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📘 Quantum Field Theory — Standard Contents

I. Introduction and Foundations

1. Motivation and historical background
2. Natural units and metric conventions
3. Relativistic quantum mechanics
4. The need for fields in relativistic quantum theory

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II. Free Field Theory

2.1 Klein–Gordon Field

• Lagrangian and equations of motion
• Canonical quantization
• One-particle states and energy–momentum tensor

2.2 Dirac Field

• Dirac equation and spinor solutions
• Canonical quantization and anticommutation relations
• Charge conjugation, parity, and time reversal

2.3 Maxwell Field

• Gauge invariance and gauge fixing
• Quantization of the electromagnetic field
• Photon polarization and physical degrees of freedom

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III. Interacting Fields and Feynman Rules

1. Interaction picture and Dyson expansion
2. Time-ordered products and the S-matrix
3. Wick’s theorem and contractions
4. Feynman diagrams and Feynman rules
5. Symmetries and conserved currents (Noether theorem)

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IV. Perturbation Theory and Renormalization

1. Scattering amplitudes and cross sections
2. Divergences and regularization
   • Cutoff and dimensional regularization
   • Anomalous dimensions and gamma-matrix algebra
3. Renormalization constants and counterterms
4. Beta function and renormalization group equations
5. Effective field theory perspective

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V. Gauge Field Theories

1. Local gauge symmetry
2. Abelian gauge theory (QED)
3. Non-Abelian gauge theory (Yang–Mills)
4. Gauge fixing and Faddeev–Popov ghosts
5. BRST symmetry and renormalization consistency
6. Spontaneous symmetry breaking and the Higgs mechanism

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VI. Advanced Topics

1. Path-integral formulation
   • From operator formalism to functional integrals
   • Fermionic path integrals and Grassmann variables
2. Ward identities and Slavnov–Taylor identities
3. Effective field theory (EFT) and low-energy expansion
   • Heavy-quark effective theory (HQET)
   • Chiral perturbation theory (ChPT)
4. Quantum anomalies (chiral and gauge)
5. Vacuum structure and instantons

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VII. Applications and Modern Developments

1. Quantum Chromodynamics (QCD) and asymptotic freedom
2. Electroweak unification
3. Symmetry breaking and particle mass generation
4. Renormalization-group flows and critical phenomena
5. Finite-temperature and thermal field theory
6. Applications in condensed matter and cosmology

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Appendix

• A. Notation and conventions
• B. Gamma-matrix identities and spinor algebra
• C. Lie-group and Lie-algebra formulas
• D. Momentum-space integrals and delta-function tricks
• E. Natural units and physical constants