Electromagnetic Field Quantization (QED)

Core Idea

Quantizing the electromagnetic field promotes the vector potential A^mu to an operator. Gauge invariance introduces complications: unphysical degrees of freedom must be removed or constrained. The result is a theory of photons -- massless spin-1 particles with two physical polarization states.

Explainer

The classical electromagnetic field is described by the four-vector potential A^mu = (phi, A), with the electric and magnetic fields given by E = -grad phi - dA/dt and B = curl A. The Lagrangian density is L = -(1/4)F_{mu nu}F^{mu nu}, where F_{mu nu} = partial_mu A_nu - partial_nu A_mu is the field strength tensor. Gauge invariance -- the fact that A^mu and A^mu + partial^mu Lambda describe the same physics -- is both the defining feature of electrodynamics and the source of all technical complications in quantization.

The problem is that gauge invariance means A^mu has redundant degrees of freedom. A massive vector field would have three physical polarizations, but the massless photon has only two (the two transverse polarizations). You must somehow eliminate the unphysical degrees of freedom. In Coulomb gauge (div A = 0), the two transverse components of A are the dynamical variables, and quantization proceeds cleanly: each transverse mode with wave vector k and polarization lambda is a harmonic oscillator with creation operator a_{k,lambda}-dagger. The photon is a quantum of this oscillator. The drawback is that Coulomb gauge is not manifestly Lorentz covariant.

In covariant gauges (like Lorenz gauge, partial_mu A^mu = 0), all four components of A^mu participate, preserving manifest Lorentz invariance. But this introduces unphysical states -- timelike and longitudinal photons with negative norm. The Gupta-Bleuler method handles this by restricting the physical Hilbert space: only states satisfying the gauge condition (as an operator equation on kets) are physical, and the unphysical polarizations cancel in all physical matrix elements. More modern approaches use the BRST formalism, which introduces ghost fields that systematically cancel the unphysical contributions.

After quantization, the electromagnetic field describes photons: massless spin-1 particles with two polarization states (left and right circular, or equivalently, two linear polarizations). The field operator A^mu(x) creates and destroys photons at spacetime point x. The vacuum has no photons but is not empty -- quantum fluctuations of E and B produce measurable effects. Coupling the quantized photon field to the quantized Dirac field via the interaction term e psi-bar gamma^mu psi A_mu gives quantum electrodynamics (QED), the most precisely tested theory in all of physics.