Quantum Field Theory I: Basics in Mathematics and Physics - A Bridge between Mathematicians and Physicists

Preface

Contents

Prologue

1. Historical Introduction

1.1 The Revolution of Physics

1.2 Quantization in a Nutshell

1.3 The Role of Göttingen

1.4 The Göttingen Tragedy

1.5 Highlights in the Sciences

1.6 The Emergence of Physical Mathematics – a New Dimension of Mathematics

1.7 The Seven Millennium Prize Problems of the Clay Mathematics Institute

2. Phenomenology of the Standard Model for Elementary Particles

2.1 The System of Units

2.2 Waves in Physics

2.3 Historical Background

2.4 The Standard Model in Particle Physics

2.5 Magic Formulas

2.6 Quantum Numbers of Elementary Particles

2.7 The Fundamental Role of Symmetry in Physics

2.8 Symmetry Breaking

2.9 The Structure of Interactions in Nature

3. The Challenge of Different Scales in Nature

3.1 The Trouble with Scale Changes

3.2 Wilson’s Renormalization Group Theory in Physics

3.3 Stable and Unstable Manifolds

3.4 A Glance at Conformal Field Theories

4. Analyticity

4.1 Power Series Expansion

4.2 Deformation Invariance of Integrals

4.3 Cauchy’s Integral Formula

4.4 Cauchy’s Residue Formula and Topological Charges

4.5 The Winding Number

4.6 Gauss’ Fundamental Theorem of Algebra

4.7 Compacti.cation of the Complex Plane

4.8 Analytic Continuation and the Local-Global Principle

4.9 Integrals and Riemann Surfaces

4.10 Domains of Holomorphy

4.11 A Glance at Analytic S-Matrix Theory

4.12 Important Applications

5. A Glance at Topology

5.1 Local and Global Properties of the Universe

5.2 Bolzano’s Existence Principle

5.3 Elementary Geometric Notions

5.4 Manifolds and Diffeomorphisms

5.5 Topological Spaces, Homeomorphisms, and Deformations

5.6 Topological Quantum Numbers

5.7 Quantum States

5.8 Perspectives

6. Many-Particle Systems in Mathematics and Physics

6.1 Partition Function in Statistical Physics

6.2 Euler’s Partition Function

6.3 Discrete Laplace Transformation

6.4 Integral Transformations

6.5 The Riemann Zeta Function

6.6 The Casimir Effect in Quantum Field Theory and the Epstein Zeta Function

6.7 Appendix: The Mellin Transformation and Other Useful Analytic Techniques by Don Zagier

7. Rigorous Finite-Dimensional Magic Formulas of Quantum Field Theory

7.1 Geometrization of Physics

7.2 Ariadne’s Thread in Quantum Field Theory

7.3 Linear Spaces

7.4 Finite-Dimensional Hilbert Spaces

7.5 Groups

7.6 Lie Algebras

7.7 Lie’s Logarithmic Trick for Matrix Groups

7.8 Lie Groups

7.9 Basic Notions in Quantum Physics

7.10 Fourier Series

7.11 Dirac Calculus in Finite-Dimensional Hilbert Spaces

7.12 The Trace of a Linear Operator

7.13 Banach Spaces

7.14 Probability and Hilbert’s Spectral Family of an Observable

7.15 Transition Probabilities, S-Matrix, and Unitary Operators

7.16 The Magic Formulas for the Green’s Operator

7.17 The Magic Dyson Formula for the Retarded Propagator

7.18 The Magic Dyson Formula for the S-Matrix

7.19 Canonical Transformations

7.20 Functional Calculus

7.21 The Discrete Feynman Path Integral

7.22 Causal Correlation Functions

7.23 The Magic Gaussian Integral

7.24 The Rigorous Response Approach to Finite Quantum Fields

7.25 The Discrete .4-Model and Feynman Diagrams

7.26 The Extended Response Approach

7.27 Complex-Valued Fields

7.28 The Method of Lagrange Multipliers

7.29 The Formal Continuum Limit

Problems

8. Rigorous Finite-Dimensional Perturbation Theory

8.1 Renormalization

8.2 The Rellich Theorem

8.3 The Trotter Product Formula

8.4 The Magic Baker–Campbell–Hausdorff Formula

8.5 Regularizing Terms

9. Fermions and the Calculus for Grassmann Variables

9.1 The Grassmann Product

9.2 Differential Forms

9.3 Calculus for One Grassmann Variable

9.4 Calculus for Several Grassmann Variables

9.5 The Determinant Trick

9.6 The Method of Stationary Phase

9.7 The Fermionic Response Model

10. Infinite-Dimensional Hilbert Spaces

10.1 The Importance of Infinite Dimensions in Quantum Physics

10.2 The Hilbert Space

10.3 Harmonic Analysis

10.4 The Dirichlet Problem in Electrostatics as a Paradigm

Problems

11. Distributions and Green’s Functions

11.1 Rigorous Basic Ideas

11.2 Dirac’s Formal Approach

11.3 Laurent Schwartz’s Rigorous Approach

11.4 Hadamard’s Regularization of Integrals

11.5 Renormalization of the Anharmonic Oscillator

11.6 The Importance of Algebraic Feynman Integrals

11.7 Fundamental Solutions of Differential Equations

11.8 Functional Integrals

11.9 A Glance at Harmonic Analysis

11.10 The Trouble with the Euclidean Trick

12. Distributions and Physics

12.1 The Discrete Dirac Calculus

12.2 Rigorous General Dirac Calculus

12.3 Fundamental Limits in Physics

12.4 Duality in Physics

12.5 Microlocal Analysis

12.6 Multiplication of Distributions

Problems

13. Basic Strategies in Quantum Field Theory

13.1 The Method of Moments and Correlation Functions

13.2 The Power of the S-Matrix

13.3 The Relation Between the S-Matrix and the Correlation Functions

13.4 Perturbation Theory and Feynman Diagrams

13.5 The Trouble with Interacting Quantum Fields

13.6 External Sources and the Generating Functional

13.7 The Beauty of Functional Integrals

13.8 Quantum Field Theory at Finite Temperature

14. The Response Approach

14.1 The Fourier–Minkowski Transform

14.2 The .4-Model

14.3 A Glance at Quantum Electrodynamics

Problems

15. The Operator Approach

15.1 The .4-Model

15.2 A Glance at Quantum Electrodynamics

15.3 The Role of Effective Quantities in Physics

15.4 A Glance at Renormalization

15.5 The Convergence Problem in Quantum Field Theory

15.6 Rigorous Perspectives

16. Peculiarities of Gauge Theories

16.1 Basic Difficulties

16.2 The Principle of Critical Action

16.3 The Language of Physicists

16.4 The Importance of the Higgs Particle

16.5 Integration over Orbit Spaces

16.6 The Magic Faddeev–Popov Formula and Ghosts

16.7 The BRST Symmetry

16.8 The Power of Cohomology

16.9 The Batalin–Vilkovisky Formalism

16.10 A Glance at Quantum Symmetries

17. A Panorama of the Literature

17.1 Introduction to Quantum Field Theory

17.2 Standard Literature in Quantum Field Theory

17.3 Rigorous Approaches to Quantum Field Theory

17.4 The Fascinating Interplay between Modern Physics and Mathematics

17.5 The Monster Group, Vertex Algebras, and Physics

17.6 Historical Development of Quantum Field Theory

17.7 General Literature in Mathematics and Physics

17.8 Encyclopedias

17.9 Highlights of Physics in the 20th Century

17.10 Actual Information

Appendix

A.1 Notation

A.2 The International System of Units

A.3 The Planck System

A.4 The Energetic System

A.5 The Beauty of Dimensional Analysis

A.6 The Similarity Principle in Physics

Epilogue

References

List of Symbols

Index