Class Notes

• Lie Groups in QFT
• Introduction - First day Relativistic notation, QED interactions, field Lagrangians, the Klein–Gordon equation, and Noether’s theorem

  Natural units, and four-vectors, a basic QED scattering diagram, causality and antiparticles, the Lagrangian and Hamiltonian formulations of scalar fields, and the connection between continuous symmetries and conserved currents.

• Noether’s theorem & Canonical Quantization of a Free Scalar Field How a classical Klein–Gordon field becomes a quantum field whose excitations are spin-0 particles.
  • Continuous symmetries and conserved currents, including the U(1) charge current of a complex scalar field
  • Spacetime translations and the stress-energy tensor
  • Canonical commutation relations for a scalar field and its conjugate momentum
  • Fourier expansion of the Klein–Gordon field
  • Interpreting each momentum mode as a harmonic oscillator
  • Creation and annihilation operators
  • The scalar-field Hamiltonian, vacuum energy, and one-particle states
  • The momentum operator of the quantized field
• Canonical Quantization of the Dirac Field

  The Dirac-field mode expansions, fermionic anticommutation relations, particles and antiparticles, the Hamiltonian and momentum operators, negative-energy problems, and causality of local observables.

• Dirac Propagator and Parity How is the Feynman propagator for the Dirac field constructed, and how does the Dirac field transform under parity?

  Construction of the Dirac Green function and time-ordered Feynman propagator, including the pole prescription, followed by the parity transformation of Dirac creation and annihilation operators and the opposite intrinsic parities of particles and antiparticles.

• Discrete symmetries and combined CPT Symmetry

  Time reversal, charge conjugation, the transformation table and CPT, followed by applying the symmetries to the Dirac Lagrangian and deriving the symmetry properties of relativistic spin-1/2 particles and their interactions.

• Interacting Vacua, Time-Ordered Products, and Wick’s Theorem

  How correlation functions in an interacting quantum field theory can be rewritten using the free vacuum and the interaction picture. They introduce the Gell-Mann–Low formula, normal ordering, field contractions, the Feynman propagator, and Wick’s theorem as the foundation of perturbation theory and Feynman-diagram calculations.

• Field Contractions and the Two-Point Feynman Propagator

  We split a scalar field into positive- and negative-frequency parts, use normal ordering to rearrange creation and annihilation operators, and define the contraction of two fields. We then show that the contraction equals the vacuum expectation value of the time-ordered product, namely the Feynman propagator, providing the basic two-field case underlying Wick’s theorem.

• Perturbative Correlation Functions and Feynman Rules

  We develop perturbative quantum field theory from correlation functions and Wick’s theorem, then derive the Feynman rules for scalar φ⁴ theory, Yukawa theory, and quantum electrodynamics. We also introduce connected and amputated diagrams, scattering amplitudes, the S-matrix, fermion propagators, and photon interactions.