The QED Path Integral
A full lecture reconstruction showing how bosonic and fermionic coherent-state path integrals combine into the QED generating functional, and how changes of variables turn the Coulomb-gauge Hamiltonian path integral into a manifestly Lorentz-invariant Maxwell–Dirac action with sources.
Introduction
The previous lecture introduced path integrals in a deliberately modest way. It did not claim that they solve interacting quantum mechanics exactly. It claimed something more useful: they provide a Lagrangian-based representation of quantum time evolution, and for weakly interacting systems they generate perturbation theory in a form that is often far cleaner than the Hamiltonian operator method.
This lecture now applies that machinery to quantum electrodynamics.
That is a big step, because QED contains both: - bosonic degrees of freedom, namely the electromagnetic field, - and fermionic degrees of freedom, namely electrons and positrons.
A full lecture reconstruction introducing path integrals as a Lagrangian reformulation of quantum mechanics, starting from the Feynman–Hibbs phase-space path integral, then developing coherent-state path integrals for bosons and Grassmann coherent-state path integrals for fermions as preparation for Lagrangian perturbation theory in QFT.
A full lecture reconstruction showing how Lorentz-invariant perturbative QED emerges from the QED path integral as variational calculus on a Gaussian generating functional, and how the surviving terms organize into Feynman diagrams and Feynman rules.