In Quantum Electrodynamics (QED), Maxwell's equations are incorporated as classical equations describing the behavior of the electromagnetic field, while the quantum nature of the electromagnetic interaction is introduced through the concept of quantum fields and their interactions.
In QED, the electromagnetic field is treated as a quantum field, known as the photon field. The dynamics of this field and its interaction with charged particles, such as electrons, are described using a quantized version of Maxwell's equations. These quantized equations incorporate the principles of quantum mechanics and provide a framework for understanding the behavior of photons and their interactions with charged particles.
The quantized version of Maxwell's equations in QED gives rise to the fundamental interactions between electrons and photons. It describes the emission, absorption, and scattering of photons by charged particles, as well as the propagation of electromagnetic waves. The equations govern the behavior of the photon field and the probability amplitudes associated with different electromagnetic processes.
Through the use of Feynman diagrams, which are graphical representations of the interactions between particles in quantum field theory, QED allows for the calculation of probabilities and amplitudes associated with various electromagnetic processes. These calculations involve summing over all possible Feynman diagrams corresponding to the interaction processes, taking into account factors such as the charges and momenta of the particles involved.
In summary, in QED, Maxwell's equations serve as a starting point, describing the classical behavior of the electromagnetic field. However, they are quantized to account for the quantum nature of the electromagnetic interaction, leading to the concept of the photon field and the formulation of quantum field theory for describing electromagnetic phenomena at the fundamental level.