Maxwell's equations describe classical electromagnetism and were formulated before the development of quantum electrodynamics (QED). Maxwell's equations can be derived from QED in certain limits or approximations, but a direct derivation of Maxwell's equations from the full formalism of QED is not straightforward.
QED is a quantum field theory that describes the interactions between charged particles and electromagnetic fields. It is based on the principles of quantum mechanics and special relativity. In QED, the fundamental entities are quantum fields that describe the behavior of particles (such as electrons) and the electromagnetic field.
Maxwell's equations, on the other hand, describe the classical behavior of electromagnetic fields and their interactions with charged particles. They are derived from classical electrodynamics, which does not incorporate quantum effects.
In certain limits, such as the classical limit or the limit of large quantum numbers, the quantum behavior described by QED converges to classical electrodynamics. In these limits, the equations of motion for the quantum fields can be approximated by the classical equations of motion for the electromagnetic field, leading to the emergence of Maxwell's equations.
However, it's important to note that the full formalism of QED is considerably more complex than classical electrodynamics, involving quantum field operators, Feynman diagrams, and renormalization techniques. The connection between QED and classical electrodynamics is established through appropriate approximations and limit transitions.
So while Maxwell's equations can be recovered in specific limits of QED, directly deriving Maxwell's equations from the full formalism of QED is not a straightforward or direct process.