Quantum Electrodynamics: Gauge Symmetry, Scattering, and Precision
Twelve research-grade QED units covering Dirac fields, local gauge symmetry, physical photons, amplitudes, Compton scattering, Ward identities, vacuum polarization, coupling running, anomalous moments and cavity QED. Original native mathematical derivations, explicit worked calculations, lecture resources and selected licensed figures accompany each topic.
Published syllabus
Course Units
Relativistic wave equations
Relativistic wave equations must reconcile quantum propagation with Lorentz invariance and energy positivity. The Klein–Gordon equation is second order, and a consistent one-particle probability interpretation is difficult. The Dirac equation is first order in time and naturally carries spin-half states, but full particle-antiparticle physics uses second quantization.
Dirac spinors and gamma matrices
Dirac spinors transform under Lorentz spinor representations rather than as ordinary four-vectors. Gamma matrices satisfy the Clifford anticommutation algebra, ensuring the squared Dirac operator reproduces relativistic dispersion. Positive- and negative-energy plane-wave solutions combine into quantized electron and positron fields.
Electromagnetic gauge symmetry
Local U(1) invariance requires replacing ∂mu with a covariant derivative Dmu=∂mu+ieA_mu in a consistent charge convention. The electromagnetic gauge field transforms to compensate local spinor phase rotations. Gauge invariance constrains allowed couplings and currents; it is a redundancy in description, not a direct measurable transformation.
Quantizing the electromagnetic field
The free electromagnetic field decomposes into photon modes with physical transverse polarizations after gauge constraints are handled. Canonical gauge-fixed quantization may temporarily introduce unphysical polarizations that cancel from observables. Photon number and field quadratures describe different measurements, and a coherent laser mode need not have a definite photon number.
Electron-photon vertex
The electron-photon vertex is fixed by the minimal-coupling interaction ψbarγ^mu A_muψ. It conserves charge and connects spinor currents to photon polarization. Its use in perturbation theory includes external spinor normalization, momentum conservation and gauge-invariant cancellation among diagrams.
Feynman rules and scattering amplitudes
Scattering amplitudes are expanded in powers of e; cross sections arise from squared amplitudes summed or averaged over appropriate spins and polarizations, integrated over Lorentz-invariant phase space. External particles are modeled as asymptotic states. Conservation laws constrain the kinematics before any matrix algebra is attempted.
Compton, Moller, and Bhabha processes
Compton scattering describes photon-electron scattering; Møller scattering involves two electrons; Bhabha scattering involves an electron and positron. The different external particles determine which diagram channels contribute and what exchange antisymmetry signs appear. Relativistic kinematics and flux factors cannot be reused indiscriminately among these processes.
Ward identities
Ward–Takahashi identities link vertex and propagator corrections through gauge symmetry. Contracting an on-shell photon amplitude with its momentum should enforce decoupling of unphysical longitudinal polarizations. These identities constrain renormalization factors and provide strong checks on symbolic perturbation calculations.
Loops and vacuum polarization
Virtual charged particles contribute loop corrections to photon propagation, producing vacuum polarization. Ultraviolet divergences require regularization and renormalization; gauge symmetry constrains the tensor structure. Low-energy polarization of the quantum vacuum is not the same as a classical dielectric permittivity without identifying the response regime.
Renormalization and running alpha
Renormalization relates measured electric charge to a scale-dependent effective coupling. QED's coupling increases slowly with energy in perturbation theory due to charged vacuum fluctuations. The running depends on active charged species and renormalization scheme, while low-energy precision predictions combine multiple diagram orders.
Precision tests of QED
Precision QED connects loop calculations with anomalous magnetic moments, Lamb shifts and scattering measurements. Confronting theory with data requires bound-state effects, hadronic contributions and propagated constant uncertainties where relevant. Agreement at extraordinary precision is a composite achievement of theory, metrology and error accounting.
QED in materials, cavities, and quantum optics
Cavity QED and circuit QED enhance light-matter interaction through confinement of electromagnetic modes. The Jaynes–Cummings model captures a rotating-wave-coupled two-level system and one oscillator mode, leading to vacuum Rabi splitting and excitation exchange under appropriate approximations.