Lecture notes

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.

Schrödinger, Heisenberg and interaction (Dirac) pictures. Exponential representations. Derivation of the Dyson series, time ordering, the T-exponential, definition of the S-matrix.

Cross sections in Bjorken–Drell conventions. Perturbative expansion of the S-matrix and Wick’s theorem.

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.

Overview, the interaction vertex, classification of processes. Muon pair production, e⁺e⁻ → μ⁺μ⁻.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

The Lorentz group and its Lie algebra: infinitesimal transformations, rotation and boost generators, the su(2)⊕su(2) structure of the complexified algebra and the finite-dimensional representations (j₊, j₋). The Poincaré algebra, the Pauli–Lubanski vector and the Casimir operators P² and W² — mass and spin or helicity. The (½, 0) and (0, ½) representations: Weyl spinors, the 2-to-1 homomorphism SL(2,C) → SO⁺(1,3), the sigma matrices σ^μ, σ̄^μ and the transformation laws of every spinor index position.

Spinor products and the index staircase rule; Lorentz invariance of χη and χ̄η̄; commuting versus anticommuting spinors; the complete list of bilinear covariants and the Fierz identities. Four-component spinors: gamma matrices in the chiral representation, the Clifford algebra, γ⁵ and chirality, the Dirac adjoint, the Dirac equation from the coupled Weyl system, charge conjugation and Majorana spinors. Free Lagrangians for Dirac, Weyl and Majorana fermions and the energy-momentum tensor. End of Chapter 9.

Why supersymmetry still matters: the hierarchy problem, gauge coupling unification in the MSSM, neutralino dark matter. The free Wess–Zumino model: SUSY transformations of φ and χ, invariance of the action, the auxiliary field F and off-shell closure. The SUSY algebra derived from commutators of variations, {Q, Q̄} = σ^μP_μ and [Q, P_μ] = 0; the Hamiltonian as a sum of squares, H ≥ 0; the supersymmetry current and the supercharges.

Massless supermultiplets built by helicity raising and lowering: chiral (scalar and Weyl fermion), vector (gauge boson and gaugino), graviton and gravitino. Boson–fermion degeneracy, hence broken supersymmetry; why only N = 1 for the Standard Model. The most general renormalizable interaction: holomorphy of the superpotential W from a Fierz cancellation, Wⁱ = ∂W/∂φ_i, elimination of the auxiliary fields and V = |Wⁱ|², mass degeneracy within a supermultiplet, a worked single-chiral example, the non-renormalization theorem. End of Chapter 10 and of the course.