Kazuhiko Nishijima Masud Chaichian Anca Tureanu Editors Quantum Field Theory By Academician Prof. Kazuhiko Nishijima A Classic in Theoretical Physics Quantum Field Theory Kazuhiko Nishijima Quantum Field Theory By Academician Prof. Kazuhiko Nishijima - A Classic in Theoretical Physics Masud Chaichian • Anca Tureanu Editors Author KazuhikoNishijima UniversityofTokyo Tokyo,Japan Editors MasudChaichian AncaTureanu UniversityofHelsinki UniversityofHelsinki Helsinki,Finland Helsinki,Finland Translatedby YukiSato TokuyamaCollege NationalInstituteofTechnology Yamaguchi,Japan ISBN978-94-024-2189-7 ISBN978-94-024-2190-3 (eBook) https://doi.org/10.1007/978-94-024-2190-3 OriginallypublishedbyKinokuniyaCompanyLtd.,Tokyo,Japan,1987 Japanese original edition BA NO RIRON by Kazuhiko Nishijima published in 1987 by Kinokuniya CompanyLtd.,Tokyo ©SpringerNatureB.V.2023 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewhole orpart ofthematerial isconcerned, specifically therights oftranslation, reprinting, reuse ofillustrations, recitation, broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. BackCover:PhotoofProf.Nishijima,2008(courtesyoflateMrs.HidekoNishijima) ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureB.V. Theregisteredcompanyaddressis:VanGodewijckstraat30,3311GXDordrecht,TheNetherlands Foreword In the Preface, Professors Masud Chaichian and Anca Tureanu describe in vivid detail how the English version of this book on quantum field theory by the great Japanese physicist Kazuhiko Nishijma evolved as a labor of love on their parts. Nishijima enjoyed visiting Helsinki and Finland (Who doesn’t? I certainly do.) regularly and was often seen consulting a physics book in Japanese, of which he was actually the author. (I can tell you from personal experience that authors like to read their own books; I certainly find my books ridiculously clear and easy to understand.) Chaichian and Tureanu urged Nishijima to have the book translated into English. After Nishijima’s death in 2009, they found a translator, haditproofread,improvedtheEnglish,arrangedforapublisher,andinvitedoneof themosteminenttheoreticalphysicistsofourtimes,ProfessorYoichiroNambu,to writeaForeword. AfterNambu’suntimelydeathin2015,theyaskedmetostepin,perhapsbecause Ihavealsowrittenatextbookonquantumfieldtheory.Anyreaderqualifiedtoread this book knows full well that I am far from the level of achievements attained by Yoichiro Nambu, for whom I have the greatest admiration as a person and as a physicist. Nevertheless, out of respect for Nishijma, Nambu, Chaichian, and Tureanu,Iagreedtodoso. IbelievethatDickFeynmansaidthatlearningphysicsissomethinglikepainting a house. (In case you have no experience painting houses, one coat is never enough. Only the most shoddy house painters would do that.) John Wheeler told mesomethingsimilar:oneshouldnotimbibephysicslikesippingwaterfromacup, but more like drinking from a fire hydrant.Anybodywho tells you that he or she could learn quantumfield theory by reading a single textbookis not only kidding you,butalsohimselforherself.Duringmyfouryearsingraduateschool,Iattended Julian Schwinger’s course on quantum field theory four times, not because I was especially stupid, but because quantum field theory is a profound subject. In this spirit, I strongly urge students, and also physics professors who claim that they understand quantum field theory, to read this book in addition to whatever other books they have read or possibly in parallel with other books they are reading. I v vi Foreword alsofindthatthisbooktreatssometopicsthatarenotdiscussedindetail,oromitted entirely,inmorerecentbooksonquantumfieldtheory. SantaBarbara,CA,USA AnthonyZee July2022 Preface to the English Edition During several years of our collaboration with Prof. Kazuhiko Nishijma, we had noticedthat always he was carryinga bookin Japanese, which he consultedfrom timetotimeondifferentsubjects.Prof.NishijimausedtovisitusattheUniversity of Helsinki regularly in the years 1985–2008.During one of his visits, he finally left a copy of that book with us in Helsinki to keep it for him for his future use. Notknowingwhich bookit was, we asked aboutit and he explainedthatit was a bookonQuantumField Theoryhehadwritten.Uponourquestionwhetheritwill betranslatedintoEnglish,Prof.Nishijimasaidthatitwashiswishtotranslatethe bookhimself. After Prof. Nishijima passed away in Tokyo on 15 February 2009, the book and Prof. Nishijima’s wish sadly came to our mind and we reached a wishful thoughtofgettingthebookbyallmeanstranslatedintoEnglishasatributetoProf. Nishijima.Prof.YoichiroNambu,the2008laureateoftheNobelPrizeinPhysics, knew Prof. Nishijima very well and in the early 1960’s was nominating him for the Nobel Prize for the discovery of the strangeness quantum number. Knowing this,wediscussedtheplanwithProf.Nambu,andhesupportedtheideawithgreat enthusiasm,mentioningthatheknewthebookwellandthathewouldliketowritea forewordforitsEnglishedition.However,healsomentionedaboutthewell-known difficultyoftranslatingfromJapanesetoanyotherlanguageofanygroupofEnglish kind, and that it was for this reason that Nishijima wanted to translate his book himself. We contacted Springer-Verlag about publishing the translated book and they consulted Prof. Nambu for a review. Prof. Nambu wrote a very strong evaluation, mentioningthetransparencyofthebookand,inseveralparts,theoriginalityofthe presentationsofthesubject.Thenexttaskwastofindapersonwhocouldtranslate thebook,beinga physicistwithagoodgraspofEnglish,especiallyasregardsthe terminologyusedinthebook.ItwasonlythankstotheeffortsofProf.Nambu,who advertised the projectin Japan, asking for volunteers, that a translator was found. Eventually,a young researcher in particle physics at Nagoya University, Dr. Yuki Sato,wrotetousmentioningthathewouldliketotakethetaskmerelyasasignof vii viii PrefacetotheEnglishEdition respecttoProf.Nishijima,oneofJapan’sgreatestphysicistswhomadesignificant contributionstotheoreticalphysics. The translation of the bookbyDr. YukiSato, togetherwith us the undersigned as editors going through it and checking the text and all the formulas again and again, as well as adding the reference list, took a few years and meanwhile Prof. Nambu passed away. Instead, Prof. Anthony Zee from University of California, SantaBarbara,mostkindlyacceptedtowritetheforewordforthebook. After the translationof the book,anotherprofessionalnative Englishtranslator andaphysicistbyeducation,Mr.StephenLyle,wentthroughthewholetranslated text and performed a more complete polishing of the text into a fine English language. We express our utmost thanks to Mr. Stephen Lyle for his devotion to perfection.We would like to thank Dr. Ramon Khanna, executiveeditor, and Ms. ChristinaFehling,editorialassistant,atSpringerNaturefortheirkindsupportand adviceduringthe whole publicationprocess,and also for their unlimitedpatience withseveraldelaysinourpreparingtheEnglisheditionofthebooktosendforprint, withtheaimthatitwillbecomeagenuinetributetoProf.Nishijima. Duringthe preparationofthe Englisheditionofthe bookwe havereceivedthe encouragementof severalphysicistsin Japan, to whom we are mostgrateful.Our specialgratitudegoestoProf.MisaoSasakifromKyotoUniversity,andProf.Kei- IchiKondofromChibaUniversity,fortheirconstantsupportandmuchhelpduring severalyears. Helsinki,Finland MosheM.Chaichian July2022 AncaTureanu Preface of the Author One of the important concepts in quantum mechanics is the dual particle-wave natureofmatter.Thesameobjectistreatedasaparticleinonecaseandasawave in another case, which is not the case in classical physics. This was one of the launchpadsofquantummechanics.Itstypicalexampleislight:inclassicalphysics it was mainly described by the wave theory,althoughthere existed also Newton’s particle theory. However, once Einstein’s photon hypothesis was proposed at the beginning of this century,1 one could no longer claim that light is a wave or a particle. In this way quantum mechanics made its appearance. In its framework, the electron, which is thoroughly treated as a particle in the classical theory, is always describedas a wave. With that the followingproblemcomes up:How can we introduce the particle-wave duality in the same theory? Quantum field theory answersthisquestion. Quantumfieldtheorywasdevelopednolaterthanthebirthofquantummechan- ics. It has experiencedmany changesup to the present. In the early stage, studies were entirely restricted to the area of electrodynamics,and turned out to be quite successful.Nevertheless,earlyquantumfieldtheoryreachedadeadlockcalledthe problem of divergences. However, with the advent of the renormalisation theory afterthewar,quantumelectrodynamicsbasedonperturbationtheoryreachedcom- pletion. On the other hand, after pions were created by accelerators, the struggles offieldtheorystartedagain.Thereasonswasthattheperturbativeapproachcannot beusedforstronginteractions,andoneencounteredthedifficultyofextendingthe renormalisationtheorytoanapproximationmethodbeyondtheperturbationtheory. Asaconsequence,fieldtheorywasstagnantforawhile;instead,theS-matrixtheory based on dispersion relations became pre-eminent. In addition, after hundreds of “elementary”particleswerecreatedbyaccelerators,itbecamenecessarytoanswer the question:What on earth are fundamentalparticles? And this is how the quark ThisistheonlyPrefacewrittenbyProf.NishijimainJapanesetothe1steditionpublishedin1987 inTokyo;the8theditionwithextendedchapterswaspublishedin1997inTokyo. 1“Thiscentury”meansthetwentiethcentury. ix x PrefaceoftheAuthor model appeared. On the other hand, from the studies of weak interactions, gauge theorieswereintroduced.Thisbookcoversasurveyofquantumfieldtheoryrelated tothedevelopmentsabove. After the emergence of the gauge theory, there have been quite many devel- opments toward new directions. For example, the study of monopoles, solitons, and instantonsrelated to the topologicalnature of classical gauge theories; lattice gaugetheory,in which gaugeinvarianceis giventhe leadingrole to the detriment of Lorentz invariance; applications of grand unified theories to cosmology; more recently, supersymmetric theories, supergravity theories and superstring theories, andsoon.These theoriesare developing;each ofthemissufficientlyvastalready astorequireanindependentbook.Imyselfdonotyetunderstandthemverywell, sothatIhaveomittedtheminthisbook,consideringthatmoreadequatepeoplewill writeaboutthem. AlthoughIpublished“FundamentalParticles”(1963)[1]and“FieldsandParti- cles”(1969)[2]withtheW.A.BenjaminCorporationastextbooksforelementary particle theory and quantum field theory, the contents of those books are by now outdated,andthetopicstobeemphasizedhavealsochangedin time.However,as fas as the fundamentallyinvariable parts are concerned,this book is based on the two books above. Additionally, since I have had a reluctance to write a book in theMathematicalLibraryinignoranceofmathematics,Ihavedecidedtowritethe bookbasednotonthe mathematicallogic butonthe physicallogic,onthe advice ofProfessorSeizoIto.Ithasbeenlongtimesincetherequesttowritethisbook,but ProfessorItohaspatientlywaited,andIwouldliketoexpressmyappreciationfor him. Inthe61yearoftheShowaera(1986),atRakuhoku, Tokyo,Japan KazuhikoNishijima