Path Integrals
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.
Introduction
The previous lectures pushed Hamiltonian perturbation theory in QED far enough to reveal both its power and its limitations.
On the one hand, the Hamiltonian formalism gave a fully defined quantum theory: - photons, electrons, and positrons are operator excitations in one Hilbert space, - the interaction Hamiltonian is explicit, - and in principle every process can be computed by ordinary quantum-mechanical perturbation theory.
On the other hand, even very simple-looking processes become algebraically ugly. The lecture reviews Compton scattering and reminds us that at leading order in \(\alpha\), the amplitude comes from four separate time-ordered operator terms. Even worse, those four “processes” are not Lorent
A full lecture reconstruction showing why naive low-order perturbative QED corrections to particle energies are problematic, why Compton scattering is the first clean finite order-α process, and how Lorentz-invariant scattering amplitudes emerge only after coherent addition of all electron and positron intermediate-state processes.
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.