ProblemBook inQuantumField Theory Voja Radovanovic´ Problem Book in Quantum Field Theory Second Edition ABC VojaRadovanovic´ FacultyofPhysics UniversityofBelgrade Studentskitrg12-16 11000Belgrade SerbiaandMontenegro E-mail:[email protected] LibraryofCongressControlNumber:2007940156 ISBN978-3-540-77013-8 SpringerBerlinHeidelbergNewYork ISBN978-3-540-29062-91sted.SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright. Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember 9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:2)c Springer-VerlagBerlinHeidelberg2008 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:bytheauthorusingaSpringerLATEXmacropackage Coverdesign:eStudioCalamar,Spain Printedonacid-freepaper SPIN:12197873 543210 To my daughter Natalija Preface This Problem Book is based on the exercises and lectures which I have given to undergraduateandgraduatestudents ofthe Faculty ofPhysics,University ofBelgradeovermanyyears.Nowadays,there area lot ofexcellentQuantum Field Theory textbooks. Unfortunately, there is a shortageof ProblemBooks inthisfield,oneoftheexceptionsbeingtheProblemBookofChengandLi[7]. The overlapbetweenthis ProblemBookand [7] is verysmall,since the latter mostly deals with gauge field theory and particle physics. Textbooks usually contain problems without solutions. As in other areas of physics doing more problems in full details improves both understanding and efficiency. So, I feel that the absence of such a book in Quantum Field Theory is a gap in the literature. This was my main motivation for writing this Problem Book. To students: You cannot start to do problems without previous study- ing your lecture notes and textbooks. Try to solve problems without using solutions; they should help you to check your results. The level of this Prob- lem Book correspondsto the textbooks of Mandl and Show [15]; Greiner and Reinhardt [11] and Peskin and Schroeder [16]. Each Chapter begins with a short introduction aimed to define notation. The first Chapter is devoted to the LorentzandPoincar´esymmetries.Chapters2,3 and4 dealwith the rela- tivistic quantummechanicswithaspecialemphasisonthe Diracequation.In Chapter 5 we present problems related to the Euler-Lagrange equations and the Noether theorem. The following Chapters concern the canonical quanti- zation of scalar, Dirac and electromagnetic fields. In Chapter 10 we consider tree level processes, while the last Chapter deals with renormalization and regularization. There are many colleagues whom I would like to thank for their support and help. Professors Milutin Blagojevi´c and Maja Buri´c gave many useful ideasconcerningproblemsandsolutions.IamgratefultotheAssistantsatthe Faculty of Physics, University of Belgrade: Marija Dimitrijevi´c, Duˇsko Latas and Antun Balaˇz who checked many of the solutions. Duˇsko Latas also drew all the figures in the Problem Book. I would like to mention the contribution of the students: Branislav Cvetkovi´c,Bojan Nikoli´c, Mihailo Vanevi´c, Marko VIII Preface Vojinovi´c, Aleksandra Stojakovi´c, Boris Grbi´c, Igor Salom, Irena Kneˇzevi´c, Zoran Ristivojevi´c and Vladimir Juriˇci´c. Branislav Cvetkovi´c, Maja Buri´c, MilutinBlagojevi´candDejanStojkovi´chavecorrectedmyEnglishtranslation of the Problem Book. I thank them all, but it goes without saying that all the errorsthat have creptin are my own.I wouldbe gratefulfor any readers’ comments. Belgrade Voja Radovanovi´c August 2005 Contents Part I Problems 1 Lorentz and Poincar´e symmetries.......................... 3 2 The Klein–Gordon equation ............................... 9 3 The γ–matrices............................................ 13 4 The Dirac equation ........................................ 17 5 Classical field theory and symmetries ..................... 25 6 Green functions ........................................... 31 7 Canonical quantization of the scalar field .................. 35 8 Canonical quantization of the Dirac field .................. 43 9 Canonical quantization of the electromagnetic field ........ 49 10 Processes in the lowest order of perturbation theory....... 55 11 Renormalization and regularization........................ 61 Part II Solutions 1 Lorentz and Poincar´e symmetries.......................... 67 2 The Klein–Gordon equation ............................... 77 3 The γ–matrices............................................ 85 X Contents 4 The Dirac equation ........................................ 93 5 Classical fields and symmetries ............................121 6 Green functions ...........................................131 7 Canonical quantization of the scalar field .................141 8 Canonical quantization of the Dirac field ..................161 9 Canonical quantization of the electromagnetic field ........179 10 Processes in the lowest order of the perturbation theory...191 11 Renormalization and regularization........................211 References.....................................................239 Index..........................................................241 Part I Problems