Coulomb Gauge

Core Idea

The Coulomb gauge (∇·A = 0) simplifies calculations in the non-relativistic limit and atomic physics. In this gauge, the scalar potential φ satisfies Poisson's equation ∇²φ = -ρ/ε₀ (instantaneous Coulomb interaction), while the vector potential A satisfies a wave equation with a source term. This gauge naturally separates Coulomb (instantaneous) interactions from radiation effects.

Explainer

From gauge transformations you know that the scalar potential φ and vector potential A⃗ are not uniquely determined by the physical fields E⃗ and B⃗. You can add ∇λ to A⃗ and subtract ∂λ/∂t from φ for any scalar function λ, leaving E⃗ and B⃗ unchanged. A gauge choice is a condition that fixes λ and thereby picks a unique representative pair (φ, A⃗) from each equivalence class of physically identical potentials. The Coulomb gauge imposes ∇·A⃗ = 0 — the vector potential is divergence-free.

The payoff for this choice is that Poisson's equation ∇²φ = −ρ/ε₀ drops out immediately for the scalar potential. You already solved Poisson's equation in electrostatics: its solution is the familiar Coulomb integral φ(r⃗, t) = (1/4πε₀) ∫ ρ(r⃗', t)/|r⃗ − r⃗'| d³r'. Notice that t appears only as a parameter — the scalar potential in the Coulomb gauge responds to the charge distribution *instantaneously*, as if electrostatics applied at every moment. This apparent violation of relativity is a gauge artifact, not physics: the physically measurable fields E⃗ and B⃗ still propagate at c, and no signal actually travels faster than light. The instantaneous φ and the wave-equation terms in A⃗ conspire to give causal fields.

The separation of Coulomb and radiation physics is the Coulomb gauge's great practical advantage. In atomic and molecular physics, the dominant interaction between an electron and a nucleus is the Coulomb attraction, which is captured entirely by φ. The radiation field — light absorbed or emitted during transitions — lives in the transverse part of A⃗ (the part satisfying ∇·A⃗ = 0). Perturbation theory for atomic transitions can therefore be structured cleanly: compute the unperturbed atomic states from the Coulomb potential, then treat the transverse vector potential as a perturbation responsible for photon emission and absorption. This is the standard approach in non-relativistic quantum electrodynamics and quantum optics.

The Coulomb gauge trades manifest Lorentz covariance for simplicity in the non-relativistic domain. The Lorenz gauge (∂_μ A^μ = 0) is the preferred choice in relativistic calculations because it treats space and time symmetrically and makes the covariance of Maxwell's equations explicit. But for atoms, molecules, and condensed matter systems — where velocities are far below c and the primary interaction is electrostatic — the Coulomb gauge is the natural language. Recognizing which gauge is most convenient for a given problem, and knowing what can and cannot depend on the gauge choice, is a core skill in advanced electrodynamics.