Quantum Field Theory I: Foundations and Abelian and Non-Abelian Gauge Theories PDF
Preview Quantum Field Theory I: Foundations and Abelian and Non-Abelian Gauge Theories
Graduate Texts in Physics Edouard B. Manoukian Quantum Field Theory I Foundations and Abelian and Non-Abelian Gauge Theories Graduate Texts in Physics Serieseditors KurtH.Becker,PolytechnicSchoolofEngineering,Brooklyn,USA SadriHassani,IllinoisStateUniversity,Normal,USA BillMunro,NTTBasicResearchLaboratories,Atsugi,Japan RichardNeeds,UniversityofCambridge,Cambridge,UK Jean-MarcDiMeglio,UniversitéParisDiderot,Paris,France WilliamT.Rhodes,FloridaAtlanticUniversity,BocaRaton,USA SusanScott,AustralianNationalUniversity,Acton,Australia H.EugeneStanley,BostonUniversity,Boston,USA MartinStutzmann,TUMünchen,Garching,Germany AndreasWipf,Friedrich-Schiller-UnivJena,Jena,Germany GraduateTexts inPhysics Graduate Texts in Physics publishes core learning/teachingmaterial for graduate- andadvanced-levelundergraduatecoursesontopicsofcurrentandemergingfields within physics, both pure and applied. These textbooks serve students at the MS- or PhD-level and their instructors as comprehensive sources of principles, definitions,derivations,experimentsandapplications(asrelevant)fortheirmastery and teaching, respectively. International in scope and relevance, the textbooks correspondtocoursesyllabisufficientlytoserveasrequiredreading.Theirdidactic style, comprehensiveness and coverage of fundamental material also make them suitable as introductions or references for scientists entering, or requiring timely knowledgeof,aresearchfield. Moreinformationaboutthisseriesathttp://www.springer.com/series/8431 Edouard B. Manoukian Quantum Field Theory I Foundations and Abelian and Non-Abelian Gauge Theories 123 EdouardB.Manoukian TheInstituteforFundamentalStudy NaresuanUniversity Phitsanulok,Thailand ISSN1868-4513 ISSN1868-4521 (electronic) GraduateTextsinPhysics ISBN978-3-319-30938-5 ISBN978-3-319-30939-2 (eBook) DOI10.1007/978-3-319-30939-2 LibraryofCongressControlNumber:2016935720 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland Preface to Volume I This textbook is based on lectures given in quantum field theory (QFT) over the yearsto graduatestudents in theoretical and experimentalphysics. The writing of thebookspreadoverthreecontinents:NorthAmerica(Canada),Europe(Ireland), andAsia(Thailand).QFTwasbornabout90yearsago,whenquantummechanics met relativity, and is still going strong. The book covers, pedagogically,the wide spectrum of developments in QFT emphasizing, however, those parts which are reasonablywellunderstoodandforwhichsatisfactorytheoreticaldescriptionshave beengiven. The legendary Richard Feynman in his 1958 Cornell, 1959–1960 Cal Tech lectures on QFT of fundamentalprocesses, the first statement he makes, the very first one, is that the lectures cover all of physics.1 One quickly understands what Feynmanmeant by coveringall of physics. The role of fundamentalphysics is to describe the basic interactions of Nature and QFT, par excellence, is supposed to do just that. Feynman’s statement is obviously more relevant today than it was then, since the recent common goal is to provide a unified description of all the fundamentalinteractionsinnature. The book requires as background a good knowledge of quantum mechanics, includingrudimentsoftheDiracequation,aswellaselementsoftheKlein-Gordon equation, and the reader would benefit much by reading relevant sections of my earlierbook:QuantumTheory:AWideSpectrum(2006),Springerinthisrespect. ThisbookdiffersfromQFTbooksthathaveappearedinrecentyears2inseveral respectsand,inparticular,itofferssomethingnewinitsapproachtothesubject,and the reader hasplenty of opportunityto be exposedto manytopicsnotcovered,or 1R.P.Feynman,TheTheoryofFundamentalProcesses,TheBenjamin/CummingsPublishingCo., MenloPark,California.6thPrinting(1982),page1. 2Some of the fine books that I am familiar with are: L. H. Ryder, Quantum Field Theory; S. Weinberg, The Quantum Theory of Fields I (1995) & II (1996), Cambridge: Cambridge UniversityPress;M.PeskinandD.V.Schroeder,AnIntroductiontoQuantumFieldTheory,New York:WestviewPress(1995);B.DeWitt,TheGlobalApproachtoQuantumFieldTheory,Oxford: OxfordUniversityPress(2014). v vi PrefacetoVolumeI justtouchedupon,instandardreferences.Somenotabledifferencesareseen,partly, fromuniquefeaturesinthefollowingmaterialincludedinours: (cid:129) The very elegant functional differential approach of Schwinger, referred to as the quantum dynamical (action) principle, and its underlying theory are used systematically in generating the so-called vacuum-to-vacuum transition amplitudeofbothabelianandnon-abeliangaugetheories,inadditiontothewell- knownfunctionalintegralapproachofFeynman,referredtoasthepath-integral approach, which are simply related by functional Fourier transforms and delta functionals. (cid:129) Transitionamplitudesarereadilyextractedbyadirectexpansionofthevacuum- to-vacuumtransitionamplitudeintermsofaunitaritysum,whichismostclosely relatedtoactualexperimentalsetupswithparticlesemittedanddetectedpriorand after a given process and thus represent the underlying physics in the clearest possibleway. (cid:129) Particular emphasis is put on the concept of a quantum field and its particle content,bothphysicallyandtechnically,asprovidinganappropriatedescription ofphysicalprocessesatsufficientlyhighenergies,forwhichrelativitybecomes the indispensable language to do physics and explains the exchange that takes placebetweenenergyandmatter,allowingthecreationofanunlimitednumber of particles such that the number of particles need not be conserved, and for which a variable number of particles may be created or destroyed. Moreover, quantummechanicsimpliesthat a wavefunctionrenormalizationarises in QFT fieldindependentofanyperturbationtheory–apointnotsufficientlyemphasized intheliterature. (cid:129) Therationaleofthestationaryactionprincipleandemergenceoffieldequations, viafieldvariationsoftransformationfunctionsandgeneratorsoffieldvariations. Theintroductionofsuchgeneratorslead,selfconsistently,tothefieldequations. Suchquestionsareaddressedas:“Whyisthevariationoftheaction,withinthe boundariesoftransformationfunctions,setequaltozerowhicheventuallyleads totheEuler-Lagrangeequations?”,“HowdoestheLagrangiandensityappearin the formalism?” “What is the significance in commuting/anti-commutingfield componentswithintheinteractionLagrangiandensityinatheoryinvolvingfield operators?”Thesearesomeofthequestionsmanystudentsseemtoworryabout. (cid:129) Apanoramaofallthefieldsencounteredinpresenthigh-energyphysics,together withthedetailsoftheunderlyingderivationsaregiven. (cid:129) Schwinger’s point splitting method of currents is developed systematically in studyingabelianandnon-abeliangaugetheoriesanomalies.Moreover,anexplicit experimentaltestofthepresenceofananomalyisshownbyanexample. (cid:129) DerivationoftheSpin&StatisticsconnectionandCPTsymmetry,emphasizing for the latter that the invariance of the action under CPT transformation is not sufficientforCPTsymmetry,butonehasalsotoconsidertherolesofincoming andoutgoingparticles. PrefacetoVolumeI vii (cid:129) Thefine-structureeffectivecoupling ˛ ' 1=128 athighenergycorresponding tothemassoftheneutral Z0 vectorbosonbasedonallthechargedleptonsand allthosecontributingquarksofthethreegenerations. (cid:129) Emphasis is put on renormalization theory, including its underlying general subtractionsscheme,oftenneglectedintreatmentsofQFT. (cid:129) Elementary derivation of Faddeev-Popov factors directly from the functional differential formalism, with constraints, and their modifications, and how they mayevenariseinsomeabeliangaugetheories. (cid:129) A fairly detailed presentation is given of “deep inelastic” experiments as a fundamentalapplicationofquantumchromodynamics. (cid:129) Schwingerlineintegrals,originofWilsonloops,lattices,andquarkconfinement. (cid:129) Neutrino oscillations,3 neutrino masses, neutrino mass differences, and the “seesawmechanism.” (cid:129) QCDjetsandpartonsplitting,includinggluonsplittingtogluons. (cid:129) Equal importance is put on both abelian and non-abelian gauge theories, witnessingthewealthofinformationalsostoredintheabeliancase.4 (cid:129) A most important, fairly detailed, and semi-technical introductory chapter is given which traces the development of QFT since its birth in 1926 without tears, in abelian and non-abelian gauge theories, including aspects of quantum gravity,aswellasexaminingtheimpactofsupersymmetry,stringtheory,andthe developmentofthetheoryofrenormalization,asapedagogicalstrategyforthe readertobeabletomasterthebasicideasofthesubjectattheoutsetbeforethey areencounteredinglorioustechnicaldetailslater. (cid:129) Solutionstoalltheproblemsaregivenrightattheendofthebook. With the mathematical rigor that renormalization has met over the years and thereasonableagreementbetweengaugetheoriesandexperiments,theunderlying theories are in pretty good shape. This volume is organized as follows. The first introductory chapter traces the subject of QFT since its birth, elaborating on many of its important developments which are conveniently described in a fairly simplelanguageandwillbequiteusefulforunderstandingtheunderlyingtechnical details of the theory covered in later chapters including those in Volume II. A preliminarychapterfollowswhichincludesthestudyofsymmetrytransformations in the quantum world, as well as of intricacies of functional differentiation and functional integration which are of great importance in field theory. Chapter 3 deals with quantum field theory methods of spin 1/2 culminating in the study of anomaliesinthequantumworld.Thelatterreferstothefactthataconservationlaw 3Itisratherinterestingtopointoutthatthetheoryofneutrinooscillationswaswrittenupinthis bookmuchearlierthanthe2015NobelPrizeinPhysicswasannouncedonneutrinooscillations. 4Withthedevelopmentofnon-abeliangaugetheories,unfortunately,itseemsthatsomestudents arenotevenexposedtosuchderivationsasofthe“Lambshift”andofthe“anomalousmagnetic momentoftheelectron”inQED. viii PrefacetoVolumeI in classical physics does not necessarily hold in the quantum world. Chapter 4, a critical one, deals with the concept of a quantum field, the Poincaré algebra, and particle states. Particular attention is given to the stationary action principle as well as in developing the solutions of QFT via the quantum dynamical principle. This chapter includes the two celebrated theorems dealing with CPT symmetry and of the Spin & Statistics connection. A detailed section is involved with the basicquantumfieldsoneencountersinpresentdayhigh-energy/elementary-particle physicsandshouldprovideausefulreferencesourceforthereader.Chapter5treats abelian gauge theories (QED, scalar boson electrodynamics) in quite details and includes, in particular, the derivations of two of the celebrated results of QED which are the anomalous magnetic moment of the electron and the Lamb shift. Chapter 6 is involvedwith non-abeliangaugetheories (electroweak,QCD, Grand unification).5 Such important topics are included as “asymptotic freedom,” “deep inelastic”scattering,QCDjets,partonsplittings,neutrinooscillations,the“seesaw mechanism”andneutrinomasses,Schwinger-lineintegrals,Wilsonloops,lattices, and quark confinement. Unification of coupling parameters of the electroweak theoryandofQCDarealsostudied,aswellasofspontaneoussymmetrybreaking inbothabelianandnon-abeliangaugetheories,andofrenormalizabilityaspectsof bothgaugetheories,emphasizingtheso-calledBRS transformationsforthe latter. We make it a point, pedagogically, to derive things in detail, and some of such details are relegated to appendices at the end of the respective chapters with the main results given in the sections in question. Five generalappendices,at the end of this volume, cover some additional important topics and/or technical details. In particular, I have included an appendix covering some aspects of the general theory of renormalization and its underlying subtractions scheme itself which is oftenneglectedinbooksonQFT.Fortunately,myearlierbook,withproofsnotjust words, devoted completely to renormalization theory – Renormalization (1983), Academic Press – may be consulted for more details. The problems given at the end of the chapters form an integral part of the book, and many developmentsin thetextdependontheproblemsandmayinclude,inturn,additionalmaterial.They shouldbeattemptedbyeveryseriousstudent.Solutionstoalltheproblemsaregiven right at the end of the book for the convenience of the reader. The introductory chaptertogetherwiththeintroductionstoeachchapterprovidethemotivationand thepedagogicalmeanstohandlethetechnicalitiesthatfollowtheminthetexts. I hope this book will be useful for a wide range of readers. In particular, I hope that physics graduate students, not only in quantum field theory and high- energy physics, but also in other areas of specializations will also benefit from it as,accordingtomyexperience,theyseemtohavebeenleftoutofthisfundamental area of physics, as well as instructors and researchers in theoretical physics. The contentofthis volumemaybe coveredin one-year(two semesters) quantumfield theorycourses. 5QEDandQCDstand,respectively,forquantumelectrodynamicsandquantumchromodynamics. PrefacetoVolumeI ix In Volume II, the reader is introducedto quantum gravity,supersymmetry,and string theory,6 which although may, to some extent, be independently read by a reader with a good background in field theory, the present volume sets up the language,thenotation,providesadditionalbackgroundforintroducingthesetopics, andwillcertainlymakeitmucheasierforthereadertofollow.Inthistwo-volume set, aimingforcompletenessincoveringthe basicsofthesubject,I haveincluded topics from the so-called conventionalfield theory (the classics) to ones from the modern or the new physics which I believe that every serious graduate student studyingquantumfieldtheoryshouldbeexposedto. Withoutfurtherado,andwithallduerespecttothelegendarysongwriterCole Porter,letusfindout“whatisthisthingcalledQFT?” EdouardB.Manoukian 6Entitled:QuantumFieldTheoryII:IntroductionstoQuantumGravity,Supersymmetry,andString Theory”(2016),Springer.
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