sselidee(wiebereitsvonFierzundScharfvor-
geschlagen), diese Zeitentwicklung zwischen zeitlich variierenden Fock Ra¨umen umzusetzen.
Wir zeigen, dass diese Umsetzung bis auf eine Phase eindeutig ist. Alle sich dadurch ergeben-
denU¨bergangsratensindeindeutigundendlich.
IneinemletztenTeilgebenwirausunseremBlickwinkeleinenkurzenAusblickaufeinediver-
genzfreie, elektrodynamische Theorie u¨ber Punktladungen, welche beide Pha¨nomene, sowohl
Strahlungsru¨ckwirkungalsauchPaarerzeugungbeschreibt.SiebasiertaufderIdee,dassderDi-
racSeedasvonWheelerundFeynmaneingefu¨hrteAbsorbermediumdarstellt.Diepra¨sentierten
mathematischenResultateko¨nntenalsersteSchritteindieseRichtungangesehenwerden.
Acknowledgements
This work would not have been possible without the help of many people who contributed in
manyandvariousways. Itisapleasuretospelloutmygratitude.
Myfirstthanksbelongtomyparents,Rolf-RainerDeckertandWaltraud-EmmaGo¨hde-Deckert,
who never missed a chance to spark my interest in nature. This together with their loving and
alsofinancialsupportpavedmywaytophysics.
MyverynextthanksareduetoProf. Dr. DetlefDu¨rrwhohasbeenmypatientteacherinphysics
andmathematicsoverthelasttenyears. DuringthistimehehasbeenalighthouseontheGreek
coastline guiding through the shallow waters of modern natural philosophy. I am very grateful
heintroducedmetoWheeler-Feynmanelectrodynamicsaswellasforhisuntiringlysupervision.
TheGermanwordDoktorvaterexpressesverywellwhathemeanstome.
At this place I also want to thank Michaela Shakina Changazi for her love and her enduring
encouragement. Duringthelastfiveyearsshecreatedtheenvironmentthatallowedmetofocus
onmyworkwhileshewasproofreadingallofit.
PartIofthisworkwasco-supervisedbyProf. Dr. GernotBauerwhogavetheideaofreformu-
lating Wheeler-Feynman electrodynamics as an initial value problem. Numerous enlightening
discussionswithhim,hiscommentsandcriticismandhismanyinvitationstoMu¨nsterprovided
thebackboneofthispartofthework, whichIthereforeconsideracollaborationwithProf. Dr.
GernotBauerandProf. Dr. DetlefDu¨rr. AtthisplaceIneedtothankalsoDr. MartinKolbfor
manyvaluablediscussionsaboutfunctionaldifferentialequations,weightedfunctionspacesand
forproofreadingtheseparts.
The mathematical results of Part II of this work emerged from a collaboration with Prof. Dr.
Detlef Du¨rr, Prof. Dr. Franz Merkl and Prof. Dr. Martin Schottenloher. In this regard I espe-
ciallyneedtothankProf. Dr. FranzMerklforourregularcomputationsessionsonWednesday
afternoons. ThanksalsotoSo¨renPetratandTorbenKru¨gerfordiscussionsandforproofreading.
Furthermore, I am very grateful to Prof. Dr. Herbert Spohn for valuable discussions about
classical and quantum field theory and his support during the last five years, as well as to Prof.
Martin Faessler and Dr. Reiner Geyer for a wonderful and enlightening stay at CERN in the
COMPASSgroup.
Many thanks go also to: BayEFG of the Universita¨t Bayern for partly funding, Maaneli Der-
akhshani for pointing out to me Barut’s works on quantum electrodynamics and its similarities
to Wheeler-Feynman electrodynamics, Veronika Du¨rr for tender care while Prof. Dr. Du¨rr and
IwereworkingtogetherinLandsbergandforhercompanyduringtheMu¨nstervisits,Matthias
Jungeforbeinganextraordinarygoodhighschoolteacherinphysicsandmathematics(intimes
of little appreciation for non-profit use of mathematics) and also for providing the motivation
for my studies, and finally our work group here in Munich (roughly time-ordered): My former
room mate Dr. Tilo Moser, the master of counterexamples, for great times and joint discov-
eries of well-known facts in the big world of mathematics, Dr. Angelo Bassi for our past and
present collaborations, for being a friend and his great Italian dinners, Dr. Peter Pickl, the Mr.
Sunshine and our former senior officer, for our past and present collaborations and for creating
a cheerful group atmosphere (Losinj!), Dr. Martin Kolb, the living bibliography, for being a
fantasticcolleague,forpresentcollaborationandmanyinvaluablemathematicalinsights,Sarah
Ro¨mer for her steady chocolate supply during hard times, Nicola Vona for his plenty ideas on
Wheeler-Feynmanelectrodynamics(nowadaysweareknownasthedynamicduo!) andkeeping
ourroomsotidy, RobertGrummtforourdiscussionsandhisgreatcompanyontheconference
inDresden, andfinallyChristianBeck, NiklasBoers, DustinLazarovici, TorbenKru¨ger, So¨ren
PetratandGeorgVolkertforinterestingdiscussionsandforbeinggreatfellowstudents.
Style of Writing
Althoughthisdoctoralthesisiswrittenbyonlyoneauthor,thechosenformofwritingemploys
theuseoffirstpersonpluralthroughouttheworkfortworeasons: First,researchisneverdoneby
asinglepersonalone. Inthissensephraseslike“weconclude”areusedtorecallallpeoplewho
contributed to a “conclusion” in one way or another. Second, for an interested reader phrases
like“weprove”arealsomeantinthesensethattheauthorandthereadergothrougha“proof”
togethertocheckifitiscorrect.
Contents
1 Preface 1
1.1 InteractionbetweenLightandMatter. . . . . . . . . . . . . . . . . . . . . . . 1
1.2 ScopeofthisWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
I RadiationReaction 5
2 CompleteAbsorptionandRadiationReaction 7
3 Maxwell-LorentzequationsofMotion 15
3.1 ChapterOverviewandResults . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 SolutionstotheMaxwell-LorentzequationsofMotion . . . . . . . . . . . . . 18
3.3 AGlobalExistenceandUniquenessResultonBanachSpaces . . . . . . . . . 21
3.4 TheSpacesofWeightedSquareIntegrableFunctions . . . . . . . . . . . . . . 25
3.5 ProofofMainTheoremandRegularity. . . . . . . . . . . . . . . . . . . . . . 30
3.6 ConclusionandOutlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Wheeler-FeynmanEquationsofMotion 43
4.1 ChapterOverviewandResults . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 FunctionalDifferentialEquations . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.1 Wheeler-FeynmanToyModel . . . . . . . . . . . . . . . . . . . . . . 53
4.2.2 ReformulationinTermsofanInitialValueProblem . . . . . . . . . . . 55
4.3 Wheeler-FeynmanInitialFields . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3.1 SolutionstotheMaxwellequations . . . . . . . . . . . . . . . . . . . 60
4.3.2 UniqueIdentificationofWheeler-FeynmanSolutions . . . . . . . . . . 70
4.4 ExistenceofWheeler-FeynmanInitialFields . . . . . . . . . . . . . . . . . . 73
4.4.1 TheMaxwellFieldsoftheMaxwell-LorentzDynamics . . . . . . . . . 73
4.4.2 NewtonianCauchyData: Wheeler-FeynmanInteractionforFiniteTimes 75
4.5 ConclusionandOutlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5 AppendixforPartI 95
5.1 EquationsofMotioninaSpecialReferenceFrame . . . . . . . . . . . . . . . 95
5.2 MissingProofsandComputationsforSection3.2 . . . . . . . . . . . . . . . . 97
5.3 MissingComputationforSection4.3.1 . . . . . . . . . . . . . . . . . . . . . . 97
5.4 MissingLemmasandProofsforSection4.4.2 . . . . . . . . . . . . . . . . . . 100
II PairCreation 103
6 AbsorberSubsystemsandEffectiveFields 105
7 Time-EvolutionofDiracSeasinanExternalField 111
7.1 ChapterOverviewandResults . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.2 ExplicitConstructionoftheAbsorber . . . . . . . . . . . . . . . . . . . . . . 116
7.2.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.2.2 OperationsfromtheLeftandfromtheRight . . . . . . . . . . . . . . 125
7.2.3 LiftCondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.3 TheTime-EvolutionofDiracSeas . . . . . . . . . . . . . . . . . . . . . . . . 130
7.3.1 One-ParticleTime-Evolution . . . . . . . . . . . . . . . . . . . . . . . 130
7.3.2 IdentificationofPolarizationClasses. . . . . . . . . . . . . . . . . . . 144
7.3.3 TheSecondQuantizedTime-Evolution . . . . . . . . . . . . . . . . . 150
7.4 PairCreationProbabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.5 GaugeTransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.6 ConclusionandOutlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
III Outlook 157
8 ElectrodynamicAbsorberTheory 159
8.1 ShortReviewofStepsTowardsanAbsorberQuantumElectrodynamics . . . . 159
Notation 167
Bibliography 171