Quantum Field Theory - An Introduction

Contents
1 Prologue
2 Many Particle Physics as a Quantum Field Theory
	2.1 Introduction
	2.2 Non-relativistic Particles
		2.2.1 Identical and Indistinguishable Particles
		2.2.2 The Example of Weakly Interacting Particles
		2.2.3 Hamiltonian and Stationary States
		2.2.4 Particles with Spin
	2.3 Second Quantization in the Schrödinger Picture
	2.4 Second Quantization in the Heisenberg Picture
3 Degenerate Fermi and Bose Gases
	3.1 The Limit of Weakly Interacting Particles
	3.2 Degenerate Fermi Gas
		3.2.1 The Ground State |mathcalO>
		3.2.2 Particles and Holes
		3.2.3 The Grand Canonical Free Energy
	3.3 Degenerate Bose Gas
		3.3.1 Landau's Criterion for Superfluidity
		3.3.2 Vacuum Expectation Value
	3.4 Spontaneous Symmetry Breaking
4 The Action Principle and Noether's Theorem
	4.1 The Action
		4.1.1 The Euler–Lagrange Equations
	4.2 Canonical Momenta, Poisson Brackets and Commutation Relations
	4.3 Noether's Theorem
		4.3.1 Conservation Laws and Continuity Equations
		4.3.2 Definition of Symmetry
		4.3.3  Examples of Symmetries
		4.3.4 Proof of Noether's Theorem
	4.4 Phase Symmetry and the Conservation of Particle Number
5 Non-relativistic Space–Time Symmetries
	5.1 Translation Invariance and the Stress Tensor
	5.2 Galilean Symmetry
	5.3 Scale Invariance
		5.3.1 Improving the Stress Tensor
		5.3.2 The Consequences of Scale Invariance
	5.4 Special Schrödinger Symmetry
	5.5 Summary
6 Space–Time Symmetry and Relativistic Field Theory
	6.1 Quantum Mechanics and Special Relativity
	6.2 Coordinates
	6.3 Scalars, Vectors, Tensors
	6.4 The Metric
	6.5 Symmetry of Space–Time
	6.6 The Symmetries of Minkowski Space
	6.7 Natural Units
	6.8 Relativistic Fields
7 The Real Scalar Quantum Field Theory
	7.1 Constructing a Relativistic Lagrangian Density
	7.2 Field Equation and Commutation Relations
	7.3 Noether's Theorem and Poincare Symmetry
	7.4 Correlation Functions of the Real Scalar Field
	7.5 The Free Scalar Field
	7.6 Consequences of Spacetime Symmetry
	7.7 Spectral Theorem
	7.8 Normalization of the Spectral Function
	7.9 Analyticity
		7.9.1 The Reeh–Schlieder Theorem
	7.10 Conformal Symmetry
8 Emergent Relativistic Symmetry
	8.1 Phonons
	8.2 The Debye Theory of Solids
	8.3 Relativistic Fermions in Graphene
9 The Dirac Field Theory
	9.1 The Dirac Equation
	9.2 Solving the Dirac Equation
	9.3 Lorentz Invariance of the Dirac Equation
	9.4 Spin of the Dirac Field
	9.5 Phase Symmetry and the Conservation of Charge
		9.5.1 Conserved Number Current
		9.5.2 Relativistic Noether's Theorem for the Dirac Equation
		9.5.3 Alternative Proof of Noether's Theorem
	9.6 Spacetime Symmetry
		9.6.1 Translation Invariance and the Stress Tensor
		9.6.2 Lorentz Transformations
		9.6.3 Stress Tensor and Killing Vectors
10 Photons
	10.1 Relativistic Classical Electrodynamics
	10.2 Quantization
		10.2.1 Negative Normed States
		10.2.2 Physical State Condition
		10.2.3 Null States and the Equivalence Relation
	10.3 Space–Time Symmetries of the Photon
	10.4 Massive Photon
	10.5 Quantum Electrodynamics
		10.5.1 C, P and T
11 Functional Methods
	11.1 Functional Derivative
	11.2 Functional Integral
	11.3 Generating Functional for Free Scalar Fields
		11.3.1 Wick's Theorem for Scalar Fields
		11.3.2  Generating Functional as a Functional Integral
	11.4 The Interacting Real Scalar Field
12 More Functional Integrals
	12.1 Functional Integrals for the Photon Field
	12.2 Functional Methods for Fermions
	12.3 Generating Functionals for Non-relativistic Fermions
		12.3.1 Interacting Non-relativistic Fermions
	12.4 The Dirac Field
		12.4.1 2 Point Function for the Dirac Field
		12.4.2 Generating Functional for the Dirac Field
		12.4.3 Functional Integral for the Dirac Field
	12.5 Functional Quantum Electrodynamics
13 The Weakly Coupled Real Scalar Field
	13.1 Counterterms
	13.2 Computation of the 2 Point Function
	13.3 Feynman Diagrams
	13.4 Simplifications of Feynman Diagrams
	13.5 Computation of a One-Loop Feynman Integral
		13.5.1 Dimensional Regularization
		13.5.2 Wick Rotation
		13.5.3 Feynman Parameters
		13.5.4 Integration in 2ω-Dimensions
		13.5.5 Asymptotic Expansion at 2ωsim4
		13.5.6 Inverse Wick Rotation
		13.5.7 The Mass Tadpole
		13.5.8 Euclidean Quantum Field Theory
		13.5.9 The 2 Point and 4 Point Functions
	13.6 Subtraction Schemes
	13.7 Renormalization Group
	13.8 Appendix: Integration Formulae
		13.8.1 Euler's Gamma Function
		13.8.2 Feynman Parameter Formula
		13.8.3 Dimensional Regularization Integral
14 More Theory of the Real Scalar Field
	14.1 The S Matrix
		14.1.1 The  T Matrix
	14.2 The LSZ Formula
	14.3 Elastic Two-Particle Scattering
	14.4 Connected and Irreducible Generating Functionals
		14.4.1 Connected Correlation Functions and the Linked Cluster Theorem
		14.4.2 Connected Correlation Functions
		14.4.3 Cancelation of Vacuum Diagrams
		14.4.4 Irreducible Correlation Function
	14.5 Derivation of the LSZ Formula
15 Perturbative Quantum Electrodynamics
	15.1 Counterterms
	15.2 The Generating Functional in Perturbation Theory
		15.2.1 Wick's Theorem for Photons and Electrons
	15.3 Feynman Diagrams
	15.4 Feynman Rules
	15.5 The Electron 2 Point Function
	15.6 Feynman Rules in Momentum Space
	15.7 The Photon 2 Point Function
	15.8 Quantum Corrections of the Coulomb Potential
	15.9 The Electron 2 Point Function
	15.10 Radiative Correction of the Vertex
		15.10.1  Electromagnetic form Factors
		15.10.2  Anomalous Magnetic Moment
	15.11 Photon Production, the Soft Photon Theorem
	15.12 Furry's Theorem
	15.13 The Ward–Takahashi Identities
16 Epilogue
Index