Quantum Field Theory II: Quantum Electrodynamics Eberhard Zeidler Quantum Field Theory II: Quantum Electrodynamics A Bridge between Mathematicians and Physicists EberhardZeidler MaxPlanckInstitute forMathematicsintheSciences Inselstr.22-26 04103Leipzig Germany ISBN 978-3-540-85376-3 e-ISBN 978-3-540-85377-0 DOI 10.1007/978-3-540-85377-0 LibraryofCongressControlNumber:2006929535 MathematicsSubjectClassification(2000):35-XX,47-XX,49-XX,51-XX,55-XX,81-XX,82-XX (cid:2)c 2009Springer-VerlagBerlinHeidelberg Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liabletoprosecutionundertheGermanCopyrightLaw. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective lawsandregulationsandthereforefreeforgeneraluse. Coverdesign:WMXDesignGmbH Printedonacid-freepaper 9 8 7 6 5 4 3 2 1 springer.com TO FRIEDRICH HIRZEBRUCH IN GRATITUDE Preface And God said, Let there be light; and there was light. Genesis 1,3 Lightisnotonlythebasisofourbiologicalexistence,butalsoanessential source of our knowledge about the physical laws of nature, ranging from the seventeenth century geometrical optics up to the twentieth century theory of general relativity and quantum electrodynamics. Folklore Don’t give us numbers: give us insight! A contemporary natural scientist to a mathematician The present book is the second volume of a comprehensive introduction to themathematicalandphysicalaspectsofmodernquantumfieldtheorywhich comprehends the following six volumes: Volume I: Basics in Mathematics and Physics Volume II: Quantum Electrodynamics Volume III: Gauge Theory Volume IV: Quantum Mathematics Volume V: The Physics of the Standard Model Volume VI: Quantum Gravitation and String Theory. It is our goal to build a bridge between mathematicians and physicists based on the challenging question about the fundamental forces in • macrocosmos (the universe) and • microcosmos (the world of elementary particles). The six volumes address a broad audience of readers, including both under- graduate and graduate students, as well as experienced scientists who want to become familiar with quantum field theory, which is a fascinating topic in modern mathematics and physics. For students of mathematics, it is shown that detailed knowledge of the physical background helps to motivate the mathematical subjects and to VIII Preface discover interesting interrelationships between quite different mathematical topics. For students of physics, fairly advanced mathematics are presented, which is beyond the usual curriculum in physics. The strategies and the structure of the six volumes are thoroughly discussed in the Prologue to Volume I. In particular, we will try to help the reader to understand the basic ideas behind the technicalities. In this connection, the famous ancient story of Ariadne’s thread is discussed in the Preface to Volume I. In terms of this story, we want to put the beginning of Ariadne’s thread in quantum field theory into the hands of the reader. The present volume is devoted to the physics and mathematics of light. It contains the following material: Part I: Introduction • Chapter 1: Mathematical Principles of Modern Natural Philosophy • Chapter 2: The Basic Strategy of Extracting Finite Information from Infinities – Ariadne’s Thread in Renormalization Theory • Chapter 3: The Power of Combinatorics • Chapter 4: The Strategy of Equivalence Classes in Mathematics Part II: Basic Ideas in Classical Mechanics • Chapter 5: Geometrical Optics • Chapter 6:The PrincipleofCriticalActionandthe Harmonic Oscilla- tor as a Paradigm Part III: Basic Ideas in Quantum Mechanics • Chapter7:QuantizationoftheHarmonicOscillator–Ariadne’sThread in Quantization • Chapter 8:Quantum Particlesonthe RealLine –Ariadne’sThreadin Scattering Theory • Chapter 9: A Glance at General Scattering Theory. Part IV: Quantum Electrodynamics (QED) • Chapter 10: Creation and Annihilation Operators • Chapter 11: The Basic Equations in Quantum Electrodynamics • Chapter 12: The Free Quantum Fields of Electrons, Positrons, and Photons • Chapter 13: The Interacting Quantum Field, and the Magic Dyson Series for the S-Matrix • Chapter 14: The Beauty of Feynman Diagrams in QED • Chapter 15: Applications to Physical Effects Part V: Renormalization • Chapter 16: The Continuum Limit • Chapter 17: Radiative Corrections of Lowest Order • Chapter18:AGlanceatRenormalizationtoallOrdersofPerturbation Theory • Chapter 19: Perspectives Preface IX We try to find the right balance between the mathematical theory and its applications to interesting physical effects observed in experiments. In par- ticular, we do not consider purely mathematical models in this volume. It is our philosophy that the reader should learn quantum field theory by studying a realistic model, as given by quantum electrodynamics. Let us discuss the main structure of the present volume. In Chapters 1 through 4, we consider topics from classical mathematics which are closely related to modern quantum field theory. This should help the reader to un- derstand the basic ideas behind quantum field theory to be considered in this volume and the volumes to follow. In Chapter 1 on the mathematical principles of modern natural philosophy, we discuss • the infinitesimal strategy due to Newton and Leibniz, • the optimality principle for processes in nature (the principle of critical action) and the calculus of variations due to Euler and Lagrange, which leads to the fundamental differential equations in classical field theory, • the propagation of physical effects and the method of the Green’s function, • harmonicanalysis and theFourier method forcomputing theGreen’sfunctions, • Laurent Schwartz’s theory of generalized functions (distributions) which is re- latedtotheideathatthemeasurementofphysicalquantitiesbydevicesisbased on averaging, • global symmetry and conservation laws, • local symmetry and the basic ideas of modern gauge field theory, and • the Planck quantum of action and the idea of quantizing classical field theories. Gauge field theory is behind both • the Standard Model in elementary particle physics and • Einstein’s theory of gravitation (i.e., the theory of general relativity). Inquantumfieldtheory,acrucialroleisplayedbyrenormalization.Interms of physics, this is based on the following two steps: • the regularization of divergent integrals, and • the computation of effective physical parameters measured in experiments (e.g., the effective mass and the effective electric charge of the electron). Renormalization is a highly technical subject. For example, the full proof on the renormalizability of the electroweak sector of the Standard Model in particle physics needs 100 pages. This can be found in: E.Kraus,Renormalizationoftheelectroweakstandardmodeltoallorders, Annals of Physics 262 (1998), 155–259. Whoever wants to understand quantum field theory has to understand the procedure of renormalization. Therefore, the different aspects of renormal- ization theory will be studied in all of the six volumes of this series of mono- graphs. This ranges from • resonancephenomenafortheanharmonicoscillator(classicalbifurcationtheory), • the Poincar´e–Lindstedt series (including small divisors) in celestial mechanics, X Preface • and the Kolmogorov–Arnold–Moser (KAM) theory for perturbed quasi-periodic oscillations (e.g., in celestial mechanics) based on sophisticated iterative tech- niques (the hard implicit function theorem) to the following fairly advanced subjects: • the Feynman functional integral (the Faddeev–Popov approach), • the Wiener functional integral (the Glimm–Jaffe approach), • thetheoryofhigher-dimensionalAbelianintegrals(algebraicFeynmanintegrals), • Hopf algebras and Rota–Baxter algebras in combinatorics (the modern vari- ant of the Bogoliubov–Parasiuk–Hepp–Zimmermann (BPHZ) approach due to Kreimer), • the Riemann–Hilbert problem and the Birkhoff decomposition (the Connes– Kreimer approach), • Hopf superalgebras (the Brouder–Fauser–Frabetti–Oeckl (BFFO) approach), • characterizationofphysicalstatesbycohomologyandalgebraicrenormalization (the Becchi–Rouet–Stora–Tyutin (BRST) approach), • the Riesz–Gelfand theory of distribution-valued meromorphic functions (con- struction of the Green’s functions), • wavefrontsetsandH¨ormander’smultiplicationofdistributions(theStueckelberg– Bogoliubov–Epstein–Glaser–Scharf approach), • the Master Ward identity as a highly non-trivial renormalization condition and thegeneralizedDyson–Schwingerequation(theDu¨tsch–Fredenhagenapproach), • q-deformed quantum field theory (the Wess–Majid–Wachter–Schmidt approach basedontheq-deformedPoincar´egroup,quantumgroups,andtheq-analysison specific classes of q-deformed quantum spaces), • deformationofbundlesandquantization(theWeyl–Flato–Sternheimer–Fedosov– Kontsevich approach), • microlocalanalysisandrenormalizationoncurvedspace-times(theRadzikowski– Brunetti–Fredenhagen–K¨ohler approach), • renormalized operator products on curved space-times (the Wilson–Hollands– Wald approach to quantum field theory), • natural transformations of functors in category theory and covariant quantum field theory on curved space-time manifolds (the Brunetti–Fredenhagen–Verch approach), as well as • one-parameter Lie groups and the renormalization group, • attractors of dynamical systems in the space of physical theories (the Wilson– Polchinski–Kopper–Rivasseau approach to renormalization based on the renor- malization group), • the Master Ward Identity and the Stueckelberg–Petermann renormalization group (the Du¨tsch–Fredenhagen approach), • motives in number theory and algebraic geometry, the Tannakian category, and the cosmic Galois group as a universal (motivic) renormalization group (the Connes–Marcolli approach), • noncommutative geometry and renormalization (the Grosse–Wulkenhaar ap- proach). TherecentworkofAlainConnes, DirkKreimer,andMatildeMarcollishows convincingly that renormalization is rooted in highly nontrivial mathemati- cal structures. We also want to emphasize that the theory of many-particle systems(consideredinstatisticalphysicsandquantumfieldtheory)isdeeply rooted in the theory of operator algebras. This concerns Preface XI • von Neumann algebras (the von Neumann approach), • C∗-algebras (the Gelfand–Naimark–Segal approach), • local nets of operator algebras (the Haag–Kastler approach) and, • noncommutative geometry (the Connes approach). As a warmup, we show in Chapter 2 that the regularization of divergent expressions represents a main subject in the history of mathematics starting with Euler in the eighteenth century. In this connection, we will consider • the regularization of divergent series, and • the regularization of divergent integrals. In particular, in Sect. 2.1.3, we will discuss the classical Mittag–Leffler theo- remonmeromorphicfunctionsf.Ifthefunctionf hasmerelyafinitenumber ofpoles,thenthemethod ofpartialfractiondecompositionworkswell.How- ever, as a rule, this method fails if the function f has an infinite number of poles. In this case, Mittag–Leffler showed in the late nineteenth century that onehastosubtractspecialterms,whicharecalledsubtractionsbyphysicists. The subtractions force the convergence of the infinite series. This is the prototype of the method of iteratively adding subtractions in the Bogoliubov–Parasiuk–Hepp–Zimmermann(BPHZ)approachtorenormaliza- tion theory. The corresponding iterative algorithm (called the Bogoliubov R-operation) has to be constructed carefully (because of nasty overlapping divergences).ThiswasdonebyNikolaiBogoliubovinthe1950s.Aningenious explicitsolutionformulaforthisiterativemethodwasfoundbyWolfhartZim- mermann in 1969. This is the famous Zimmermann forest formula. In the late 1990s, it was discovered by Dirk Kreimer that the sophisticated combi- natorics of the Zimmermann forest formula can be understood best in terms of a Hopf algebra generated by Feynman diagrams. By this discovery, the modern formulation of the BPHZ approach is based on both Hopf algebras and Rota–Baxter algebras. As a warmup, in Chapter 3, we give an introduction to the modern com- binatorial theory, which was founded by Gian-Carlo Rota (MIT, Cambridge, Massachusetts) in the 1960s. This includes both Hopf algebras and Rota– Baxter algebras. Surprisingly enough, it turns out that the Zimmermann forest for- mula is closely related to methods developed by Lagrange in the eigh- teenth century when studying the solution of the Kepler equation for the motion of planets in celestial mechanics. In modern terminology, the Lagrange inversion formula for power series ex- pansions is based on the so-called Fa`a di Bruno Hopf algebra.1 This will be studied in Sect. 3.4.3. 1 The Italian priest and mathematician Francesco Fa`a di Bruno (1825–1888) was beatified in 1988.