Zeidler Quantum Field Theory II: Quantum Electrodynamics

A Bridge between Mathematicians and Physicists
1. Auflage 2008
ISBN: 978-3-540-85377-0
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 1101 Seiten

And God said, Let there be light; and there was light. Genesis 1,3 Light is not only the basis of our biological existence, but also an essential 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 themathematicalandphysicalaspectsofmodernquantum?eldtheorywhich 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 und- graduate and graduate students, as well as experienced scientists who want to become familiar with quantum ?eld theory, which is a fascinating topic in modern mathematics and physics.

The author, Prof. Dr. Dr. h.c. Eberhard Zeidler, is retired director of the Max Planck Institute for Mathematics in the Sciences in Leipzig (Germany). In 1996 he was one of the founding directors of this institute. He is a member of the National Academy of Natural Scientists Leopoldina. In 2006 he was awarded the 'Alfried Krupp Wissenschaftspreis' of the Alfried Krupp von Bohlen und Halbach-Stiftung.  He is author of the following books:(a) E. Zeidler, Nonlinear Functional Analysis and its Applications, Vols. I-IV,Springer Verlag New York, 1984-1988 (third edition 1998).(b) E. Zeidler, Applied Functional Analysis, Vol. 1: Applications to Mathematical Physics, 2nd edition, 1997, Springer Verlag, New York.(c) E. Zeidler, Applied Functional Analysis, Vol. 2: Main Principles and Their Applications, Springer-Verlag, New York, 1995.(d) E. Zeidler, Oxford Users' Guide to Mathematics, Oxford University Press, 2004 (translated from German).

Autoren/Hrsg.

Zeidler, Eberhard

Inhaltsverzeichnis

1;Preface;7
2;Contents;25
3;Part I. Introduction;1
3.1;Prologue;38
3.2;Mathematical Principles of Modern Natural Philosophy;48
3.2.1;Basic Principles;49
3.2.2;The Infinitesimal Strategy and Differential Equations;51
3.2.3;The Optimality Principle;51
3.2.4;The Basic Notion of Action in Physics and the Idea ofQuantization;52
3.2.5;The Method of the Green's Function;54
3.2.6;Harmonic Analysis and the Fourier Method;58
3.2.7;The Method of Averaging and the Theory of Distributions;63
3.2.8;The Symbolic Method;65
3.2.9;Gauge Theory -- Local Symmetry and the Description of Interactions by Gauge Fields;71
3.2.10;The Challenge of Dark Matter;83
3.3;The Basic Strategy of Extracting Finite Information from Infinities -- Ariadne's Thread in Renormalization Theory;84
3.3.1;Renormalization Theory in a Nutshell;84
3.3.1.1;Effective Frequency and Running Coupling Constant of an Anharmonic Oscillator;85
3.3.1.2;The Zeta Function and Riemann's Idea of Analytic Continuation;91
3.3.1.3;Meromorphic Functions and Mittag-Leffler's Ideaof Subtractions;93
3.3.1.4;The Square of the Dirac Delta Function;95
3.3.2;Regularization of Divergent Integrals in Baby Renormalization Theory;97
3.3.2.1;Momentum Cut-off and the Method of Power-Counting;97
3.3.2.2;The Choice of the Normalization Momentum;100
3.3.2.3;The Method of Differentiating Parameter Integrals;100
3.3.2.4;The Method of Taylor Subtraction;101
3.3.2.5;Overlapping Divergences;102
3.3.2.6;The Role of Counterterms;104
3.3.2.7;Euler's Gamma Function;104
3.3.2.8;Integration Tricks;106
3.3.2.9;Dimensional Regularization via Analytic Continuation;110
3.3.2.10;Pauli--Villars Regularization;113
3.3.2.11;Analytic Regularization;114
3.3.2.12;Application to Algebraic Feynman Integrals inMinkowski Space;117
3.3.2.13;Distribution-Valued Meromorphic Functions;118
3.3.2.14;Application to Newton's Equation of Motion;124
3.3.2.15;Hints for Further Reading.;129
3.3.3;Further Regularization Methods in Mathematics;130
3.3.3.1;Euler's Philosophy;130
3.3.3.2;Adiabatic Regularization of Divergent Series;131
3.3.3.3;Adiabatic Regularization of Oscillating Integrals;132
3.3.3.4;Regularization by Averaging;133
3.3.3.5;Borel Regularization;135
3.3.3.6;Hadamard's Finite Part of Divergent Integrals;137
3.3.3.7;Infinite-Dimensional Gaussian Integrals and the Zeta Function Regularization;138
3.3.4;Trouble in Mathematics;139
3.3.4.1;Interchanging Limits;139
3.3.4.2;The Ambiguity of Regularization Methods;141
3.3.4.3;Pseudo-Convergence;141
3.3.4.4;Ill-Posed Problems;142
3.3.5;Mathemagics;146
3.4;The Power of Combinatorics;152
3.4.1;Algebras;152
3.4.2;The Algebra of Multilinear Functionals;154
3.4.3;Fusion, Splitting, and Hopf Algebras;159
3.4.3.1;The Bialgebra of Linear Differential Operators;160
3.4.3.2;The Definition of Hopf Algebras;165
3.4.4;Power Series Expansion and Hopf Algebras;168
3.4.4.1;The Importance of Cancellations;168
3.4.4.2;The Kepler Equation and the LagrangeInversion Formula;169
3.4.4.3;The Composition Formula for Power Series;171
3.4.4.4;The Faà di Bruno Hopf Algebra for the FormalDiffeomorphism Group of the Complex Plane;173
3.4.4.5;The Generalized Zimmermann Forest Formula;175
3.4.4.6;The Logarithmic Function and Schur Polynomials;177
3.4.4.7;Correlation Functions in Quantum Field Theory;178
3.4.4.8;Random Variables, Moments, and Cumulants;180
3.4.5;Symmetry and Hopf Algebras;183
3.4.5.1;The Strategy of Coordinatization in Mathematics and Physics;183
3.4.5.2;The Coordinate Hopf Algebra of a Finite Group;185
3.4.5.3;The Coordinate Hopf Algebra of an Operator Group;187
3.4.5.4;The Tannaka--Krein Duality for Compact Lie Groups;189
3.4.6;Regularization and Rota--Baxter Algebras;191
3.4.6.1;Regularization of the Laurent Series;194
3.4.6.2;Projection Operators;195
3.4.6.3;The q-Integral;195
3.4.6.4;The Volterra--Spitzer Exponential Formula;197
3.4.6.5;The Importance of the Exponential Function inMathematics and Physics;198
3.4.7;Partially Ordered Sets and Combinatorics;199
3.4.7.1;Incidence Algebras and the Zeta Function;199
3.4.7.2;The Möbius Function as an Inverse Function;200
3.4.7.3;The Inclusion--Exclusion Principle in Combinatorics;201
3.4.7.4;Applications to Number Theory;203
3.4.8;Hints for Further Reading;204
3.5;The Strategy of Equivalence Classes in Mathematics;212
3.5.1;Equivalence Classes in Algebra;215
3.5.1.1;The Gaussian Quotient Ring and the QuadraticReciprocity Law in Number Theory;215
3.5.1.2;Application of the Fermat--Euler Theorem in Coding Theory;219
3.5.1.3;Quotient Rings, Quotient Groups, and Quotient Fields;221
3.5.1.4;Linear Quotient Spaces;225
3.5.1.5;Ideals and Quotient Algebras;227
3.5.2;Superfunctions and the Heaviside Calculus in Electrical Engineering;228
3.5.3;Equivalence Classes in Geometry;231
3.5.3.1;The Basic Idea of Geometry Epitomized by Klein's Erlangen Program;231
3.5.3.2;Symmetry Spaces, Orbit Spaces, and Homogeneous Spaces;231
3.5.3.3;The Space of Quantum States;236
3.5.3.4;Real Projective Spaces;237
3.5.3.5;Complex Projective Spaces;240
3.5.3.6;The Shape of the Universe;241
3.5.4;Equivalence Classes in Topology;242
3.5.4.1;Topological Quotient Spaces;242
3.5.4.2;Physical Fields, Observers, Bundles, and Cocycles;245
3.5.4.3;Generalized Physical Fields and Sheaves;253
3.5.4.4;Deformations, Mapping Classes, and Topological Charges;256
3.5.4.5;Poincaré's Fundamental Group;260
3.5.4.6;Loop Spaces and Higher Homotopy Groups;262
3.5.4.7;Homology, Cohomology, and Electrodynamics;264
3.5.4.8;Bott's Periodicity Theorem;264
3.5.4.9;K-Theory;265
3.5.4.10;Application to Fredholm Operators;270
3.5.4.11;Hints for Further Reading;272
3.5.5;The Strategy of Partial Ordering;274
3.5.5.1;Feynman Diagrams;275
3.5.5.2;The Abstract Entropy Principle in Thermodynamics;276
3.5.5.3;Convergence of Generalized Sequences;277
3.5.5.4;Inductive and Projective Topologies;278
3.5.5.5;Inductive and Projective Limits;280
3.5.5.6;Classes, Sets, and Non-Sets;282
3.5.5.7;The Fixed-Point Theorem of Bourbaki--Kneser;284
3.5.5.8;Zorn's Lemma;285
3.5.6;Leibniz's Infinitesimals and Non-Standard Analysis;285
3.5.6.1;Filters and Ultrafilters;287
3.5.6.2;The Full-Rigged Real Line;288
4;Part II. Basic Ideas in Classical Mechanics;300
4.1;Geometrical Optics;300
4.1.1;Ariadne's Thread in Geometrical Optics;301
4.1.2;Fermat's Principle of Least Time;305
4.1.3;Huygens' Principle on Wave Fronts;307
4.1.4;Carathéodory's Royal Road to Geometrical Optics;308
4.1.5;The Duality between Light Rays and Wave Fronts;311
4.1.5.1;From Wave Fronts to Light Rays;312
4.1.5.2;From Light Rays to Wave Fronts;313
4.1.6;The Jacobi Approach to Focal Points;313
4.1.7;Lie's Contact Geometry;316
4.1.7.1;Basic Ideas;316
4.1.7.2;Contact Manifolds and Contact Transformations;320
4.1.7.3;Applications to Geometrical Optics;321
4.1.7.4;Equilibrium Thermodynamics and LegendreSubmanifolds;322
4.1.8;Light Rays and Non-Euclidean Geometry;326
4.1.8.1;Linear Symplectic Spaces;327
4.1.8.2;The Kähler Form of a Complex Hilbert Space;332
4.1.8.3;The Refraction Index and Geodesics;334
4.1.8.4;The Trick of Gauge Fixing;336
4.1.8.5;Geodesic Flow;336
4.1.8.6;Hamilton's Duality Trick and Cogeodesic Flow;337
4.1.8.7;The Principle of Minimal Geodesic Energy;338
4.1.9;Spherical Geometry;339
4.1.9.1;The Global Gauss--Bonnet Theorem;340
4.1.9.2;De Rham Cohomology and the Chern Class ofthe Sphere;342
4.1.9.3;The Beltrami Model;345
4.1.10;The Poincaré Model of Hyperbolic Geometry;351
4.1.10.1;Kähler Geometry and the Gaussian Curvature;355
4.1.10.2;Kähler--Einstein Geometry;360
4.1.10.3;Symplectic Geometry;360
4.1.10.4;Riemannian Geometry;361
4.1.11;Ariadne's Thread in Gauge Theory;370
4.1.11.1;Parallel Transport of Physical Information -- the Key to Modern Physics;371
4.1.11.2;The Phase Equation and Fiber Bundles;374
4.1.11.3;Gauge Transformations and Gauge-InvariantDifferential Forms;375
4.1.11.4;Perspectives;378
4.1.12;Classification of Two-Dimensional Compact Manifolds;380
4.1.13;The Poincaré Conjecture and the Ricci Flow;383
4.1.14;A Glance at Modern Optimization Theory;385
4.1.15;Hints for Further Reading;385
4.2;The Principle of Critical Action and the HarmonicOscillator -- Ariadne's Thread in Classical Mechanics;396
4.2.1;Prototypes of Extremal Problems;397
4.2.2;The Motion of a Particle;401
4.2.3;Newtonian Mechanics;403
4.2.4;A Glance at the History of the Calculus of Variations;407
4.2.5;Lagrangian Mechanics;409
4.2.5.1;The Harmonic Oscillator;410
4.2.5.2;The Euler--Lagrange Equation;412
4.2.5.3;Jacobi's Accessory Eigenvalue Problem;413
4.2.5.4;The Morse Index;414
4.2.5.5;The Anharmonic Oscillator;415
4.2.5.6;The Ginzburg--Landau Potential and the Higgs Potential;417
4.2.5.7;Damped Oscillations, Stability, and EnergyDissipation;419
4.2.5.8;Resonance and Small Divisors;419
4.2.6;Symmetry and Conservation Laws;420
4.2.6.1;The Symmetries of the Harmonic Oscillator;421
4.2.6.2;The Noether Theorem;421
4.2.7;The Pendulum and Dynamical Systems;427
4.2.7.1;The Equation of Motion;427
4.2.7.2;Elliptic Integrals and Elliptic Functions;428
4.2.7.3;The Phase Space of the Pendulum and Bundles;433
4.2.8;Hamiltonian Mechanics;439
4.2.8.1;The Canonical Equation;441
4.2.8.2;The Hamiltonian Flow;441
4.2.8.3;The Hamilton--Jacobi Partial Differential Equation;442
4.2.9;Poissonian Mechanics;443
4.2.9.1;Poisson Brackets and the Equation of Motion;444
4.2.9.2;Conservation Laws;444
4.2.10;Symplectic Geometry;444
4.2.10.1;The Canonical Equations;445
4.2.10.2;Symplectic Transformations;446
4.2.11;The Spherical Pendulum;448
4.2.11.1;The Gaussian Principle of Critical Constraint;448
4.2.11.2;The Lagrangian Approach;449
4.2.11.3;The Hamiltonian Approach;451
4.2.11.4;Geodesics of Shortest Length;452
4.2.12;The Lie Group SU(E3) of Rotations;453
4.2.12.1;Conservation of Angular Momentum;453
4.2.12.2;Lie's Momentum Map;456
4.2.13;Carathéodory's Royal Road to the Calculus of Variations;456
4.2.13.1;The Fundamental Equation;456
4.2.13.2;Lagrangian Submanifolds in Symplectic Geometry;458
4.2.13.3;The Initial-Value Problem for the Hamilton--Jacobi Equation;460
4.2.13.4;Solution of Carathéodory's Fundamental Equation;460
4.2.14;Hints for Further Reading;461
5;Part III. Basic Ideas in Quantum Mechanics;464
5.1;Quantization of the Harmonic Oscillator -- Ariadne's Thread in Quantization;464
5.1.1;Complete Orthonormal Systems;467
5.1.2;Bosonic Creation and Annihilation Operators;469
5.1.3;Heisenberg's Quantum Mechanics;477
5.1.3.1;Heisenberg's Equation of Motion;480
5.1.3.2;Heisenberg's Uncertainty Inequality for the Harmonic Oscillator;483
5.1.3.3;Quantization of Energy;484
5.1.3.4;The Transition Probabilities;486
5.1.3.5;The Wightman Functions;488
5.1.3.6;The Correlation Functions;493
5.1.4;Schrödinger's Quantum Mechanics;496
5.1.4.1;The Schrödinger Equation;496
5.1.4.2;States, Observables, and Measurements;499
5.1.4.3;The Free Motion of a Quantum Particle;501
5.1.4.4;The Harmonic Oscillator;504
5.1.4.5;The Passage to the Heisenberg Picture;510
5.1.4.6;Heisenberg's Uncertainty Principle;512
5.1.4.7;Unstable Quantum States and the Energy-Time Uncertainty Relation;513
5.1.4.8;Schrödinger's Coherent States;515
5.1.5;Feynman's Quantum Mechanics;516
5.1.5.1;Main Ideas;517
5.1.5.2;The Diffusion Kernel and the Euclidean Strategy in Quantum Physics;524
5.1.5.3;Probability Amplitudes and the Formal Propagator Theory;525
5.1.6;Von Neumann's Rigorous Approach;532
5.1.6.1;The Prototype of the Operator Calculus;533
5.1.6.2;The General Operator Calculus;536
5.1.6.3;Rigorous Propagator Theory;542
5.1.6.4;The Free Quantum Particle as a Paradigm ofFunctional Analysis;546
5.1.6.5;The Free Hamiltonian;561
5.1.6.6;The Rescaled Fourier Transform;569
5.1.6.7;The Quantized Harmonic Oscillator and the Maslov Index;571
5.1.6.8;Ideal Gases and von Neumann's Density Operator;577
5.1.7;The Feynman Path Integral;584
5.1.7.1;The Basic Strategy;584
5.1.7.2;The Basic Definition;586
5.1.7.3;Application to the Free Quantum Particle;587
5.1.7.4;Application to the Harmonic Oscillator;589
5.1.7.5;The Propagator Hypothesis;592
5.1.7.6;Motivation of Feynman's Path Integral;592
5.1.8;Finite-Dimensional Gaussian Integrals;596
5.1.8.1;Basic Formulas;597
5.1.8.2;Free Moments, the Wick Theorem, and FeynmanDiagrams;601
5.1.8.3;Full Moments and Perturbation Theory;604
5.1.9;Rigorous Infinite-Dimensional Gaussian Integrals;607
5.1.9.1;The Infinite-Dimensional Dispersion Operator;608
5.1.9.2;Zeta Function Regularization and Infinite-Dimensional Determinants;609
5.1.9.3;Application to the Free Quantum Particle;611
5.1.9.4;Application to the Quantized Harmonic Oscillator;613
5.1.9.5;The Spectral Hypothesis;616
5.1.10;The Semi-Classical WKB Method;617
5.1.11;Brownian Motion;621
5.1.11.1;The Macroscopic Diffusion Law;621
5.1.11.2;Einstein's Key Formulas for the Brownian Motion;622
5.1.11.3;The Random Walk of Particles;622
5.1.11.4;The Rigorous Wiener Path Integral;623
5.1.11.5;The Feynman--Kac Formula;625
5.1.12;Weyl Quantization;627
5.1.12.1;The Formal Moyal Star Product;628
5.1.12.2;Deformation Quantization of the Harmonic Oscillator;629
5.1.12.3;Weyl Ordering;633
5.1.12.4;Operator Kernels;636
5.1.12.5;The Formal Weyl Calculus;639
5.1.12.6;The Rigorous Weyl Calculus;643
5.1.13;Two Magic Formulas;645
5.1.13.1;The Formal Feynman Path Integral for the Propagator Kernel;648
5.1.13.2;The Relation between the Scattering Kernel and the Propagator Kernel;651
5.1.14;The Poincaré--Wirtinger Calculus;653
5.1.15;Bargmann's Holomorphic Quantization;654
5.1.16;The Stone--Von Neumann Uniqueness Theorem;658
5.1.16.1;The Prototype of the Weyl Relation;658
5.1.16.2;The Main Theorem;663
5.1.16.3;C*-Algebras;664
5.1.16.4;Operator Ideals;666
5.1.16.5;Symplectic Geometry and the Weyl QuantizationFunctor;667
5.1.17;A Glance at the Algebraic Approach to Quantum Physics;670
5.1.17.1;States and Observables;670
5.1.17.2;Gleason's Extension Theorem -- the Main Theorem of Quantum Logic;674
5.1.17.3;The Finite Standard Model in Statistical Physics as a Paradigm;675
5.1.17.4;Information, Entropy, and the Measure of Disorder;677
5.1.17.5;Semiclassical Statistical Physics;682
5.1.17.6;The Classical Ideal Gas;685
5.1.17.7;Bose--Einstein Statistics;686
5.1.17.8;Fermi--Dirac Statistics;687
5.1.17.9;Thermodynamic Equilibrium and KMS-States;688
5.1.17.10;Quasi-Stationary Thermodynamic Processes and Irreversibility;689
5.1.17.11;The Photon Mill on Earth;691
5.1.18;Von Neumann Algebras;691
5.1.18.1;Von Neumann's Bicommutant Theorem;692
5.1.18.2;The Murray--von Neumann Classification of Factors;695
5.1.18.3;The Tomita--Takesaki Theory and KMS-States;696
5.1.19;Connes' Noncommutative Geometry;697
5.1.20;Jordan Algebras;699
5.1.21;The Supersymmetric Harmonic Oscillator;700
5.1.22;Hints for Further Reading;704
5.2;Quantum Particles on the Real Line -- Ariadne's Thread in Scattering Theory;736
5.2.1;Classical Dynamics Versus Quantum Dynamics;736
5.2.2;The Stationary Schrödinger Equation;740
5.2.3;One-Dimensional Quantum Motion in a Square-WellPotential;741
5.2.3.1;Free Motion;741
5.2.3.2;Scattering States and the S-Matrix;742
5.2.3.3;Bound States;747
5.2.3.4;Bound-State Energies and the Singularities of theS-Matrix;749
5.2.3.5;The Energetic Riemann Surface, Resonances, and the Breit--Wigner Formula;750
5.2.3.6;The Jost Functions;755
5.2.3.7;The Fourier--Stieltjes Transformation;756
5.2.3.8;Generalized Eigenfunctions of the Hamiltonian;757
5.2.3.9;Quantum Dynamics and the Scattering Operator;759
5.2.3.10;The Feynman Propagator;763
5.2.4;Tunnelling of Quantum Particles and Radioactive Decay;764
5.2.5;The Method of the Green's Function in a Nutshell;766
5.2.5.1;The Inhomogeneous Helmholtz Equation;767
5.2.5.2;The Retarded Green's Function, and the Existence and Uniqueness Theorem;768
5.2.5.3;The Advanced Green's Function;773
5.2.5.4;Perturbation of the Retarded and Advanced Green's Function;774
5.2.5.5;Feynman's Regularized Fourier Method;776
5.2.6;The Lippmann--Schwinger Integral Equation;780
5.2.6.1;The Born Approximation;780
5.2.6.2;The Existence and Uniqueness Theorem via Banach's Fixed Point Theorem;781
5.2.6.3;Hypoellipticity;782
5.3;A Glance at General Scattering Theory;784
5.3.1;The Formal Basic Idea;786
5.3.2;The Rigorous Time-Dependent Approach;788
5.3.3;The Rigorous Time-Independent Approach;790
5.3.4;Applications to Quantum Mechanics;791
5.3.5;A Glance at Quantum Field Theory;794
5.3.6;Hints for Further Reading;795
6;Part IV. Quantum Electrodynamics (QED);808
6.1;Creation and Annihilation Operators;808
6.1.1;The Bosonic Fock Space;808
6.1.1.1;The Particle Number Operator;811
6.1.1.2;The Ground State;811
6.1.2;The Fermionic Fock Space and the Pauli Principle;816
6.1.3;General Construction;821
6.1.4;The Main Strategy of Quantum Electrodynamics;825
6.2;The Basic Equations in Quantum Electrodynamics;830
6.2.1;The Classical Lagrangian;830
6.2.2;The Gauge Condition;833
6.3;The Free Quantum Fields of Electrons, Positrons,and Photons;836
6.3.1;Classical Free Fields;836
6.3.1.1;The Lattice Strategy in Quantum Electrodynamics;836
6.3.1.2;The High-Energy Limit and the Low-Energy Limit;839
6.3.1.3;The Free Electromagnetic Field;840
6.3.1.4;The Free Electron Field;843
6.3.2;Quantization;848
6.3.2.1;The Free Photon Quantum Field;849
6.3.2.2;The Free Electron Quantum Field and Antiparticles;851
6.3.2.3;The Spin of Photons;856
6.3.3;The Ground State Energy and the Normal Product;859
6.3.4;The Importance of Mathematical Models;861
6.3.4.1;The Trouble with Virtual Photons;862
6.3.4.2;Indefinite Inner Product Spaces;863
6.3.4.3;Representation of the Creation and Annihilation Operators in QED;863
6.3.4.4;Gupta--Bleuler Quantization;868
6.4;The Interacting Quantum Field, and the MagicDyson Series for the S-Matrix;872
6.4.1;Dyson's Key Formula;872
6.4.2;The Basic Strategy of Reduction Formulas;878
6.4.3;The Wick Theorem;883
6.4.4;Feynman Propagators;893
6.4.4.1;Discrete Feynman Propagators for Photons and Electrons;893
6.4.4.2;Regularized Discrete Propagators;899
6.4.4.3;The Continuum Limit of Feynman Propagators;901
6.4.4.4;Classical Wave Propagation versus Feynman Propagator;907
6.5;The Beauty of Feynman Diagrams in QED;912
6.5.1;Compton Effect and Feynman Rules in Position Space;913
6.5.2;Symmetry Properties;918
6.5.3;Summary of the Feynman Rules in Momentum Space;919
6.5.4;Typical Examples;922
6.5.5;The Formal Language of Physicists;927
6.5.6;Transition Probabilities and Cross Sections of ScatteringProcesses;928
6.5.7;The Crucial Limits;931
6.5.8;Appendix: Table of Feynman Rules;933
6.6;Applications to Physical Effects;936
6.6.1;Compton Effect;936
6.6.1.1;Duality between Light Waves and Light Particles in the History of Physics;939
6.6.1.2;The Trace Method for Computing Cross Sections;940
6.6.1.3;Relativistic Invariance;949
6.6.2;Asymptotically Free Electrons in an ExternalElectromagnetic Field;951
6.6.2.1;The Key Formula for the Cross Section;951
6.6.2.2;Application to Yukawa Scattering;952
6.6.2.3;Application to Coulomb Scattering;952
6.6.2.4;Motivation of the Key Formula via S-Matrix;953
6.6.2.5;Perspectives;958
6.6.3;Bound Electrons in an External ElectromagneticField;959
6.6.3.1;The Spontaneous Emission of Photons by the Atom;959
6.6.3.2;Motivation of the Key Formula;960
6.6.3.3;Intensity of Spectral Lines;962
6.6.4;Cherenkov Radiation;963
7;Part V. Renormalization;982
7.1;The Continuum Limit;982
7.1.1;The Fundamental Limits;982
7.1.2;The Formal Limits Fail;983
7.1.3;Basic Ideas of Renormalization;984
7.1.3.1;The Effective Mass and the Effective Charge of the Electron;984
7.1.3.2;The Counterterms of the Modified Lagrangian;984
7.1.3.3;The Compensation Principle;985
7.1.3.4;Fundamental Invariance Principles;986
7.1.3.5;Dimensional Regularization of Discrete AlgebraicFeynman Integrals;986
7.1.3.6;Multiplicative Renormalization;987
7.1.4;The Theory of Approximation Schemes in Mathematics;988
7.2;Radiative Corrections of Lowest Order;990
7.2.1;Primitive Divergent Feynman Graphs;990
7.2.2;Vacuum Polarization;991
7.2.3;Radiative Corrections of the Propagators;992
7.2.3.1;The Photon Propagator;993
7.2.3.2;The Electron Propagator;993
7.2.3.3;The Vertex Correction and the Ward Identity;994
7.2.4;The Counterterms of the Lagrangian and the Compensation Principle;994
7.2.5;Application to Physical Problems;995
7.2.5.1;Radiative Correction of the Coulomb Potential;995
7.2.5.2;The Anomalous Magnetic Moment of the Electron;996
7.2.5.3;The Anomalous Magnetic Moment of the Muon;998
7.2.5.4;The Lamb Shift;999
7.2.5.5;Photon-Photon Scattering;1001
7.3;A Glance at Renormalization to all Orders ofPerturbation Theory;1004
7.3.1;One-Particle Irreducible Feynman Graphs andDivergences;1007
7.3.2;Overlapping Divergences and Manoukian's EquivalencePrinciple;1009
7.3.3;The Renormalizability of Quantum Electrodynamics;1012
7.3.4;Automated Multi-Loop Computations in PerturbationTheory;1014
7.4;Perspectives;1016
7.4.1;BPHZ Renormalization;1018
7.4.1.1;Bogoliubov's Iterative R-Method;1018
7.4.1.2;Zimmermann's Forest Formula;1021
7.4.1.3;The Classical BPHZ Method;1022
7.4.2;The Causal Epstein--Glaser S-Matrix Approach;1024
7.4.3;Kreimer's Hopf Algebra Revolution;1027
7.4.3.1;The History of the Hopf Algebra Approach;1028
7.4.3.2;Renormalization and the Iterative BirkhoffFactorization for Complex Lie Groups;1030
7.4.3.3;The Renormalization of QuantumElectrodynamics;1033
7.4.4;The Scope of the Riemann--Hilbert Problem;1034
7.4.4.1;The Gaussian Hypergeometric Differential Equation;1035
7.4.4.2;The Confluent Hypergeometric Function and theSpectrum of the Hydrogen Atom;1041
7.4.4.3;Hilbert's 21th Problem;1041
7.4.4.4;The Transport of Information in Nature;1044
7.4.4.5;Stable Transport of Energy and Solitons;1044
7.4.4.6;Ariadne's Thread in Soliton Theory;1046
7.4.4.7;Resonances;1051
7.4.4.8;The Role of Integrable Systems in Nature;1051
7.4.5;The BFFO Hopf Superalgebra Approach;1053
7.4.6;The BRST Approach and Algebraic Renormalization;1056
7.4.7;Analytic Renormalization and Distribution-ValuedAnalytic Functions;1059
7.4.8;Computational Strategies;1060
7.4.8.1;The Renormalization Group;1060
7.4.8.2;Operator Product Expansions;1061
7.4.8.3;Binary Planar Graphs and the Renormalizationof Quantum Electrodynamics;1063
7.4.9;The Master Ward Identity;1064
7.4.10;Trouble in Quantum Electrodynamics;1064
7.4.10.1;The Landau Inconsistency Problem in QuantumElectrodynamics;1064
7.4.10.2;The Lack of Asymptotic Freedom in QuantumElectrodynamics;1066
7.4.11;Hints for Further Reading;1066
8;Epilogue;1082
9;References;1086
10;List of Symbols;1098
11;Index;1106