Lecture notes

• Free Fields — a recap
  §1–§1.6
  Real and complex scalar fields, the Dirac field, the photon field: mode expansions, commutation relations, normalisation conventions. Feynman propagators. Noether’s theorem and conserved currents. Distributed as prerequisite reading, not as a lecture.

• Lecture 1 — Pictures of quantum dynamics; the Dyson series
  §2–§3.4 · 10 March 2026
  Schrödinger, Heisenberg and interaction (Dirac) pictures. Exponential representations. Derivation of the Dyson series, time ordering, the T-exponential, definition of the S-matrix.
• Lecture 2 — Cross sections and Feynman rules
  §3.5–§4.2 · 13 March 2026
  Cross sections in Bjorken–Drell conventions. Perturbative expansion of the S-matrix and Wick’s theorem.
• Lecture 3 — Feynman rules, continued
  §4.3–§4.8 · 17 March 2026
  Scalar φ⁴ theory. First-order QED and why those diagrams vanish. Second-order contributions. Momentum-space rules. The fermion minus sign. Summary of the QED Feynman rules.
• Lecture 4 — QED scattering processes; muon pair production
  §5–§5.2 · 20 March 2026
  Overview, the interaction vertex, classification of processes. Muon pair production, e⁺e⁻ → μ⁺μ⁻.
• Lecture 5 — Muon pairs completed; Bhabha scattering
  §5.2–§5.3 · 24 March 2026
  BD normalisation in the cross section and the cancellation of the (2m) factors. Centre-of-mass kinematics, ultrarelativistic limit. Differential and total cross section for e⁺e⁻ → μ⁺μ⁻. Helicity: chirality versus helicity, selection rules, angular distributions from Wigner d-matrices. Bhabha scattering: s- and t-channel diagrams, relative sign from Fermi statistics, the X_ss and X_tt calculations.
• Lecture 6 — Bhabha scattering, continued
  §5.3 · 27 March 2026
  The interference term X_st: Fierz rearrangement and trace evaluation. Full spin-averaged cross section. Forward divergence and its regularisation. Comparison with muon pair production via crossing symmetry. Helicity amplitude analysis. Experimental applications: luminosity monitoring at colliders, precision tests.
• Lecture 7 — Compton scattering
  §5.4 · 31 March 2026
  Tree-level diagrams: s- and u-channel, and why there is no t-channel. Individual amplitudes, the tensor amplitude, the Ward identity — both the general proof from gauge invariance and explicit verification. Unpolarised cross section: spin and polarisation sums, the traces X_ss, X_uu, X_su. The Klein–Nishina formula and the Thomson limit. Compton wavelength shift, laboratory kinematics. Polarised Compton scattering: gauge choices for the polarisation vectors, Coulomb-gauge simplifications, the polarised Klein–Nishina formula.
• Lecture 8 — Scattering by an external field
  §5.5 · 7 April 2026
  The external field in the S-matrix, classical and quantised. Matrix element and modified Feynman rules: the external-field vertex, energy conservation only. Cross section for external-field scattering. Coulomb scattering and the derivation of the Mott formula. The Rutherford limit. Spin and helicity in Coulomb scattering: helicity conservation and flip, the polarisation transfer matrix.
• Lecture 9 — Bremsstrahlung; end of Chapter 5
  §5.6 · 10 April 2026
  Bremsstrahlung in a Coulomb field: diagrams and matrix element. Soft photon approximation and the factorisation theorem. The Bethe–Heitler formula. Infrared divergence with a photon mass regulator. Comparison with the virtual vertex correction — both O(α) relative to elastic. Bloch–Nordsieck cancellation of the IR divergences, and the inclusive cross section as the physical observable.
• Lecture 10 — φ⁴ review; renormalisation begins
  §4.3 review, §6.1 · 17 April 2026
  Review of scalar λφ⁴ theory: Lagrangian, Feynman rules, tree-level φφ→φφ scattering, one-loop corrections, the loop integral. Start of Chapter 6: renormalisation of λφ⁴ as a warm-up. Dimensional regularisation — d = 4−ε, the scale μ, the master integral J_n(Δ), the Γ function, Laurent expansion, UV poles as 1/ε.
• Lecture 11 — Counterterms; vacuum polarisation
  §6.1–§6.2 · 21 April 2026
  The tadpole diagram: one-loop propagator correction, mass counterterm δ_m, renormalised self-energy, physical mass. The bubble diagram and coupling renormalisation. The constants Z_φ, Z_m, Z_λ at one loop. Renormalisation schemes: MS-bar versus on-shell, scheme dependence of finite parts and scheme independence of m_phys. Photon self-energy in QED: screening of the bare charge, the Lamb shift, the contribution to muon g−2. Transversality from the Ward identity; modified photon propagator by Dyson resummation.
• Lecture 12 — Electron propagator renormalisation
  §6.3 · 24 April 2026
  One-loop self-energy Σ(p), Lorentz decomposition, Taylor expansion around p̸ = m. Physical mass condition, wave-function renormalisation Z₂ = 1/(1−B), renormalised propagator with unit residue. Worked example in both the on-shell and the MS-bar scheme.
• Lecture 13 — Vertex correction to the end of Chapter 6
  §6.4–§6.7 · 28 April 2026
  One-loop vertex correction and its renormalisation. The Ward–Takahashi identity: algebraic derivation, one-loop verification, all-orders telescoping-sum proof; hence Z₁ = Z₂. Charge renormalisation and the universality of electric charge. Counterterm reformulation of the QED Lagrangian. Form factors F₁, F₂, F₃; the Thomson condition F₁(0) = 1 and the anomalous magnetic moment as F₂(0). External line renormalisation and the Mandl–Shaw adiabatic-switching derivation. Systematic renormalisation of QED with gauge fixing. General theory: superficial degree of divergence, the master formula, super-renormalisable / renormalisable / non-renormalisable interactions. Only three primitively divergent 1PI amplitudes in QED; Furry’s theorem; light-by-light finiteness. Subdivergences and BPHZ. Effective field theories, Euler–Heisenberg, Wilson’s renormalisation group and the floating cutoff; relevant, marginal and irrelevant operators. Accidental symmetries.
• Lecture 14 — General methods for interacting fields
  §8–§8.5 · 12 May 2026
  From free to interacting fields; Green functions as the practical route to S-matrix elements. Properties of physical states: forward light cone, unique vacuum, stable one-particle states, mass gap, vanishing vacuum expectation value, asymptotic completeness — and Haag’s theorem as a caveat. In-fields and the asymptotic condition, the Yang–Feldman construction, the weak asymptotic condition. Out-fields and the S-matrix: unitarity, Lorentz invariance. Comparison with the interaction-picture approach.
• Lecture 15 — The Källén–Lehmann spectral representation
  §8.6 · 15 May 2026
  A complete set of physical states inserted between two Heisenberg field operators. One-particle pole at m² plus the multi-particle continuum above threshold. The spectral density ρ(σ²), its positivity and support. Spectral representation of the exact Feynman propagator. The one-particle pole with residue Z, the sum rule 1 = Z + ∫ρ_cont, hence 0 ≤ Z ≤ 1. Free-theory limit ρ(σ²) = δ(σ²−m²), Z = 1.
• Lecture 16 — LSZ reduction and Green’s functions
  §8.7–§8.10 · 19 May 2026
  The LSZ reduction formula for scalar fields: removing a particle from the in-state and from the out-state, conversion to a four-dimensional integral, surface terms. The general reduction formula. LSZ at work on tree-level φ⁴ scattering. The reduction formula for Dirac fields — incoming and outgoing electrons and positrons, each with its spinor factor — and its extension to photons. Green’s functions and the S-matrix: the Gell-Mann–Low formula, connected versus disconnected diagrams, the linked-cluster theorem, the Dyson equation and 1PI diagrams.
• Lecture 17
  22 May 2026
• Lecture 18
  26 May 2026
• Lecture 19
  27 May 2026
• Lecture 20
  29 May 2026

Lectures 17 to 20 have no entry in the lecture log, so no topic summary is given here rather than an invented one. The notes themselves are complete.