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Quantum Field Theory I: Basics in Mathematics and Physics - A Bridge between Mathematicians and Physicists
Eberhard Zeidler
Verlag Springer-Verlag, 2007
ISBN 9783540347644 , 1051 Seiten
Format PDF, OL
Kopierschutz Wasserzeichen
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