Quantum Field Theory I: Basics in Mathematics and Physics (eBook)

Prof. Dr. Dr. h.c. Eberhard Zeidler works at 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 Academy of Natural Scientists Leopoldina. In 2006 he was awarded the "Alfried Krupp Wissenschaftspreis" of the Alfried Krupp von Bohlen und Halbach-Stiftung. The author wrote 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 6
Contents 12
Prologue 24
1. Historical Introduction 44
1.1 The Revolution of Physics 45
1.2 Quantization in a Nutshell 50
1.3 The Role of Göttingen 83
1.4 The Göttingen Tragedy 90
1.5 Highlights in the Sciences 92
1.6 The Emergence of Physical Mathematics – a New Dimension of Mathematics 98
1.7 The Seven Millennium Prize Problems of the Clay Mathematics Institute 100
2. Phenomenology of the Standard Model for Elementary Particles 102
2.1 The System of Units 103
2.2 Waves in Physics 104
2.3 Historical Background 120
2.4 The Standard Model in Particle Physics 150
2.5 Magic Formulas 163
2.6 Quantum Numbers of Elementary Particles 166
2.7 The Fundamental Role of Symmetry in Physics 185
2.8 Symmetry Breaking 201
2.9 The Structure of Interactions in Nature 206
3. The Challenge of Different Scales in Nature 209
3.1 The Trouble with Scale Changes 209
3.2 Wilson’s Renormalization Group Theory in Physics 211
3.3 Stable and Unstable Manifolds 228
3.4 A Glance at Conformal Field Theories 229
4. Analyticity 230
4.1 Power Series Expansion 231
4.2 Deformation Invariance of Integrals 233
4.3 Cauchy’s Integral Formula 233
4.4 Cauchy’s Residue Formula and Topological Charges 234
4.5 The Winding Number 235
4.6 Gauss’ Fundamental Theorem of Algebra 236
4.7 Compacti.cation of the Complex Plane 238
4.8 Analytic Continuation and the Local-Global Principle 239
4.9 Integrals and Riemann Surfaces 240
4.10 Domains of Holomorphy 244
4.11 A Glance at Analytic S-Matrix Theory 245
4.12 Important Applications 246
5. A Glance at Topology 247
5.1 Local and Global Properties of the Universe 247
5.2 Bolzano’s Existence Principle 248
5.3 Elementary Geometric Notions 250
5.4 Manifolds and Diffeomorphisms 254
5.5 Topological Spaces, Homeomorphisms, and Deformations 255
5.6 Topological Quantum Numbers 261
5.7 Quantum States 285
5.8 Perspectives 295
6. Many-Particle Systems in Mathematics and Physics 296
6.1 Partition Function in Statistical Physics 298
6.2 Euler’s Partition Function 302
6.3 Discrete Laplace Transformation 304
6.4 Integral Transformations 308
6.5 The Riemann Zeta Function 310
6.6 The Casimir Effect in Quantum Field Theory and the Epstein Zeta Function 318
6.7 Appendix: The Mellin Transformation and Other Useful Analytic Techniques by Don Zagier 324
7. Rigorous Finite-Dimensional Magic Formulas of Quantum Field Theory 343
7.1 Geometrization of Physics 343
7.2 Ariadne’s Thread in Quantum Field Theory 344
7.3 Linear Spaces 346
7.4 Finite-Dimensional Hilbert Spaces 353
7.5 Groups 358
7.6 Lie Algebras 360
7.7 Lie’s Logarithmic Trick for Matrix Groups 363
7.8 Lie Groups 365
7.9 Basic Notions in Quantum Physics 367
7.10 Fourier Series 373
7.11 Dirac Calculus in Finite-Dimensional Hilbert Spaces 377
7.12 The Trace of a Linear Operator 381
7.13 Banach Spaces 384
7.14 Probability and Hilbert’s Spectral Family of an Observable 386
7.15 Transition Probabilities, S-Matrix, and Unitary Operators 388
7.16 The Magic Formulas for the Green’s Operator 390
7.17 The Magic Dyson Formula for the Retarded Propagator 399
7.18 The Magic Dyson Formula for the S-Matrix 408
7.19 Canonical Transformations 410
7.20 Functional Calculus 413
7.21 The Discrete Feynman Path Integral 434
7.22 Causal Correlation Functions 442
7.23 The Magic Gaussian Integral 446
7.24 The Rigorous Response Approach to Finite Quantum Fields 456
7.25 The Discrete .4-Model and Feynman Diagrams 477
7.26 The Extended Response Approach 495
7.27 Complex-Valued Fields 501
7.28 The Method of Lagrange Multipliers 505
7.29 The Formal Continuum Limit 510
Problems 511
8. Rigorous Finite-Dimensional Perturbation Theory 514
8.1 Renormalization 514
8.2 The Rellich Theorem 523
8.3 The Trotter Product Formula 524
8.4 The Magic Baker–Campbell–Hausdorff Formula 525
8.5 Regularizing Terms 526
9. Fermions and the Calculus for Grassmann Variables 531
9.1 The Grassmann Product 531
9.2 Differential Forms 532
9.3 Calculus for One Grassmann Variable 532
9.4 Calculus for Several Grassmann Variables 533
9.5 The Determinant Trick 534
9.6 The Method of Stationary Phase 535
9.7 The Fermionic Response Model 535
10. Infinite-Dimensional Hilbert Spaces 537
10.1 The Importance of Infinite Dimensions in Quantum Physics 537
10.2 The Hilbert Space 541
10.3 Harmonic Analysis 548
10.4 The Dirichlet Problem in Electrostatics as a Paradigm 556
Problems 587
11. Distributions and Green’s Functions 590
11.1 Rigorous Basic Ideas 594
11.2 Dirac’s Formal Approach 604
11.3 Laurent Schwartz’s Rigorous Approach 622
11.4 Hadamard’s Regularization of Integrals 633
11.5 Renormalization of the Anharmonic Oscillator 640
11.6 The Importance of Algebraic Feynman Integrals 649
11.7 Fundamental Solutions of Differential Equations 659
11.8 Functional Integrals 666
11.9 A Glance at Harmonic Analysis 675
11.10 The Trouble with the Euclidean Trick 681
12. Distributions and Physics 683
12.1 The Discrete Dirac Calculus 683
12.2 Rigorous General Dirac Calculus 689
12.3 Fundamental Limits in Physics 696
12.4 Duality in Physics 704
12.5 Microlocal Analysis 717
12.6 Multiplication of Distributions 743
Problems 746
13. Basic Strategies in Quantum Field Theory 752
13.1 The Method of Moments and Correlation Functions 755
13.2 The Power of the S-Matrix 758
13.3 The Relation Between the S-Matrix and the Correlation Functions 759
13.4 Perturbation Theory and Feynman Diagrams 760
13.5 The Trouble with Interacting Quantum Fields 761
13.6 External Sources and the Generating Functional 762
13.7 The Beauty of Functional Integrals 764
13.8 Quantum Field Theory at Finite Temperature 770
14. The Response Approach 777
14.1 The Fourier–Minkowski Transform 782
14.2 The .4-Model 785
14.3 A Glance at Quantum Electrodynamics 801
Problems 816
15. The Operator Approach 824
15.1 The .4-Model 825
15.2 A Glance at Quantum Electrodynamics 857
15.3 The Role of Effective Quantities in Physics 858
15.4 A Glance at Renormalization 859
15.5 The Convergence Problem in Quantum Field Theory 871
15.6 Rigorous Perspectives 873
16. Peculiarities of Gauge Theories 887
16.1 Basic Difficulties 887
16.2 The Principle of Critical Action 888
16.3 The Language of Physicists 894
16.4 The Importance of the Higgs Particle 896
16.5 Integration over Orbit Spaces 896
16.6 The Magic Faddeev–Popov Formula and Ghosts 898
16.7 The BRST Symmetry 900
16.8 The Power of Cohomology 901
16.9 The Batalin–Vilkovisky Formalism 913
16.10 A Glance at Quantum Symmetries 914
17. A Panorama of the Literature 916
17.1 Introduction to Quantum Field Theory 916
17.2 Standard Literature in Quantum Field Theory 919
17.3 Rigorous Approaches to Quantum Field Theory 920
17.4 The Fascinating Interplay between Modern Physics and Mathematics 922
17.5 The Monster Group, Vertex Algebras, and Physics 928
17.6 Historical Development of Quantum Field Theory 933
17.7 General Literature in Mathematics and Physics 934
17.8 Encyclopedias 935
17.9 Highlights of Physics in the 20th Century 935
17.10 Actual Information 937
Appendix 940
A.1 Notation 940
A.2 The International System of Units 943
A.3 The Planck System 945
A.4 The Energetic System 951
A.5 The Beauty of Dimensional Analysis 953
A.6 The Similarity Principle in Physics 955
Epilogue 963
References 967
List of Symbols 999
Index 1003