The Canonical Operator in Many-Particle Problems and Quantum Field Theory PDF
Preview The Canonical Operator in Many-Particle Problems and Quantum Field Theory
† VictorP.MaslovandOlegYu.Shvedov TheCanonicalOperatorinMany-ParticleProblemsandQuantumFieldTheory De Gruyter Expositions in Mathematics | Editedby LevBirbrair,Fortaleza,Brazil VictorP.Maslov,Moscow,Russia WalterD.Neumann,NewYorkCity,NewYork,USA MarkusJ.Pflaum,Boulder,Colorado,USA DierkSchleicher,Bremen,Germany KatrinWendland,Freiburg,Germany Volume 71 Victor P. Maslov and Oleg Yu. Shvedov† The Canonical Operator in Many-Particle Problems and Quantum Field Theory | MathematicsSubjectClassification2020 Primary:81Q20,81T15,81T70;Secondary:53D12,30H20 Authors † Prof.VictorP.Maslov OlegYu.Shvedov MoscowInst.ofElectron.&Mathematics Dept.ofAppliedMathematics Moscow109028 RussianFederation [email protected] ISBN978-3-11-076238-9 e-ISBN(PDF)978-3-11-076270-9 e-ISBN(EPUB)978-3-11-076274-7 ISSN0938-6572 LibraryofCongressControlNumber:2022934383 BibliographicinformationpublishedbytheDeutscheNationalbibliothek TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie; detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de. ©2022WalterdeGruyterGmbH,Berlin/Boston Typesetting:VTeXUAB,Lithuania Printingandbinding:CPIbooksGmbH,Leck www.degruyter.com | Inmemoryof LudwigFaddeev Preface to the English edition IhavedecidedtodedicatetheEnglishtranslationofthisbook,writtenjointlywith mylatepupil,OlegShvedov,tothememoryoftheremarkableexpertinmathematical physicsandquantumtheory,myclosefriendLudwigFaddeev.Thetwoofusalways exchangedournewideasandunderstoodeachotherperfectly.Inthepresentmono- graph,thenotionknownasthe“Faddeevtransformation”issubstantiallyused. Here,Iwouldliketociteafragmentofmyarticle“MyDearLudwig”publishedin thejournal,MathematicalNotes,in2017(Vol.101,no.6). AlthoughIknewthathewasterminallyill,thenewsofLudwigFaddeev’sdeathcameas ashocktome.Thefloorboardsspununderme,losingbalanceItriedandfailedtofindsupport fromthewallsoftheroom… Webecamefriendslongago,in1958.ThisishowFaddeevhimselfdescribedit:“Concerning conferences,Iwasveryluckyatthetime—attheendofthe1950sfunctionalanalysiswasflour- ishing,nationwideconferencesonthesubjectwereheldeveryyearinMoscow,Odessa,Kharkov. Participationinthemwasanexcellentchancetolearnfrommyeldersandmeetmycontempo- raries.AsImentionedbefore,inOdessaImetV.P.Maslov,webecamefriendsandhavesupported eachothereversince”(L.D.Faddeev,30yearsinmathematicalphysics,TrudyMat.Inst.Steklov 176,4–29,1987) Iamlookingthroughhisletters.And,forthefirsttime,understandthemeaningofthe Russiansaying“Whatiswrittenbypencan’tberubbedoutbyaxe.”Forinstance,Ilovehis handwriting—itreflectshischaracter,ashandwritingexpertsarecorrectinsaying.Beautifully handwrittenmathematicalformulas—suchawonder,somethingIcannotdo. Iwasthefirstwhodiscerned,inhisveryfirstpapers,histalent—tocreateabeautifulmath- ematicalframeandfillitwithphysicalmeaning.Mostimportantwasmathematicalbeautyand elegance.HewouldalwaysobjecttomewheneverImentionedhisremarkablephysicalintuition, stressingthatmostimportantforhimwasthebeautyofmathematicalstructure. In1967,FaddeevvisitedmeinMoscowandtoldmeabouthisunpublishedjointworkwith VictorPopov.Hisenthusiasmabouttheworkwascontagious.Itsmathematicalbeautywasinde- scribable,whiletheunderlyingphysicsweresovirtual,thattheobjectsitsauthorshaddiscovered becameknownas“Faddeev–Popovghosts.”These“ghosts”continuedtoroaminsidemyhouse themorningafterhisdeparture,anditbecamecleartomethatsuchbeautycannotbeaccidental. ThatnightwhenFaddeevwasdescribinghispaperwithPopov,heassertedthattheYang–Mills theorymustberelatedtonewparticles.Iama(still)livingwitnessofthisremarkableutterance. Theideasofthisworkledtotheawardin1999oftheNobelPrizeforclarifyingthequantum structureofelectroweakinteractions.ButtheprizewasawardedtotwoDutchphysicists,while thenameofFaddeev,tothegreatsurpriseofthephysicscommunity,didnotappearonthelistof prizewinnersunderthepretextthatPopovhaddiedbythen.Physicistswereshockedandcould notunderstandwhyFaddeev’snamehadbeenthrownoutoftheshortlistfortheprize. ItiswellknownthatCaton,whenaskedwhyamonumentinhishonorhadnotbeenerected duringhislifetime,answered“preciselybecausepeoplecouldthenaskwhyamonumentfor Catonwasnoterected,ratherthanaskwhyitwaserected.” Inourtime,itisnotcustomarytoconstructmonumentsortomakememorialmedalsfor livingindividuals.Inthatrespect,MaxPlanckandLudwigFaddeevareexceptionstotherule— medalsintheirnameappearedduringtheirlifetime. Inthe1980s,webothinventednewalgebraswithverygeneralcommutationrelations.Fad- deevaskedme,“Sowhowasthefirstoneofustodiscoverthesealgebras?”Ianswered,“You https://doi.org/10.1515/9783110762709-201 VIII | PrefacetotheEnglishedition were,ofcourse,becauseyoufoundpowerfulapplicationstoyourversionofthesealgebrasin thetheoryoffields.”Indeed,Faddeev’salgebrasdevelopedrapidly,whilemymoregeneralones werementionedinapaperbyFaddeevandTakhtadzhyan,butdidnothavetheresonanceIhad hopedfor. Faddeevneverchangedhistopicofstudy,andintuitivelyfeltatwhatpointofthefrontof scienceabreakthroughwouldbeimminent,apointthroughwhichitwouldbepossibletobreak awayfromstagnation.Andtheyoungscientificreserveswouldfollowhimintothebreach! Myreactiontohisfurtherdiscoverieswasecstaticor,ashewrote,“veryenthusiastic”;I awaitedimpatientlyforhisnewresults,whichgrewexponentiallywithtime.Asforme,Iwas movingdeeperanddeeperintostatisticalphysics,whichhecouldn’tstand,justasIcouldn’tin myyoungeryears,butlaterIovercamemyaversiontothesubjectandrewrotethescienceof “sleazydynamics,”asKapitsaandLeontovichcalledit.ThisiswhatLudwigwrotetomewhenI wasawardedtheStatePrizeofRussianFederationformyworkinthermodynamics: DearVitya,mycongratulationsforthelongawaitedprize.HadIlearnedaboutitintime,I wouldhavecometoMoscowtocelebrateitwithyou.ButIhadalreadyrefusedtocomeinorderto avoidseeingthosewhoaredestroyingourscience…Concerningmyscientificopinion,Iamunable toprofessionallyassessyourrecentwork.Sincemychildhood,Ihaveastrongaversiontostatistical physics,whichofcoursehasbeendetrimentaltome…Iloveyouverymuch. Atthetime,Ludwigwasmovingtowardanewbreakthroughinfieldtheoryandparticlethe- ory.Hislifeshouldnot,andshouldneverhaveendedthen.Afteroneofhisearlierdiscoveries, whichreflectedthebeautyofcreation,hetoldme:“IambeginningtobelieveinGod!”Iamwell familiarwiththisfeeling:whenamathematicalstructureunexpectedlybecomesclear—youcan- notbelieveyoureyes.Itissoterriblyunfairthatlifecansuddenlyendbeforeyouhavehadenough timetofinishputtingthestructuretogether;andjustwhenhehadcaughthissecondwind! Archimedessaid,“Givemeapointofsupport,andIwillmovetheEarth!”Buthedidnotsay whathappenswhenthepointofsupportislost.Withthelossofmyfriend,IfeelIhavelostmy pointofsupport. Ifirstconsideredgeneralalgebras(i.e.,certainnonlinearcommutationrelations)in thepaper“Applicationofthemethodoforderedoperatorstoobtainexactsolutions” publishedinthejournalTheoreticalandMathematicalPhysicsin1977(Vol.33,no.2, pp.185–209).AlgebraswithgeneralcommutationrelationswithoutLiealgebracondi- tionswereconsideredinjointworkwithmycolleaguesV.NazaikinskiiandM.Karasev. LudwigFaddevwithcoauthors,N.ReshetikhinandL.Takhtadzhyan,inthearti- cle“QuantizationofLiegroupsandLiealgebras”(AlgebraiAnaliz,1989,Vol.1,no.1, pp.178–206)introducedaspecialclassofalgebrasthathecalledquantumanalogues ofHopfalgebras.Inthegalleyproofsofthearticle,theauthorsaddedthefollowingre- mark:“Algebraswithnonstandardnon-Liecommutationrelations”wereconsidered inaseriesofpapersbyV.P.Maslov,M.V.Karasev,andV.E.Nazaikinskii“Algebras withgeneralcommutationrelationsandtheirapplications,I,II”inthe(JournalofSo- vietMathematics,1981,Vol.15,pp.167–273,273–368). Hopf algebras have numerous applications as was noticed, in particular, by V.G. Drinfeld in the paper “Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang–Baxter equations” (Soviet Math Doklady,1983,Vol.268,no.2.pp.285–287).ThequantumanalogueofHopfalgebras constructed by Faddeev found spectacular applications as a new science, namely PrefacetotheEnglishedition | IX quantum field theory. The more general quantum groups that I introduced do not have such spectacular applications, but they are more natural and simpler in the framework of the old science. What are their potential applications? That question remainsopen. IntheEnglishversionofthepresentmonograph,therearepracticallynochanges or additions to the Russian edition of 2000, except that the bibliography has been expandedtoincludethemostimportantlaterworksofOlegShvedov,whoseuntimely deathinterruptedtheseriesofpapersdevelopingtheideasofthebook. Inconclusion,letmeexpressmydeepgratitudetomytranslators,AlexanderStoy- anovsky,MariaShishkova,andAlexeyAlimov.Iamalsogratefultomyfriend,Alexey Sossinsky, for his invariably highly qualified recommendations concerning English terminologyandhelpintheformulationofmyideasinEnglish.Specialthanksare duetoVladimirNazaikinskiiforhisscientificandstyleeditingofthewholetransla- tionofthebook.
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