Editing Maxwell's equations (section)

== Maxwell's equations as the classical limit of QED ==

Maxwell's equations and the Lorentz force law (along with the rest of classical electromagnetism) are extraordinarily successful at explaining and predicting a variety of phenomena.  However, they do not account for quantum effects, and so their domain of applicability is limited. Maxwell's equations are thought of as the classical limit of [[quantum electrodynamics]] (QED).

Some observed electromagnetic phenomena cannot be explained with Maxwell's equations if the source of the electromagnetic fields are the classical distributions of charge and current. These include  [[photon–photon scattering]] and many other phenomena related to [[photon]]s or [[virtual particle|virtual photons]], "[[nonclassical light]]" and [[quantum entanglement]] of electromagnetic fields (see ''[[Quantum optics]]''). E.g. [[quantum cryptography]] cannot be described by Maxwell theory, not even approximately. The approximate nature of Maxwell's equations becomes more and more apparent when going into the extremely strong field regime (see ''[[Euler–Heisenberg Lagrangian]]'') or to extremely small distances.

Finally, Maxwell's equations cannot explain any phenomenon involving individual [[photon]]s interacting with quantum matter, such as the [[photoelectric effect]], [[Planck's law]], the [[Duane–Hunt law]], and [[Single-photon avalanche diode|single-photon light detectors]]. However, many such phenomena may be explained using a halfway theory of quantum matter coupled to a classical electromagnetic field, either as external field or with the expected value of the charge current and density on the right hand side of Maxwell's equations. This is known as semiclassical theory or self-field QED and was initially discovered by de Broglie and Schrödinger and later fully developed by E.T. Jaynes and A.O. Barut.