Foundations and Frontiers of QFT · Vol I

Quantum Field Theory in 20 chapters — from “Why Quantum Field Theory?” to “Effective Field Theory”.

Chapters

  1. Why Quantum Field Theory?
    • A Subject Worth Its Difficulty
    • The Klein Paradox: Relativistic Quantum Mechanics Breaks
    • Dirac’s Sea and Its Trouble
    • The Lamb Shift: An Experiment That Demands QFT
    • The Vacuum Has Structure: The Casimir Effect
    • Fields vs. Particles: A Preview
    • The Road from Here
    • Problems
    • Chapter 1 Summary
  2. Classical Fields and the Mathematical Toolkit
    • The Lagrangian Perspective, Revisited
    • Special Relativity in the Notation We Will Use
    • From Particles to Fields: Classical Field Theory
    • Symmetries and Noether’s Theorem
    • The Electromagnetic Field
    • Dimensional Analysis in Field Theory
    • Summary and Look Ahead
    • Chapter 2 Summary
    • Problems
  3. Groups and Representations
    • Groups: What and Why
    • Lie Groups and Lie Algebras
    • Representations
    • The Lorentz Group
    • The Poincaré Group
    • Summary and Look Ahead
    • Problems
  4. Poincaré Invariance and One-Particle States
    • What Is a One-Particle State?
    • Wigner’s Strategy: Labels from Symmetries
    • Boost, Stabilize, Classify: The Little Group Method
    • Massive Particles and Spin
    • Massless Particles and Helicity
    • The Result: A Short List of Possible Particles
    • Why Nature Picks What She Picks
    • Summary and Look Ahead
    • Problems
  5. Many-Particle States and Fock Space
    • From One Particle to Many
    • Occupation-Number Representation
    • Creation and Annihilation Operators
    • The Fock Space
    • Working with Fock States: Some Examples
    • What Fock Space Does and Does Not Give Us
    • Summary and Look Ahead
    • Problems
  6. Why Fields? From Particles to Fields
    • What We Want: A Local, Lorentz-Invariant S-Matrix
    • Why Creation and Annihilation Operators Alone Are Not Enough
    • The Key Construction: Bundling Operators into a Field
    • Microcausality and the Necessity of Antiparticles
    • Spin-Statistics, Previewed
    • Fields with Spin
    • The Weinberg Perspective and What Comes Next
    • Summary and Look Ahead
    • Problems
  7. Canonical Quantization of the Free Scalar
    • The Faster Route
    • From Lagrangian to Hamiltonian for a Field
    • Canonical Commutation Relations
    • Mode Expansion of the Free Scalar
    • The Hamiltonian and the Vacuum
    • The Feynman Propagator
    • The Casimir Effect, Finally
    • Summary and Look Ahead
    • Problems
  8. Quantum Mechanics as 0+1D QFT
    • The Dictionary
    • The Dimensional Reduction
    • Computing the Oscillator Using the QFT Machinery
    • Euclidean Time and Thermal Physics
    • A First Look at the Renormalization Group
    • The Climb of the Dimensional Ladder
    • Summary and Look Ahead
    • Problems
  9. Interactions and Perturbation Theory
    • What Is an Interaction?
    • The Interaction Picture
    • The S-Matrix
    • Wick’s Theorem, Starting from the Oscillator
    • Feynman Diagrams: What They Are and Why They Exist
    • Worked Example: Tree-Level Scattering in Phi-Fourth Theory
    • The Path from Here to (g-2)
    • Summary and Look Ahead
    • Problems
  10. From Amplitudes to Observables
    • The Disaster of Squaring a Delta Function
    • Lorentz-Invariant Phase Space
    • Master Formulas: Cross Sections and Decay Rates
    • Fully Worked Example: Scalar-Scalar Scattering
    • The Optical Theorem
    • A Concrete Sketch of (g-2)
    • Summary and Look Ahead
    • Problems
  11. The Path Integral from Scratch
    • Why a Second Formulation?
    • The Core Idea: Sum Over Paths
    • Time-Slicing: From Evolution Operator to Path Integral
    • The Euclidean Path Integral
    • The Harmonic Oscillator as a Worked Example
    • From Quantum Mechanics to Field Theory
    • The Generating Functional
    • The Classical Limit
    • Summary and Look Ahead
    • Problems
  12. Path Integral and Canonical, Compared
    • Two Languages, One Physics
    • The Two-Point Function, In Both Formulations
    • The Translation Dictionary
    • When Each Formulation Is Better
    • The Places Where the Equivalence Is Subtle
    • A Worked Example: Phi-Fourth Tree-Level Amplitude, Redux
    • Summary and Look Ahead
    • Problems
  13. Symmetries in the Quantum Theory
    • The Master Identity
    • Schwinger-Dyson Equations: Classical Equations of Motion, Quantized
    • Ward Identities: Noether’s Theorem Quantized
    • Anomalies: When the Measure Fights Back
    • Ward Identities in Action: QED, Previewed
    • A First Encounter with the Anomaly, in 1+1D
    • The End of Part III
    • Summary and Look Ahead
    • Problems
  14. First Loop Calculations
    • What Is a Loop?
    • The Tadpole Diagram
    • What the Divergence Means
    • Power Counting: Which Integrals Diverge?
    • Regularization, Briefly
    • The Program for Part IV
    • Summary and Look Ahead
    • Problems
  15. Regularization Methods Compared
    • Why We Want Another Regulator
    • Dimensional Regularization: The Idea
    • Comparison with Hard Cutoff
    • The Dimensional-Regularization Integral Formula
    • Pauli-Villars, Briefly
    • The Renormalization Scale
    • A Subtlety: Chirality and Gamma-Five
    • Summary and Look Ahead
    • Problems
  16. The Loop Calculation Toolkit
    • The One Master Integral
    • Feynman Parameters
    • Tensor Reductions
    • Example: A One-Loop Vertex Correction
    • A Preview: Multi-Loop Integrals
    • Summary and Look Ahead
    • Chapter 16 Summary
    • Problems
  17. Renormalization of Phi-Fourth: Old School and New
    • Two Views of Renormalization
    • Counter-Terms and the Bare Lagrangian
    • The One-Loop Calculation
    • The Pattern of Divergences
    • The Renormalization Conditions
    • The Wilsonian Reframe
    • Summary and Look Ahead
    • Problems
  18. The Renormalization Group
    • Why Parameters Run
    • The Beta Function
    • The Callan-Symanzik Equation
    • Fixed Points and Universality
    • Relevant, Marginal, and Irrelevant Operators
    • Asymptotic Freedom: A Preview of QCD
    • Summary and Look Ahead
    • Problems
  19. Spontaneous Symmetry Breaking
    • The Phenomenon
    • The Simplest Example
    • Goldstone’s Theorem, Derived
    • The Non-Abelian Generalization
    • The Higgs Mechanism
    • Exceptions: No Goldstones in Low Dimensions
    • Summary and Look Ahead
    • Chapter 19 Summary
    • Problems
  20. Effective Field Theory
    • The Core Insight
    • Fermi’s Theory as the Canonical Example
    • Einstein Gravity as an Effective Field Theory
    • The Decoupling Theorem
    • EFT Matching: The Procedure
    • The Standard Model as an EFT
    • A Retrospective, and the View Forward
    • Summary and Look Ahead
    • Problems