Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory

311 Pages·2011·1.566 MB·English

by  Tommy Ohlsson

RELATIVISTIC QUANTUM PHYSICS
From Advanced Quantum Mechanics to Introductory
Quantum Field Theory
Quantum physics and special relativity theory were two of the greatest break-
throughs in physics during the twentieth century and contributed to paradigm
shiftsinphysics.Thisbookcombinesthesetwodiscoveriestoprovideacomplete
descriptionofthefundamentalsofrelativisticquantumphysics,guidingthereader
effortlesslyfromrelativisticquantummechanicstobasicquantumfieldtheory.
Thebookgivesathoroughanddetailedtreatmentofthesubject,beginningwith
the classification of particles, the Klein–Gordon equation and the Dirac equation.
It then moves on to the canonical quantization procedure of the Klein–Gordon,
Dirac, and electromagnetic fields. Classical Yang–Mills theory, the LSZ formal-
ism, perturbation theory and elementary processes in QED are introduced, and
regularization, renormalization, and radiative corrections are explored. With exer-
cisesscatteredthroughthetextandproblemsattheendofmostchapters,thebook
isidealforadvancedundergraduateandgraduatestudentsintheoreticalphysics.
TOMMY OHLSSON is Professor of Theoretical Physics at the Royal Institute of
Technology(KTH),Sweden.Hismainresearchfieldistheoreticalparticlephysics,
especiallyneutrinophysicsandphysicsbeyondtheStandardModel.

RELATIVISTIC QUANTUM PHYSICS
From Advanced Quantum Mechanics to Introductory
Quantum Field Theory
TOMMY OHLSSON
RoyalInstituteofTechnology(KTH),Sweden
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,Sa˜oPaulo,Delhi,Tokyo,MexicoCity
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521767262
(cid:2)c T.Ohlsson2011
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithout
thewrittenpermissionofCambridgeUniversityPress.
Firstpublished2011
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloging-in-PublicationData
Ohlsson,Tommy,1973–
Relativisticquantumphysics:fromadvancedquantummechanicstointroductory
quantumfieldtheory/TommyOhlsson.
p. cm.
ISBN978-0-521-76726-2(Hardback)
1. Quantumtheory. I. Title.
QC174.12.O352011
530.12–dc23
2011018860
ISBN 978-0-521-76726-2Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
inthispublication,anddoesnotguaranteethatanycontentonsuch
websitesis,orwillremain,accurateorappropriate.
InmemoryofmyfatherDick

Contents
Preface pagexi
1 Introductiontorelativisticquantummechanics 1
1.1 Tensornotation 1
1.2 TheLorentzgroup 3
1.3 ThePoincarégroup 9
1.4 Casimiroperators 11
1.5 Generaldescriptionofrelativisticstates 12
1.6 IrreduciblerepresentationsofthePoincarégroup 13
1.7 One-particlerelativisticstates 16
Problems 21
Guidetoadditionalrecommendedreading 21
2 TheKlein–Gordonequation 22
2.1 Transformationproperties 24
2.2 Thecurrent 25
2.3 SolutionstotheKlein–Gordonequation 26
2.4 Chargedparticles 28
2.5 TheKleinparadox 30
2.6 Thepionicatom 34
Problems 38
Guidetoadditionalrecommendedreading 39
3 TheDiracequation 40
3.1 FreeparticlesolutionstotheDiracequation 45
3.2 Problems with the Dirac equation: the hole theory and the
Diracsea 50
3.3 Somegammagymnasticsandtracetechnology 52
3.4 Spinoperators 57
viii Contents
3.5 Orthogonalityconditionsandenergyprojectionoperators 61
3.6 RelativisticinvarianceoftheDiracequation 63
3.7 Bilinearcovariants 66
3.8 Electromagnetic structure of Dirac particles and charge
conjugation 68
3.9 Constantsofmotion 72
3.10 Centralpotentials 74
3.11 Thehydrogenicatom 77
3.12 TheWeylequation 86
3.13 Helicityandchirality 89
Problems 90
Guidetoadditionalrecommendedreading 92
4 Quantizationofthenon-relativisticstring 94
4.1 Equationofmotionforthenon-relativisticstring 94
4.2 Solutionstothewaveequation:normalmodes 97
4.3 Generalizedpositionsandmomenta 98
4.4 Quantization 99
4.5 Quantaasparticles 101
Problem 103
Guidetoadditionalrecommendedreading 104
5 Introductiontorelativisticquantumfieldtheory:
propagators,interactions,andallthat 105
5.1 Propagators 106
5.2 Lagrangians 109
5.3 Gaugeinteractions 111
5.4 ScatteringtheoryandMøllerwaveoperators 113
5.5 The S operator 115
Guidetoadditionalrecommendedreading 121
6 QuantizationoftheKlein–Gordonfield 122
6.1 Canonicalquantization 122
6.2 Fieldoperatorsandcommutators 126
6.3 Green’sfunctionsandpropagators 129
6.4 Theenergy–momentumtensor 132
6.5 Classicalexternalsources 134
6.6 ThechargedKlein–Gordonfield 135
Problems 135
Guidetoadditionalrecommendedreading 137