Quantum Field Theory II: Quantum Electrodynamics - A Bridge between Mathematicians and Physicists

Preface

Contents

Part I. Introduction

Prologue

Mathematical Principles of Modern Natural Philosophy

Basic Principles

The Infinitesimal Strategy and Differential Equations

The Optimality Principle

The Basic Notion of Action in Physics and the Idea ofQuantization

The Method of the Green's Function

Harmonic Analysis and the Fourier Method

The Method of Averaging and the Theory of Distributions

The Symbolic Method

Gauge Theory -- Local Symmetry and the Description of Interactions by Gauge Fields

The Challenge of Dark Matter

The Basic Strategy of Extracting Finite Information from Infinities -- Ariadne's Thread in Renormalization Theory

Renormalization Theory in a Nutshell

Effective Frequency and Running Coupling Constant of an Anharmonic Oscillator

The Zeta Function and Riemann's Idea of Analytic Continuation

Meromorphic Functions and Mittag-Leffler's Ideaof Subtractions

The Square of the Dirac Delta Function

Regularization of Divergent Integrals in Baby Renormalization Theory

Momentum Cut-off and the Method of Power-Counting

The Choice of the Normalization Momentum

The Method of Differentiating Parameter Integrals

The Method of Taylor Subtraction

Overlapping Divergences

The Role of Counterterms

Euler's Gamma Function

Integration Tricks

Dimensional Regularization via Analytic Continuation

Pauli--Villars Regularization

Analytic Regularization

Application to Algebraic Feynman Integrals inMinkowski Space

Distribution-Valued Meromorphic Functions

Application to Newton's Equation of Motion

Hints for Further Reading.

Further Regularization Methods in Mathematics

Euler's Philosophy

Adiabatic Regularization of Divergent Series

Adiabatic Regularization of Oscillating Integrals

Regularization by Averaging

Borel Regularization

Hadamard's Finite Part of Divergent Integrals

Infinite-Dimensional Gaussian Integrals and the Zeta Function Regularization

Trouble in Mathematics

Interchanging Limits

The Ambiguity of Regularization Methods

Pseudo-Convergence

Ill-Posed Problems

Mathemagics

The Power of Combinatorics

Algebras

The Algebra of Multilinear Functionals

Fusion, Splitting, and Hopf Algebras

The Bialgebra of Linear Differential Operators

The Definition of Hopf Algebras

Power Series Expansion and Hopf Algebras

The Importance of Cancellations

The Kepler Equation and the LagrangeInversion Formula

The Composition Formula for Power Series

The Faà di Bruno Hopf Algebra for the FormalDiffeomorphism Group of the Complex Plane

The Generalized Zimmermann Forest Formula

The Logarithmic Function and Schur Polynomials

Correlation Functions in Quantum Field Theory

Random Variables, Moments, and Cumulants

Symmetry and Hopf Algebras

The Strategy of Coordinatization in Mathematics and Physics

The Coordinate Hopf Algebra of a Finite Group

The Coordinate Hopf Algebra of an Operator Group

The Tannaka--Krein Duality for Compact Lie Groups

Regularization and Rota--Baxter Algebras

Regularization of the Laurent Series

Projection Operators

The q-Integral

The Volterra--Spitzer Exponential Formula

The Importance of the Exponential Function inMathematics and Physics

Partially Ordered Sets and Combinatorics

Incidence Algebras and the Zeta Function

The Möbius Function as an Inverse Function

The Inclusion--Exclusion Principle in Combinatorics

Applications to Number Theory

Hints for Further Reading

The Strategy of Equivalence Classes in Mathematics

Equivalence Classes in Algebra

The Gaussian Quotient Ring and the QuadraticReciprocity Law in Number Theory

Application of the Fermat--Euler Theorem in Coding Theory

Quotient Rings, Quotient Groups, and Quotient Fields

Linear Quotient Spaces

Ideals and Quotient Algebras

Superfunctions and the Heaviside Calculus in Electrical Engineering

Equivalence Classes in Geometry

The Basic Idea of Geometry Epitomized by Klein's Erlangen Program

Symmetry Spaces, Orbit Spaces, and Homogeneous Spaces

The Space of Quantum States

Real Projective Spaces

Complex Projective Spaces

The Shape of the Universe

Equivalence Classes in Topology

Topological Quotient Spaces

Physical Fields, Observers, Bundles, and Cocycles

Generalized Physical Fields and Sheaves

Deformations, Mapping Classes, and Topological Charges

Poincaré's Fundamental Group

Loop Spaces and Higher Homotopy Groups

Homology, Cohomology, and Electrodynamics

Bott's Periodicity Theorem

K-Theory

Application to Fredholm Operators

Hints for Further Reading

The Strategy of Partial Ordering

Feynman Diagrams

The Abstract Entropy Principle in Thermodynamics

Convergence of Generalized Sequences

Inductive and Projective Topologies

Inductive and Projective Limits

Classes, Sets, and Non-Sets

The Fixed-Point Theorem of Bourbaki--Kneser

Zorn's Lemma

Leibniz's Infinitesimals and Non-Standard Analysis

Filters and Ultrafilters

The Full-Rigged Real Line

Part II. Basic Ideas in Classical Mechanics

Geometrical Optics

Ariadne's Thread in Geometrical Optics

Fermat's Principle of Least Time

Huygens' Principle on Wave Fronts

Carathéodory's Royal Road to Geometrical Optics

The Duality between Light Rays and Wave Fronts

From Wave Fronts to Light Rays

From Light Rays to Wave Fronts

The Jacobi Approach to Focal Points

Lie's Contact Geometry

Basic Ideas

Contact Manifolds and Contact Transformations

Applications to Geometrical Optics

Equilibrium Thermodynamics and LegendreSubmanifolds

Light Rays and Non-Euclidean Geometry

Linear Symplectic Spaces

The Kähler Form of a Complex Hilbert Space

The Refraction Index and Geodesics

The Trick of Gauge Fixing

Geodesic Flow

Hamilton's Duality Trick and Cogeodesic Flow

The Principle of Minimal Geodesic Energy

Spherical Geometry

The Global Gauss--Bonnet Theorem

De Rham Cohomology and the Chern Class ofthe Sphere

The Beltrami Model

The Poincaré Model of Hyperbolic Geometry

Kähler Geometry and the Gaussian Curvature

Kähler--Einstein Geometry

Symplectic Geometry

Riemannian Geometry

Ariadne's Thread in Gauge Theory

Parallel Transport of Physical Information -- the Key to Modern Physics

The Phase Equation and Fiber Bundles

Gauge Transformations and Gauge-InvariantDifferential Forms

Perspectives

Classification of Two-Dimensional Compact Manifolds

The Poincaré Conjecture and the Ricci Flow

A Glance at Modern Optimization Theory

Hints for Further Reading

The Principle of Critical Action and the HarmonicOscillator -- Ariadne's Thread in Classical Mechanics

Prototypes of Extremal Problems

The Motion of a Particle

Newtonian Mechanics

A Glance at the History of the Calculus of Variations

Lagrangian Mechanics

The Harmonic Oscillator

The Euler--Lagrange Equation

Jacobi's Accessory Eigenvalue Problem

The Morse Index

The Anharmonic Oscillator

The Ginzburg--Landau Potential and the Higgs Potential

Damped Oscillations, Stability, and EnergyDissipation

Resonance and Small Divisors

Symmetry and Conservation Laws

The Symmetries of the Harmonic Oscillator

The Noether Theorem

The Pendulum and Dynamical Systems

The Equation of Motion

Elliptic Integrals and Elliptic Functions

The Phase Space of the Pendulum and Bundles

Hamiltonian Mechanics

The Canonical Equation

The Hamiltonian Flow

The Hamilton--Jacobi Partial Differential Equation

Poissonian Mechanics

Poisson Brackets and the Equation of Motion

Conservation Laws

Symplectic Geometry

The Canonical Equations

Symplectic Transformations

The Spherical Pendulum

The Gaussian Principle of Critical Constraint

The Lagrangian Approach

The Hamiltonian Approach

Geodesics of Shortest Length

The Lie Group SU(E3) of Rotations

Conservation of Angular Momentum

Lie's Momentum Map

Carathéodory's Royal Road to the Calculus of Variations

The Fundamental Equation

Lagrangian Submanifolds in Symplectic Geometry

The Initial-Value Problem for the Hamilton--Jacobi Equation

Solution of Carathéodory's Fundamental Equation

Hints for Further Reading

Part III. Basic Ideas in Quantum Mechanics

Quantization of the Harmonic Oscillator -- Ariadne's Thread in Quantization

Complete Orthonormal Systems

Bosonic Creation and Annihilation Operators

Heisenberg's Quantum Mechanics

Heisenberg's Equation of Motion

Heisenberg's Uncertainty Inequality for the Harmonic Oscillator

Quantization of Energy

The Transition Probabilities

The Wightman Functions

The Correlation Functions

Schrödinger's Quantum Mechanics

The Schrödinger Equation

States, Observables, and Measurements

The Free Motion of a Quantum Particle

The Harmonic Oscillator

The Passage to the Heisenberg Picture

Heisenberg's Uncertainty Principle

Unstable Quantum States and the Energy-Time Uncertainty Relation

Schrödinger's Coherent States

Feynman's Quantum Mechanics

Main Ideas

The Diffusion Kernel and the Euclidean Strategy in Quantum Physics

Probability Amplitudes and the Formal Propagator Theory

Von Neumann's Rigorous Approach

The Prototype of the Operator Calculus

The General Operator Calculus

Rigorous Propagator Theory

The Free Quantum Particle as a Paradigm ofFunctional Analysis

The Free Hamiltonian

The Rescaled Fourier Transform

The Quantized Harmonic Oscillator and the Maslov Index

Ideal Gases and von Neumann's Density Operator

The Feynman Path Integral

The Basic Strategy

The Basic Definition

Application to the Free Quantum Particle

Application to the Harmonic Oscillator

The Propagator Hypothesis

Motivation of Feynman's Path Integral

Finite-Dimensional Gaussian Integrals

Basic Formulas

Free Moments, the Wick Theorem, and FeynmanDiagrams

Full Moments and Perturbation Theory

Rigorous Infinite-Dimensional Gaussian Integrals

The Infinite-Dimensional Dispersion Operator

Zeta Function Regularization and Infinite-Dimensional Determinants

Application to the Free Quantum Particle

Application to the Quantized Harmonic Oscillator

The Spectral Hypothesis

The Semi-Classical WKB Method

Brownian Motion

The Macroscopic Diffusion Law

Einstein's Key Formulas for the Brownian Motion

The Random Walk of Particles

The Rigorous Wiener Path Integral

The Feynman--Kac Formula

Weyl Quantization

The Formal Moyal Star Product

Deformation Quantization of the Harmonic Oscillator

Weyl Ordering

Operator Kernels

The Formal Weyl Calculus

The Rigorous Weyl Calculus

Two Magic Formulas

The Formal Feynman Path Integral for the Propagator Kernel

The Relation between the Scattering Kernel and the Propagator Kernel

The Poincaré--Wirtinger Calculus

Bargmann's Holomorphic Quantization

The Stone--Von Neumann Uniqueness Theorem

The Prototype of the Weyl Relation

The Main Theorem

C*-Algebras

Operator Ideals

Symplectic Geometry and the Weyl QuantizationFunctor

A Glance at the Algebraic Approach to Quantum Physics

States and Observables

Gleason's Extension Theorem -- the Main Theorem of Quantum Logic

The Finite Standard Model in Statistical Physics as a Paradigm

Information, Entropy, and the Measure of Disorder

Semiclassical Statistical Physics

The Classical Ideal Gas

Bose--Einstein Statistics

Fermi--Dirac Statistics

Thermodynamic Equilibrium and KMS-States

Quasi-Stationary Thermodynamic Processes and Irreversibility

The Photon Mill on Earth

Von Neumann Algebras

Von Neumann's Bicommutant Theorem

The Murray--von Neumann Classification of Factors

The Tomita--Takesaki Theory and KMS-States

Connes' Noncommutative Geometry

Jordan Algebras

The Supersymmetric Harmonic Oscillator

Hints for Further Reading

Quantum Particles on the Real Line -- Ariadne's Thread in Scattering Theory

Classical Dynamics Versus Quantum Dynamics

The Stationary Schrödinger Equation

One-Dimensional Quantum Motion in a Square-WellPotential

Free Motion

Scattering States and the S-Matrix

Bound States

Bound-State Energies and the Singularities of theS-Matrix

The Energetic Riemann Surface, Resonances, and the Breit--Wigner Formula

The Jost Functions

The Fourier--Stieltjes Transformation

Generalized Eigenfunctions of the Hamiltonian

Quantum Dynamics and the Scattering Operator

The Feynman Propagator

Tunnelling of Quantum Particles and Radioactive Decay

The Method of the Green's Function in a Nutshell

The Inhomogeneous Helmholtz Equation

The Retarded Green's Function, and the Existence and Uniqueness Theorem

The Advanced Green's Function

Perturbation of the Retarded and Advanced Green's Function

Feynman's Regularized Fourier Method

The Lippmann--Schwinger Integral Equation

The Born Approximation

The Existence and Uniqueness Theorem via Banach's Fixed Point Theorem

Hypoellipticity

A Glance at General Scattering Theory

The Formal Basic Idea

The Rigorous Time-Dependent Approach

The Rigorous Time-Independent Approach

Applications to Quantum Mechanics

A Glance at Quantum Field Theory

Hints for Further Reading

Part IV. Quantum Electrodynamics (QED)

Creation and Annihilation Operators

The Bosonic Fock Space

The Particle Number Operator

The Ground State

The Fermionic Fock Space and the Pauli Principle

General Construction

The Main Strategy of Quantum Electrodynamics

The Basic Equations in Quantum Electrodynamics

The Classical Lagrangian

The Gauge Condition

The Free Quantum Fields of Electrons, Positrons,and Photons

Classical Free Fields

The Lattice Strategy in Quantum Electrodynamics

The High-Energy Limit and the Low-Energy Limit

The Free Electromagnetic Field

The Free Electron Field

Quantization

The Free Photon Quantum Field

The Free Electron Quantum Field and Antiparticles

The Spin of Photons

The Ground State Energy and the Normal Product

The Importance of Mathematical Models

The Trouble with Virtual Photons

Indefinite Inner Product Spaces

Representation of the Creation and Annihilation Operators in QED

Gupta--Bleuler Quantization

The Interacting Quantum Field, and the MagicDyson Series for the S-Matrix

Dyson's Key Formula

The Basic Strategy of Reduction Formulas

The Wick Theorem

Feynman Propagators

Discrete Feynman Propagators for Photons and Electrons

Regularized Discrete Propagators

The Continuum Limit of Feynman Propagators

Classical Wave Propagation versus Feynman Propagator

The Beauty of Feynman Diagrams in QED

Compton Effect and Feynman Rules in Position Space

Symmetry Properties

Summary of the Feynman Rules in Momentum Space

Typical Examples

The Formal Language of Physicists

Transition Probabilities and Cross Sections of ScatteringProcesses

The Crucial Limits

Appendix: Table of Feynman Rules

Applications to Physical Effects

Compton Effect

Duality between Light Waves and Light Particles in the History of Physics

The Trace Method for Computing Cross Sections

Relativistic Invariance

Asymptotically Free Electrons in an ExternalElectromagnetic Field

The Key Formula for the Cross Section

Application to Yukawa Scattering

Application to Coulomb Scattering

Motivation of the Key Formula via S-Matrix

Perspectives

Bound Electrons in an External ElectromagneticField

The Spontaneous Emission of Photons by the Atom

Motivation of the Key Formula

Intensity of Spectral Lines

Cherenkov Radiation

Part V. Renormalization

The Continuum Limit

The Fundamental Limits

The Formal Limits Fail

Basic Ideas of Renormalization

The Effective Mass and the Effective Charge of the Electron

The Counterterms of the Modified Lagrangian

The Compensation Principle

Fundamental Invariance Principles

Dimensional Regularization of Discrete AlgebraicFeynman Integrals

Multiplicative Renormalization

The Theory of Approximation Schemes in Mathematics

Radiative Corrections of Lowest Order

Primitive Divergent Feynman Graphs

Vacuum Polarization

Radiative Corrections of the Propagators

The Photon Propagator

The Electron Propagator

The Vertex Correction and the Ward Identity

The Counterterms of the Lagrangian and the Compensation Principle

Application to Physical Problems

Radiative Correction of the Coulomb Potential

The Anomalous Magnetic Moment of the Electron

The Anomalous Magnetic Moment of the Muon

The Lamb Shift

Photon-Photon Scattering

A Glance at Renormalization to all Orders ofPerturbation Theory

One-Particle Irreducible Feynman Graphs andDivergences

Overlapping Divergences and Manoukian's EquivalencePrinciple

The Renormalizability of Quantum Electrodynamics

Automated Multi-Loop Computations in PerturbationTheory

Perspectives

BPHZ Renormalization

Bogoliubov's Iterative R-Method

Zimmermann's Forest Formula

The Classical BPHZ Method

The Causal Epstein--Glaser S-Matrix Approach

Kreimer's Hopf Algebra Revolution

The History of the Hopf Algebra Approach

Renormalization and the Iterative BirkhoffFactorization for Complex Lie Groups

The Renormalization of QuantumElectrodynamics

The Scope of the Riemann--Hilbert Problem

The Gaussian Hypergeometric Differential Equation

The Confluent Hypergeometric Function and theSpectrum of the Hydrogen Atom

Hilbert's 21th Problem

The Transport of Information in Nature

Stable Transport of Energy and Solitons

Ariadne's Thread in Soliton Theory

Resonances

The Role of Integrable Systems in Nature

The BFFO Hopf Superalgebra Approach

The BRST Approach and Algebraic Renormalization

Analytic Renormalization and Distribution-ValuedAnalytic Functions

Computational Strategies

The Renormalization Group

Operator Product Expansions

Binary Planar Graphs and the Renormalizationof Quantum Electrodynamics

The Master Ward Identity

Trouble in Quantum Electrodynamics

The Landau Inconsistency Problem in QuantumElectrodynamics

The Lack of Asymptotic Freedom in QuantumElectrodynamics

Hints for Further Reading

Epilogue

References

List of Symbols

Index