Introduction to Quantum Field Theory - Classical Mechanics to Gauge Field Theories

Cover
Half-title
Title page
Copyright information
Contents
Detailed Contents
Preface
Organization of the Book
How to Use This Book
Acknowledgments
1 Lorentz and Poincaré Invariance
1.1 Introduction
1.1.1 Inertial Reference Frames
1.1.2 Galilean Relativity
1.2 Lorentz and Poincaré Transformations
1.2.1 Postulates of Special Relativity and Their Implications
1.2.2 Active and Passive Transformations
1.2.3 Lorentz Group
1.2.4 Poincaré Group
1.2.5 Representation-Independent Poincaré Lie Algebra
1.3 Representations of the Lorentz Group
1.3.1 Labeling Representations of the Lorentz Group
1.3.2 Lorentz Transformations of Weyl Spinors
1.4 Poincaré Group and the Little Group
1.4.1 Intrinsic Spin and the Poincaré Group
1.4.2 The Little Group
Summary
Problems
2 Classical Mechanics
2.1 Lagrangian Formulation
2.1.1 Euler-Lagrange Equations
2.1.2 Hamilton’s Principle
2.1.3 Lagrange Multipliers and Constraints
2.2 Symmetries, Noether’s Theorem and Conservation Laws
2.3 Small Oscillations and Normal Modes
2.4 Hamiltonian Formulation
2.4.1 Hamiltonian and Hamilton’s Equations
2.4.2 Poisson Brackets
2.4.3 Liouville Equation and Liouville’s Theorem
2.4.4 Canonical Transformations
2.5 Relation to Quantum Mechanics
2.6 Relativistic Kinematics
2.7 Electromagnetism
2.7.1 Maxwell’s Equations
2.7.2 Electromagnetic Waves
2.7.3 Gauge Transformations and Gauge Fixing
2.8 Analytic Relativistic Mechanics
2.9 Constrained Hamiltonian Systems
2.9.1 Construction of the Hamiltonian Approach
2.9.2 Summary of the Dirac-Bergmann Algorithm
2.9.3 Gauge Fixing, the Dirac Bracket and Quantization
2.9.4 Dirac Bracket and Canonical Quantization
Summary
Problems
3 Relativistic Classical Fields
3.1 Relativistic Classical Scalar Fields
3.2 Noether’s Theorem and Symmetries
3.2.1 Noether’s Theorem for Classical Fields
3.2.2 Stress-Energy Tensor
3.2.3 Angular Momentum Tensor
3.2.4 Intrinsic Angular Momentum
3.2.5 Internal Symmetries
3.2.6 Belinfante-Rosenfeld Tensor
3.2.7 Noether’s Theorem and Poisson Brackets
3.2.8 Generators of the Poincaré Group
3.3 Classical Electromagnetic Field
3.3.1 Lagrangian Formulation of Electromagnetism
3.3.2 Hamiltonian Formulation of Electromagnetism
Summary
Problems
4 Relativistic Quantum Mechanics
4.1 Review of Quantum Mechanics
4.1.1 Postulates of Quantum Mechanics
4.1.2 Notation, Linear and Antilinear Operators
4.1.3 Symmetry Transformations and Wigner’s Theorem
4.1.4 Projective Representations of Symmetry Groups
4.1.5 Symmetry in Quantum Systems
4.1.6 Parity Operator
4.1.7 Time Reversal Operator
4.1.8 Additive and Multiplicative Quantum Numbers
4.1.9 Systems of Identical Particles and Fock Space
4.1.10 Charge Conjugation and Antiparticles
4.1.11 Interaction Picture in Quantum Mechanics
4.1.12 Path Integrals in Quantum Mechanics
4.2 Wavepackets and Dispersion
4.3 Klein-Gordon Equation
4.3.1 Formulation of the Klein-Gordon Equation
4.3.2 Conserved Current
4.3.3 Interaction with a Scalar Potential
4.3.4 Interaction with an Electromagnetic Field
4.4 Dirac Equation
4.4.1 Formulation of the Dirac Equation
4.4.2 Probability Current
4.4.3 Nonrelativistic Limit and Relativistic Classical Limit
4.4.4 Interaction with an Electromagnetic Field
4.4.5 Lorentz Covariance of the Dirac Equation
4.4.6 Dirac Adjoint Spinor
4.4.7 Plane Wave Solutions
4.4.8 Completeness and Projectors
4.4.9 Spin Vector
4.4.10 Covariant Interactions and Bilinears
4.4.11 Poincaré Group and the Dirac Equation
4.5 P, C and T: Discrete Transformations
4.5.1 Parity Transformation
4.5.2 Charge Conjugation
4.5.3 Time Reversal
4.5.4 CPT Transformation
4.6 Chirality and Weyl and Majorana Fermions
4.6.1 Helicity
4.6.2 Chirality
4.6.3 Weyl Spinors and the Weyl Equations
4.6.4 Plane Wave Solutions in the Chiral Representation
4.6.5 Majorana Fermions
4.6.6 Weyl Spinor Notation
4.7 Additional Topics
Summary
Problems
5 Introduction to Particle Physics
5.1 Overview of Particle Physics
5.2 The Standard Model
5.2.1 Development of Quantum Electrodynamics (QED)
5.2.2 Development of Quantum Chromodynamics (QCD)
5.2.3 Development of Electroweak (EW) Theory
5.2.4 Quark Mixing and the CKM Matrix
5.2.5 Neutrino Mixing and the PMNS Matrix
5.2.6 Majorana Neutrinos and Double Beta Decay
5.3 Representations of SU(N) and the Quark Model
5.3.1 Multiplets of SU(N) and Young Tableaux
5.3.2 Quark Model
Summary
Problems
6 Formulation of Quantum Field Theory
6.1 Lessons from Quantum Mechanics
6.1.1 Quantization of Normal Modes
6.1.2 Motivation for Relativistic Quantum Field Theory
6.2 Scalar Particles
6.2.1 Free Scalar Field
6.2.2 Field Configuration and Momentum Density Space
6.2.3 Covariant Operator Formulation
6.2.4 Poincaré Covariance
6.2.5 Causality and Spacelike Separations
6.2.6 Feynman Propagator for Scalar Particles
6.2.7 Charged Scalar Field
6.2.8 Wick’s Theorem
6.2.9 Functional Integral Formulation
6.2.10 Euclidean Space Formulation
6.2.11 Generating Functional for a Scalar Field
6.3 Fermions
6.3.1 Annihilation and Creation Operators
6.3.2 Fock Space and Grassmann Algebra
6.3.3 Feynman Path Integral for Fermions
6.3.4 Fock Space for Dirac Fermions
6.3.5 Functional Integral for Dirac Fermions
6.3.6 Canonical Quantization of Dirac Fermions
6.3.7 Quantum Field Theory for Dirac Fermions
6.3.8 Generating Functional for Dirac Fermions
6.4 Photons
6.4.1 Canonical Quantization of the Electromagnetic Field
6.4.2 Fock Space for Photons
6.4.3 Functional Integral for Photons
6.4.4 Gauge-Fixing
6.4.5 Covariant Canonical Quantization for Photons
6.5 Massive Vector Bosons
6.5.1 Classical Massive Vector Field
6.5.2 Normal Modes of the Massive Vector Field
6.5.3 Quantization of the Massive Vector Field
6.5.4 Functional Integral for Massive Vector Bosons
6.5.5 Covariant Canonical Quantization for Massive Vector Bosons
Summary
Problems
7 Interacting Quantum Field Theories
7.1 Physical Spectrum of States
7.2 Källén-Lehmann Spectral Representation
7.3 Scattering Cross-Sections and Decay Rates
7.3.1 Cross-Section
7.3.2 Relating the Cross-Section to the S-Matrix
7.3.3 Particle Decay Rates
7.3.4 Two-Body Scattering (2→2) and Mandelstam Variables
7.3.5 Unitarity of the S-Matrix and the Optical Theorem
7.4 Interaction Picture and Feynman Diagrams
7.4.1 Interaction Picture
7.4.2 Feynman Diagrams
7.4.3 Feynman Rules in Momentum Space
7.5 Calculating Invariant Amplitudes
7.5.1 LSZ Reduction Formula for Scalars
7.5.2 LSZ for Fermions
7.5.3 LSZ for Photons
7.6 Feynman Rules
7.6.1 External States and Internal Lines
7.6.2 Examples of Interacting Theories
7.6.3 Example Tree-Level Results
7.6.4 Substitution Rules and Crossing Symmetry
7.6.5 Examples of Calculations of Cross-Sections
7.6.6 Unstable Particles
Summary
Problems
8 Symmetries and Renormalization
8.1 Discrete Symmetries: P, C and T
8.1.1 Parity
8.1.2 Charge Conjugation
8.1.3 Time Reversal
8.1.4 The CPT Theorem
8.1.5 Spin-Statistics Connection
8.2 Generating Functionals and the Effective Action
8.2.1 Generating Functional for Connected Green’s Functions
8.2.2 The Effective Action
8.2.3 Effective Potential
8.2.4 Loop Expansion
8.3 Schwinger-Dyson Equations
8.3.1 Derivation of Schwinger-Dyson Equations
8.3.2 Ward and Ward-Takahashi Identities
8.4 Renormalization
8.4.1 Superficial Degree of Divergence
8.4.2 Superficial Divergences in QED
8.5 Renormalized QED
8.5.1 QED Schwinger-Dyson Equations with Bare Fields
8.5.2 Renormalized QED Green’s Functions
8.5.3 Renormalization Group
8.6 Regularization
8.6.1 Regularization Methods
8.6.2 Dimensional Regularization
8.7 Renormalized Perturbation Theory
8.7.1 Renormalized Perturbation Theory for
ϕ[sup(4)]
8.7.2 Renormalized Perturbative Yukawa Theory
8.7.3 Renormalized Perturbative QED
8.7.4 Minimal Subtraction Renormalization Schemes
8.7.5 Running Coupling and Running Mass in QED
8.7.6 Renormalization Group Flow and Fixed Points
8.8 Spontaneous Symmetry Breaking
8.9 Casimir Effect
Summary
Problems
9 Nonabelian Gauge Theories
9.1 Nonabelian Gauge Theories
9.1.1 Formulation of Nonabelian Gauge Theories
9.1.2 Wilson Lines and Wilson Loops
9.1.3 Quantization of Nonabelian Gauge Theories
9.2 Quantum Chromodynamics
9.2.1 QCD Functional Integral
9.2.2 Renormalization in QCD
9.2.3 Running Coupling and Running Quark Mass
9.2.4 BRST Invariance
9.2.5 Lattice QCD
9.3 Anomalies
9.4 Introduction to the Standard Model
9.4.1 Electroweak Symmetry Breaking
9.4.2 Quarks and Leptons
Summary
Problems
Appendix
A.1 Physical Constants
A.2 Notation and Useful Results
A.2.1 Levi-Civita Tensor
A.2.2 Dirac Delta Function and Jacobians
A.2.3 Fourier Transforms
A.2.4 Cauchy’s Integral Theorem
A.2.5 Wirtinger Calculus
A.2.6 Exactness, Conservative Vector Fields and Integrating Factors
A.2.7 Tensor and Exterior Products
A.3 Dirac Algebra
A.4 Euclidean Space Conventions
A.5 Feynman Parameterization
A.6 Dimensional Regularization
A.7 Group Theory and Lie Groups
A.7.1 Elements of Group Theory
A.7.2 Lie Groups
A.7.3 Unitary Representations of Lie Groups
A.8 Results for Matrices
References
Index