Classical limit of QED

The classical limit of quantum electrodynamics is the high-occupation-number regime in which Maxwell's equations are recovered as expectation values of the underlying quantum-field operators in coherent states with macroscopic photon occupation. For a single mode of the EM field with annihilation operator â, the coherent state |α⟩ satisfies â|α⟩ = α|α⟩ with complex amplitude α, and the expectation value ⟨α| Ê |α⟩ of the electric-field operator is exactly the classical electric field of a wave with amplitude |α| and phase arg α. The relative quantum fluctuation σ_E / ⟨E⟩ scales as 1/√⟨n⟩ where ⟨n⟩ = |α|² is the mean photon number, so for a laser at full power (⟨n⟩ ≈ 10^{15}) the fluctuation is ≈ 10^{−7.5} and the classical wave description is accurate to parts per million.

The historical inversion — that classical electromagnetism is the limit of a quantum theory, not the foundation that quantum mechanics modifies — was Feynman's pedagogical organisation in Volume II of the Lectures on Physics (1963): he derives Maxwell's equations from the photon, not the other way around. Two corrections to Maxwell's equations cannot be derived from the classical theory and instead come directly from QED at one loop: the Lamb shift (1947) of hydrogen-atom energy levels by ≈ 1057 MHz, due to vacuum polarisation; and Schwinger's anomalous magnetic moment a_e = α/(2π) ≈ 0.00116, the most-precisely-tested prediction in physics. Both are quantum effects with no classical limit — they vanish as ℏ → 0 — and both signal that the classical theory is incomplete. But the classical theory is consistent, computable, and the language every successor theory has had to learn: QED is built on Maxwell's vector potential A^μ, the Standard Model gauge unification builds on QED, and whatever theory eventually subsumes the Standard Model will build on the structure already in place. The next branch of physics is QUANTUM. Maxwell's equations come along for the ride.