The simplest quantum field theory that describes the interaction between electrons and photons is known as quantum electrodynamics (QED). It is a quantum field theory based on the principles of quantum mechanics and special relativity that describes the electromagnetic interaction.
In QED, the electromagnetic interaction is mediated by virtual photons, which are quantized excitations of the electromagnetic field. The theory treats the electron as a fermionic field and the photon as a bosonic field. The interaction between electrons and photons is described by a Lagrangian, which specifies the dynamics of the fields and their interactions.
The action principle is used to derive the equations of motion and determine the behavior of the fields. The action is constructed from the Lagrangian and is minimized to obtain the equations of motion, which are the famous Dirac equation for the electron and the Maxwell's equations for the photon.
The interactions between electrons and photons in QED are captured through Feynman diagrams, which represent possible particle interactions. These diagrams allow calculations of scattering amplitudes, cross-sections, and other observables.
QED is a highly successful theory and one of the pillars of the Standard Model of particle physics. It accurately describes a wide range of phenomena, including electromagnetic radiation, atomic structure, and particle scattering processes involving electrons and photons.
It is important to note that although QED is considered the simplest quantum field theory involving electrons and photons, it still involves sophisticated mathematical techniques, such as perturbation theory and renormalization, to deal with the inherent challenges of quantum field theories.