Methods of Quantization

ofquantum?eldtheories,werehighlightedinthelecturesofBernardBakker andThomasHeinzl. Oneoftheopenproblemsoflight-conequantizationis theissueofspontaneoussymmetrybreaking. Thiscanbetracedbacktozero modeswhich,ingeneral,aresubjecttocomplicatedconstraintequations. A generalformalismforthequantizationofphysicalsystemswithconstraints waspresentedbyJohnKlauder. Theperturbativede?nitionofquantum?eld theoriesisingenerala?ictedbysingularitieswhichareovercomebyare- larizationandrenormalizationprocedure. Structuralaspectsoftherenormal- VI Preface izationprobleminthecaseofgaugeinvariant?eldtheorieswerediscussed inthelectureofKlausSibold. Areviewofthemathematicsunderlyingthe functional-integralquantizationwasgivenbyLudwigStreit. Apartfromthetopicsincludedinthisvolumetherewerealsolectures ontheKaluza–Kleinprogramforsupergravity(P. vanNieuwenhuizen),on dynamicalr-matricesandquantization(A. Alekseev),andonthequantum Liouvillemodelasaninstructiveexampleofquantumintegrablemodels(L. Faddeev). Inaddition,theschoolwascomplementedbymanyexcellents- inars. Thelistofseminarspeakersandthetopicsaddressedbythemcanbe foundattheendofthisvolume. Theinterestedreaderisrequestedtocontact thespeakersdirectlyfordetailedinformationorpertinentmaterial. Finally,wewouldliketoexpressourgratitudetothelecturersforalltheir e?ortsandtothemainsponsorsoftheschool,theAustrianMinistryofE- cation,Science,andCultureandtheGovernmentofStyria,forprovidingg- eroussupport. Wealsoappreciatethevaluableorganizationalandtechnical assistanceofthetownofSchladming,theSteyr-Daimler-PuchFahrzeugte- nik, Ricoh Austria, Styria Online, and the Hornig company. Furthermore, wethankoursecretaries,S. FuchsandE. Monschein,anumberofgra- atestudentsfromourinstitute,and,lastbutnotleast,ourcolleaguesfrom theorganizingcommitteefortheirassistanceinpreparingandrunningthe school. Graz, HeimoLatal March2001 WolfgangSchweiger Contents FormsofRelativisticDynamics BernardL. G. Bakker. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 ThePoincar´eGroup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 FormsofRelativisticDynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. 1 ComparisonofInstantForm,FrontForm,andPointForm. . . 6 4 Light-FrontDynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4. 1 RelativeMomentum,InvariantMass. . . . . . . . . . . . . . . . . . . . . . 9 4. 2 TheBoxDiagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5 Poincar´eGeneratorsinFieldTheory. . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5. 1 FermionsInteractingwithaScalarField. . . . . . . . . . . . . . . . . . . 20 5. 2 InstantForm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5. 3 FrontForm(LF). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5. 4 InteractingandNon-interactingGeneratorsonanInstant andontheLightFront. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 6 Light-FrontPerturbationTheory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6. 1 ConnectionofCovariantAmplitudes toLight-FrontAmplitudes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6. 2 Regularization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 6. 3 MinusRegularization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 7 TriangleDiagraminYukawaTheory. . . . . . . . . . . . . . . . . . . . . . . . . . . 27 7. 1 CovariantCalculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7. 2 ConstructionoftheCurrentinLFD. . . . . . . . . . . . . . . . . . . . . . . 30 7. 3 NumericalResults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3 8 FourVariat

Most of our present understanding of the elementary building blocks of matter and the forces between them is based on the quantized version of the field theories which are locally symmetric under gauge transformations. The present set of lecture notes gives both a status report and a survey of recent advances for the most important quantization methods in the field theories for elementary particle physics. The first part of the book introduces light-cone quantization as an interesting alternative to the commonly used covariant perturbation theory and functional-integral methods. Next, a general formalism for quantizing systems with constraints, the projection-operator approach, is presented and structural aspects of the renormalization problem for gauge invariant field theories are discussed. Finally, the mathematics underlying the functional-integral quantization is reviewed. Suitable as a reference for researchers in the field, the book will prove particularly useful for lecturers and graduate students in search of additional reading beyond the standard texts on quantum field theory.