Quantum Field Theory is the language of modern particle physics, condensed matter, and cosmology. Our new Quantum Field Theory course spans 16 detailed chapters that take you from the classical field equations all the way to lattice gauge theory and beyond-the-Standard-Model physics — with every derivation spelled out.
Part I: Foundations (Chapters 1–4)
The course begins with Classical Field Theory, reviewing the Lagrangian formalism for fields, Noether's theorem, and the stress-energy tensor. We derive conservation laws for scalar, spinor, and vector fields — setting the stage for quantization.
Chapter 2: Canonical Quantization promotes classical fields to operator-valued distributions. We construct the Fock space for the free scalar field, derive the equal-time commutation relations, and show how the vacuum energy divergence motivates normal ordering. The chapter closes with the quantization of the Dirac field, including the spin-statistics connection and the necessity of anticommutators.
Chapter 3: The Path Integral introduces Feynman's sum-over-histories approach. Starting from the quantum mechanical propagator, we extend to field theory, derive the generating functional Z[J], and show how functional derivatives produce Green's functions. The Gaussian integration formula is derived carefully, as it underpins every free-field computation that follows.
Chapter 4: Interacting Fields and Feynman Diagrams is where the machinery pays off. We derive the Feynman rules for φ&sup4; theory from the perturbative expansion of the path integral, explain symmetry factors with explicit examples, and compute the one-loop self-energy and vertex corrections. Every diagram is drawn inline with labeled momenta and coupling constants.
Part II: QED and Renormalization (Chapters 5–8)
Quantum Electrodynamics is the crown jewel of perturbative QFT. We derive the QED Lagrangian from local U(1) gauge invariance, obtain the Feynman rules for electron-photon vertices, and compute tree-level cross sections for Compton scattering, Bhabha scattering, and pair annihilation. The derivation of the anomalous magnetic moment (g−2) at one loop is presented in full, reproducing Schwinger's famous α/2π result.
The Renormalization chapters tackle the ultraviolet divergences head-on. We introduce dimensional regularization, compute divergent integrals using Feynman parameters, and implement the MS-bar renormalization scheme. The physical meaning of renormalization is emphasized: it is not a trick to sweep infinities under the rug, but a systematic procedure for relating bare and physical parameters.
Running Couplings and the Renormalization Group receive a dedicated chapter. We derive the beta function for QED and QCD, show asymptotic freedom in non-Abelian gauge theories, and include Python simulations that numerically integrate the RG equations so students can visualize how coupling constants evolve with energy scale.
Part III: Gauge Theories and the Standard Model (Chapters 9–12)
We build non-Abelian gauge theory from scratch — starting with the SU(N) Lie algebra, constructing the Yang–Mills Lagrangian, and deriving the Faddeev–Popov ghost procedure for path-integral quantization. The QCD chapter covers color confinement, the running of α_s, deep inelastic scattering, and the parton model.
The Electroweak Theory chapter derives the SU(2)×U(1) gauge structure, the Higgs mechanism for mass generation, and the Weinberg mixing angle. We compute the W and Z boson masses in terms of the vacuum expectation value and gauge couplings, then verify against experimental values.
Anomalies are treated rigorously. We derive the ABJ anomaly from the triangle diagram, explain its role in π&sup0; → γγ decay, and show why anomaly cancellation constrains the fermion content of the Standard Model.
Part IV: Beyond the Standard Model (Chapters 13–16)
The final part surveys Supersymmetry (SUSY), introducing the SUSY algebra, chiral and vector superfields, and the Minimal Supersymmetric Standard Model (MSSM). We explain how SUSY solves the hierarchy problem and unifies gauge couplings at the GUT scale.
Effective Field Theory provides the modern perspective on QFT as a tower of theories valid at different energy scales. We derive the Euler–Heisenberg Lagrangian as a worked example, then discuss the Standard Model itself as an EFT.
The course concludes with Lattice QFT, showing how to discretize the path integral on a spacetime lattice, implement Wilson's gauge action, and extract physical observables via Monte Carlo simulation. This chapter bridges the gap between analytic and computational approaches to quantum field theory.
Python Simulations Throughout
The course includes interactive Python simulations for running coupling evolution, Monte Carlo integration of lattice actions, and numerical evaluation of Feynman parameter integrals. All simulations run directly in the browser using Pyodide — no installation required.
Who Should Take This Course?
The course is aimed at graduate students in theoretical physics or advanced undergraduates who have completed quantum mechanics and special relativity. Familiarity with the Dirac equation and basic group theory is helpful but not strictly required — we review the essentials as needed.