Coulomb's potential, which describes the electric potential generated by a static point charge, can be derived from quantum electrodynamics (QED) by considering the behavior of the electromagnetic field and its interaction with charged particles. Here is a simplified overview of the derivation:
In QED, the electromagnetic field is described by a quantum field called the photon field. The interaction between the photon field and charged particles is mediated by the exchange of virtual photons.
Start with the QED Lagrangian: The QED Lagrangian describes the dynamics of the electromagnetic field and charged particles. It includes terms for the kinetic energy of the field, the electromagnetic interaction between the field and charged particles, and the masses and charges of the particles.
Quantize the electromagnetic field: The quantization process treats the electromagnetic field as an operator field. This means that instead of classical functions, the field is represented by operators that act on a quantum state. This allows us to describe the field in terms of creation and annihilation operators that create or destroy photons.
Compute the scattering amplitude: The scattering amplitude describes the probability amplitude for a particular scattering process to occur. In this case, we consider the scattering of a charged particle by the electromagnetic field. By calculating the relevant Feynman diagrams (perturbative diagrams representing the interaction processes), we can compute the scattering amplitude.
Extract the long-range behavior: In the case of a static point charge, we are interested in the long-range behavior of the electromagnetic field. By analyzing the scattering amplitude in the limit of large distances, we can isolate the terms that contribute to the potential energy.
Identify the Coulomb's potential term: In the long-range limit, we find that the scattering amplitude includes terms that correspond to the exchange of virtual photons between the charged particle and the electromagnetic field. By evaluating these terms, we can extract the Coulomb's potential term, which describes the electric potential due to the presence of a point charge.
It's important to note that the above derivation is a simplified overview, and the complete mathematical details of the derivation involve advanced techniques in quantum field theory. However, the fundamental idea is to consider the interaction between the electromagnetic field and charged particles in the framework of QED and extract the long-range behavior that corresponds to Coulomb's potential.