Quantum Field Theory II: Quantum Electrodynamics (eBook)

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).

Preface 7
Contents 25
Part I. Introduction 1
Prologue 38
Mathematical Principles of Modern Natural Philosophy 48
Basic Principles 49
The Infinitesimal Strategy and Differential Equations 51
The Optimality Principle 51
The Basic Notion of Action in Physics and the Idea ofQuantization 52
The Method of the Green's Function 54
Harmonic Analysis and the Fourier Method 58
The Method of Averaging and the Theory of Distributions 63
The Symbolic Method 65
Gauge Theory -- Local Symmetry and the Description of Interactions by Gauge Fields 71
The Challenge of Dark Matter 83
The Basic Strategy of Extracting Finite Information from Infinities -- Ariadne's Thread in Renormalization Theory 84
Renormalization Theory in a Nutshell 84
Effective Frequency and Running Coupling Constant of an Anharmonic Oscillator 85
The Zeta Function and Riemann's Idea of Analytic Continuation 91
Meromorphic Functions and Mittag-Leffler's Ideaof Subtractions 93
The Square of the Dirac Delta Function 95
Regularization of Divergent Integrals in Baby Renormalization Theory 97
Momentum Cut-off and the Method of Power-Counting 97
The Choice of the Normalization Momentum 100
The Method of Differentiating Parameter Integrals 100
The Method of Taylor Subtraction 101
Overlapping Divergences 102
The Role of Counterterms 104
Euler's Gamma Function 104
Integration Tricks 106
Dimensional Regularization via Analytic Continuation 110
Pauli--Villars Regularization 113
Analytic Regularization 114
Application to Algebraic Feynman Integrals inMinkowski Space 117
Distribution-Valued Meromorphic Functions 118
Application to Newton's Equation of Motion 124
Hints for Further Reading. 129
Further Regularization Methods in Mathematics 130
Euler's Philosophy 130
Adiabatic Regularization of Divergent Series 131
Adiabatic Regularization of Oscillating Integrals 132
Regularization by Averaging 133
Borel Regularization 135
Hadamard's Finite Part of Divergent Integrals 137
Infinite-Dimensional Gaussian Integrals and the Zeta Function Regularization 138
Trouble in Mathematics 139
Interchanging Limits 139
The Ambiguity of Regularization Methods 141
Pseudo-Convergence 141
Ill-Posed Problems 142
Mathemagics 146
The Power of Combinatorics 152
Algebras 152
The Algebra of Multilinear Functionals 154
Fusion, Splitting, and Hopf Algebras 159
The Bialgebra of Linear Differential Operators 160
The Definition of Hopf Algebras 165
Power Series Expansion and Hopf Algebras 168
The Importance of Cancellations 168
The Kepler Equation and the LagrangeInversion Formula 169
The Composition Formula for Power Series 171
The Faà di Bruno Hopf Algebra for the FormalDiffeomorphism Group of the Complex Plane 173
The Generalized Zimmermann Forest Formula 175
The Logarithmic Function and Schur Polynomials 177
Correlation Functions in Quantum Field Theory 178
Random Variables, Moments, and Cumulants 180
Symmetry and Hopf Algebras 183
The Strategy of Coordinatization in Mathematics and Physics 183
The Coordinate Hopf Algebra of a Finite Group 185
The Coordinate Hopf Algebra of an Operator Group 187
The Tannaka--Krein Duality for Compact Lie Groups 189
Regularization and Rota--Baxter Algebras 191
Regularization of the Laurent Series 194
Projection Operators 195
The q-Integral 195
The Volterra--Spitzer Exponential Formula 197
The Importance of the Exponential Function inMathematics and Physics 198
Partially Ordered Sets and Combinatorics 199
Incidence Algebras and the Zeta Function 199
The Möbius Function as an Inverse Function 200
The Inclusion--Exclusion Principle in Combinatorics 201
Applications to Number Theory 203
Hints for Further Reading 204
The Strategy of Equivalence Classes in Mathematics 212
Equivalence Classes in Algebra 215
The Gaussian Quotient Ring and the QuadraticReciprocity Law in Number Theory 215
Application of the Fermat--Euler Theorem in Coding Theory 219
Quotient Rings, Quotient Groups, and Quotient Fields 221
Linear Quotient Spaces 225
Ideals and Quotient Algebras 227
Superfunctions and the Heaviside Calculus in Electrical Engineering 228
Equivalence Classes in Geometry 231
The Basic Idea of Geometry Epitomized by Klein's Erlangen Program 231
Symmetry Spaces, Orbit Spaces, and Homogeneous Spaces 231
The Space of Quantum States 236
Real Projective Spaces 237
Complex Projective Spaces 240
The Shape of the Universe 241
Equivalence Classes in Topology 242
Topological Quotient Spaces 242
Physical Fields, Observers, Bundles, and Cocycles 245
Generalized Physical Fields and Sheaves 253
Deformations, Mapping Classes, and Topological Charges 256
Poincaré's Fundamental Group 260
Loop Spaces and Higher Homotopy Groups 262
Homology, Cohomology, and Electrodynamics 264
Bott's Periodicity Theorem 264
K-Theory 265
Application to Fredholm Operators 270
Hints for Further Reading 272
The Strategy of Partial Ordering 274
Feynman Diagrams 275
The Abstract Entropy Principle in Thermodynamics 276
Convergence of Generalized Sequences 277
Inductive and Projective Topologies 278
Inductive and Projective Limits 280
Classes, Sets, and Non-Sets 282
The Fixed-Point Theorem of Bourbaki--Kneser 284
Zorn's Lemma 285
Leibniz's Infinitesimals and Non-Standard Analysis 285
Filters and Ultrafilters 287
The Full-Rigged Real Line 288
Part II. Basic Ideas in Classical Mechanics 300
Geometrical Optics 300
Ariadne's Thread in Geometrical Optics 301
Fermat's Principle of Least Time 305
Huygens' Principle on Wave Fronts 307
Carathéodory's Royal Road to Geometrical Optics 308
The Duality between Light Rays and Wave Fronts 311
From Wave Fronts to Light Rays 312
From Light Rays to Wave Fronts 313
The Jacobi Approach to Focal Points 313
Lie's Contact Geometry 316
Basic Ideas 316
Contact Manifolds and Contact Transformations 320
Applications to Geometrical Optics 321
Equilibrium Thermodynamics and LegendreSubmanifolds 322
Light Rays and Non-Euclidean Geometry 326
Linear Symplectic Spaces 327
The Kähler Form of a Complex Hilbert Space 332
The Refraction Index and Geodesics 334
The Trick of Gauge Fixing 336
Geodesic Flow 336
Hamilton's Duality Trick and Cogeodesic Flow 337
The Principle of Minimal Geodesic Energy 338
Spherical Geometry 339
The Global Gauss--Bonnet Theorem 340
De Rham Cohomology and the Chern Class ofthe Sphere 342
The Beltrami Model 345
The Poincaré Model of Hyperbolic Geometry 351
Kähler Geometry and the Gaussian Curvature 355
Kähler--Einstein Geometry 360
Symplectic Geometry 360
Riemannian Geometry 361
Ariadne's Thread in Gauge Theory 370
Parallel Transport of Physical Information -- the Key to Modern Physics 371
The Phase Equation and Fiber Bundles 374
Gauge Transformations and Gauge-InvariantDifferential Forms 375
Perspectives 378
Classification of Two-Dimensional Compact Manifolds 380
The Poincaré Conjecture and the Ricci Flow 383
A Glance at Modern Optimization Theory 385
Hints for Further Reading 385
The Principle of Critical Action and the HarmonicOscillator -- Ariadne's Thread in Classical Mechanics 396
Prototypes of Extremal Problems 397
The Motion of a Particle 401
Newtonian Mechanics 403
A Glance at the History of the Calculus of Variations 407
Lagrangian Mechanics 409
The Harmonic Oscillator 410
The Euler--Lagrange Equation 412
Jacobi's Accessory Eigenvalue Problem 413
The Morse Index 414
The Anharmonic Oscillator 415
The Ginzburg--Landau Potential and the Higgs Potential 417
Damped Oscillations, Stability, and EnergyDissipation 419
Resonance and Small Divisors 419
Symmetry and Conservation Laws 420
The Symmetries of the Harmonic Oscillator 421
The Noether Theorem 421
The Pendulum and Dynamical Systems 427
The Equation of Motion 427
Elliptic Integrals and Elliptic Functions 428
The Phase Space of the Pendulum and Bundles 433
Hamiltonian Mechanics 439
The Canonical Equation 441
The Hamiltonian Flow 441
The Hamilton--Jacobi Partial Differential Equation 442
Poissonian Mechanics 443
Poisson Brackets and the Equation of Motion 444
Conservation Laws 444
Symplectic Geometry 444
The Canonical Equations 445
Symplectic Transformations 446
The Spherical Pendulum 448
The Gaussian Principle of Critical Constraint 448
The Lagrangian Approach 449
The Hamiltonian Approach 451
Geodesics of Shortest Length 452
The Lie Group SU(E3) of Rotations 453
Conservation of Angular Momentum 453
Lie's Momentum Map 456
Carathéodory's Royal Road to the Calculus of Variations 456
The Fundamental Equation 456
Lagrangian Submanifolds in Symplectic Geometry 458
The Initial-Value Problem for the Hamilton--Jacobi Equation 460
Solution of Carathéodory's Fundamental Equation 460
Hints for Further Reading 461
Part III. Basic Ideas in Quantum Mechanics 464
Quantization of the Harmonic Oscillator -- Ariadne's Thread in Quantization 464
Complete Orthonormal Systems 467
Bosonic Creation and Annihilation Operators 469
Heisenberg's Quantum Mechanics 477
Heisenberg's Equation of Motion 480
Heisenberg's Uncertainty Inequality for the Harmonic Oscillator 483
Quantization of Energy 484
The Transition Probabilities 486
The Wightman Functions 488
The Correlation Functions 493
Schrödinger's Quantum Mechanics 496
The Schrödinger Equation 496
States, Observables, and Measurements 499
The Free Motion of a Quantum Particle 501
The Harmonic Oscillator 504
The Passage to the Heisenberg Picture 510
Heisenberg's Uncertainty Principle 512
Unstable Quantum States and the Energy-Time Uncertainty Relation 513
Schrödinger's Coherent States 515
Feynman's Quantum Mechanics 516
Main Ideas 517
The Diffusion Kernel and the Euclidean Strategy in Quantum Physics 524
Probability Amplitudes and the Formal Propagator Theory 525
Von Neumann's Rigorous Approach 532
The Prototype of the Operator Calculus 533
The General Operator Calculus 536
Rigorous Propagator Theory 542
The Free Quantum Particle as a Paradigm ofFunctional Analysis 546
The Free Hamiltonian 561
The Rescaled Fourier Transform 569
The Quantized Harmonic Oscillator and the Maslov Index 571
Ideal Gases and von Neumann's Density Operator 577
The Feynman Path Integral 584
The Basic Strategy 584
The Basic Definition 586
Application to the Free Quantum Particle 587
Application to the Harmonic Oscillator 589
The Propagator Hypothesis 592
Motivation of Feynman's Path Integral 592
Finite-Dimensional Gaussian Integrals 596
Basic Formulas 597
Free Moments, the Wick Theorem, and FeynmanDiagrams 601
Full Moments and Perturbation Theory 604
Rigorous Infinite-Dimensional Gaussian Integrals 607
The Infinite-Dimensional Dispersion Operator 608
Zeta Function Regularization and Infinite-Dimensional Determinants 609
Application to the Free Quantum Particle 611
Application to the Quantized Harmonic Oscillator 613
The Spectral Hypothesis 616
The Semi-Classical WKB Method 617
Brownian Motion 621
The Macroscopic Diffusion Law 621
Einstein's Key Formulas for the Brownian Motion 622
The Random Walk of Particles 622
The Rigorous Wiener Path Integral 623
The Feynman--Kac Formula 625
Weyl Quantization 627
The Formal Moyal Star Product 628
Deformation Quantization of the Harmonic Oscillator 629
Weyl Ordering 633
Operator Kernels 636
The Formal Weyl Calculus 639
The Rigorous Weyl Calculus 643
Two Magic Formulas 645
The Formal Feynman Path Integral for the Propagator Kernel 648
The Relation between the Scattering Kernel and the Propagator Kernel 651
The Poincaré--Wirtinger Calculus 653
Bargmann's Holomorphic Quantization 654
The Stone--Von Neumann Uniqueness Theorem 658
The Prototype of the Weyl Relation 658
The Main Theorem 663
C*-Algebras 664
Operator Ideals 666
Symplectic Geometry and the Weyl QuantizationFunctor 667
A Glance at the Algebraic Approach to Quantum Physics 670
States and Observables 670
Gleason's Extension Theorem -- the Main Theorem of Quantum Logic 674
The Finite Standard Model in Statistical Physics as a Paradigm 675
Information, Entropy, and the Measure of Disorder 677
Semiclassical Statistical Physics 682
The Classical Ideal Gas 685
Bose--Einstein Statistics 686
Fermi--Dirac Statistics 687
Thermodynamic Equilibrium and KMS-States 688
Quasi-Stationary Thermodynamic Processes and Irreversibility 689
The Photon Mill on Earth 691
Von Neumann Algebras 691
Von Neumann's Bicommutant Theorem 692
The Murray--von Neumann Classification of Factors 695
The Tomita--Takesaki Theory and KMS-States 696
Connes' Noncommutative Geometry 697
Jordan Algebras 699
The Supersymmetric Harmonic Oscillator 700
Hints for Further Reading 704
Quantum Particles on the Real Line -- Ariadne's Thread in Scattering Theory 736
Classical Dynamics Versus Quantum Dynamics 736
The Stationary Schrödinger Equation 740
One-Dimensional Quantum Motion in a Square-WellPotential 741
Free Motion 741
Scattering States and the S-Matrix 742
Bound States 747
Bound-State Energies and the Singularities of theS-Matrix 749
The Energetic Riemann Surface, Resonances, and the Breit--Wigner Formula 750
The Jost Functions 755
The Fourier--Stieltjes Transformation 756
Generalized Eigenfunctions of the Hamiltonian 757
Quantum Dynamics and the Scattering Operator 759
The Feynman Propagator 763
Tunnelling of Quantum Particles and Radioactive Decay 764
The Method of the Green's Function in a Nutshell 766
The Inhomogeneous Helmholtz Equation 767
The Retarded Green's Function, and the Existence and Uniqueness Theorem 768
The Advanced Green's Function 773
Perturbation of the Retarded and Advanced Green's Function 774
Feynman's Regularized Fourier Method 776
The Lippmann--Schwinger Integral Equation 780
The Born Approximation 780
The Existence and Uniqueness Theorem via Banach's Fixed Point Theorem 781
Hypoellipticity 782
A Glance at General Scattering Theory 784
The Formal Basic Idea 786
The Rigorous Time-Dependent Approach 788
The Rigorous Time-Independent Approach 790
Applications to Quantum Mechanics 791
A Glance at Quantum Field Theory 794
Hints for Further Reading 795
Part IV. Quantum Electrodynamics (QED) 808
Creation and Annihilation Operators 808
The Bosonic Fock Space 808
The Particle Number Operator 811
The Ground State 811
The Fermionic Fock Space and the Pauli Principle 816
General Construction 821
The Main Strategy of Quantum Electrodynamics 825
The Basic Equations in Quantum Electrodynamics 830
The Classical Lagrangian 830
The Gauge Condition 833
The Free Quantum Fields of Electrons, Positrons,and Photons 836
Classical Free Fields 836
The Lattice Strategy in Quantum Electrodynamics 836
The High-Energy Limit and the Low-Energy Limit 839
The Free Electromagnetic Field 840
The Free Electron Field 843
Quantization 848
The Free Photon Quantum Field 849
The Free Electron Quantum Field and Antiparticles 851
The Spin of Photons 856
The Ground State Energy and the Normal Product 859
The Importance of Mathematical Models 861
The Trouble with Virtual Photons 862
Indefinite Inner Product Spaces 863
Representation of the Creation and Annihilation Operators in QED 863
Gupta--Bleuler Quantization 868
The Interacting Quantum Field, and the MagicDyson Series for the S-Matrix 872
Dyson's Key Formula 872
The Basic Strategy of Reduction Formulas 878
The Wick Theorem 883
Feynman Propagators 893
Discrete Feynman Propagators for Photons and Electrons 893
Regularized Discrete Propagators 899
The Continuum Limit of Feynman Propagators 901
Classical Wave Propagation versus Feynman Propagator 907
The Beauty of Feynman Diagrams in QED 912
Compton Effect and Feynman Rules in Position Space 913
Symmetry Properties 918
Summary of the Feynman Rules in Momentum Space 919
Typical Examples 922
The Formal Language of Physicists 927
Transition Probabilities and Cross Sections of ScatteringProcesses 928
The Crucial Limits 931
Appendix: Table of Feynman Rules 933
Applications to Physical Effects 936
Compton Effect 936
Duality between Light Waves and Light Particles in the History of Physics 939
The Trace Method for Computing Cross Sections 940
Relativistic Invariance 949
Asymptotically Free Electrons in an ExternalElectromagnetic Field 951
The Key Formula for the Cross Section 951
Application to Yukawa Scattering 952
Application to Coulomb Scattering 952
Motivation of the Key Formula via S-Matrix 953
Perspectives 958
Bound Electrons in an External ElectromagneticField 959
The Spontaneous Emission of Photons by the Atom 959
Motivation of the Key Formula 960
Intensity of Spectral Lines 962
Cherenkov Radiation 963
Part V. Renormalization 982
The Continuum Limit 982
The Fundamental Limits 982
The Formal Limits Fail 983
Basic Ideas of Renormalization 984
The Effective Mass and the Effective Charge of the Electron 984
The Counterterms of the Modified Lagrangian 984
The Compensation Principle 985
Fundamental Invariance Principles 986
Dimensional Regularization of Discrete AlgebraicFeynman Integrals 986
Multiplicative Renormalization 987
The Theory of Approximation Schemes in Mathematics 988
Radiative Corrections of Lowest Order 990
Primitive Divergent Feynman Graphs 990
Vacuum Polarization 991
Radiative Corrections of the Propagators 992
The Photon Propagator 993
The Electron Propagator 993
The Vertex Correction and the Ward Identity 994
The Counterterms of the Lagrangian and the Compensation Principle 994
Application to Physical Problems 995
Radiative Correction of the Coulomb Potential 995
The Anomalous Magnetic Moment of the Electron 996
The Anomalous Magnetic Moment of the Muon 998
The Lamb Shift 999
Photon-Photon Scattering 1001
A Glance at Renormalization to all Orders ofPerturbation Theory 1004
One-Particle Irreducible Feynman Graphs andDivergences 1007
Overlapping Divergences and Manoukian's EquivalencePrinciple 1009
The Renormalizability of Quantum Electrodynamics 1012
Automated Multi-Loop Computations in PerturbationTheory 1014
Perspectives 1016
BPHZ Renormalization 1018
Bogoliubov's Iterative R-Method 1018
Zimmermann's Forest Formula 1021
The Classical BPHZ Method 1022
The Causal Epstein--Glaser S-Matrix Approach 1024
Kreimer's Hopf Algebra Revolution 1027
The History of the Hopf Algebra Approach 1028
Renormalization and the Iterative BirkhoffFactorization for Complex Lie Groups 1030
The Renormalization of QuantumElectrodynamics 1033
The Scope of the Riemann--Hilbert Problem 1034
The Gaussian Hypergeometric Differential Equation 1035
The Confluent Hypergeometric Function and theSpectrum of the Hydrogen Atom 1041
Hilbert's 21th Problem 1041
The Transport of Information in Nature 1044
Stable Transport of Energy and Solitons 1044
Ariadne's Thread in Soliton Theory 1046
Resonances 1051
The Role of Integrable Systems in Nature 1051
The BFFO Hopf Superalgebra Approach 1053
The BRST Approach and Algebraic Renormalization 1056
Analytic Renormalization and Distribution-ValuedAnalytic Functions 1059
Computational Strategies 1060
The Renormalization Group 1060
Operator Product Expansions 1061
Binary Planar Graphs and the Renormalizationof Quantum Electrodynamics 1063
The Master Ward Identity 1064
Trouble in Quantum Electrodynamics 1064
The Landau Inconsistency Problem in QuantumElectrodynamics 1064
The Lack of Asymptotic Freedom in QuantumElectrodynamics 1066
Hints for Further Reading 1066
Epilogue 1082
References 1086
List of Symbols 1098
Index 1106