Hamiltonian Gauge Theory
A full lecture reconstruction showing how the gauge-invariant Maxwell–Dirac Lagrangian leads, after constraints and gauge fixing, to the Hamiltonian formulation of electrodynamics in Coulomb gauge, with only the two transverse dynamical photon degrees of freedom remaining.
Introduction
Up to this point in the course, gauge invariance has mostly appeared in Lagrangian language. The Maxwell–Dirac theory was introduced as a local \(U(1)\)-invariant field theory, and that local symmetry determined the form of the electromagnetic interaction. But to build canonical quantum field theory, we need a **Hamiltonian**.
That is where the trouble begins.
For ordinary unconstrained fields, the Hamiltonian formalism is straightforward: identify the canonical coordinates, compute their conjugate momenta, perform the Legendre transform, and obtain a phase-space dynamics. But for a gauge field, things are different. The gauge potential \(A_\mu\) has four components, so naively one might think it gives four coordinates and four momenta, making eight phase-space variables. The lecture’s main point is that this is wrong. Local gauge symmetry removes part of this apparent phase space.
A full lecture reconstruction showing how the inertial vacuum of a quantum field appears thermal to a uniformly accelerating observer, using a massless scalar field in 1+1 dimensions, two-mode squeezed states, Rindler coordinates, and the emergence of the Unruh temperature.
A full lecture reconstruction showing how the free electromagnetic field in Coulomb gauge becomes an infinite set of quantum harmonic oscillators whose quanta are photons, and why gauge-invariant observables still preserve relativistic causality in photon detection.