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When two charges are separated there is an electrostatic field between two. Quantum mechanically, is it actually the discrete energy packet or photons that's travelling from one charge to another? If yes, what is its wavelength? How can I better understand the interaction between two charges with the perspective of quantum theory?

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The interaction of two charges quantum mechanically are described by Feynman diagrams of many orders, where the dominant in the calculation are the first order ones. The diagrams depend on the quantum field theoretical framework .

Here is a first order diagram of electron electron interaction:

e-e-

No matter how far away the two electrons are , this is the quantum mechanical interaction. The internal line is called a virtual particle, because it is off mass shell. It has the quantum numbers of a photon but the mass under the integration depends on the bounds on the integral, the specific interaction. Thus one cannot talk of "frequency" for this virtual photon. The integral will reproduce the coulomb potential forces between the two charges. See this for a general view of exchanged virtual particles for static potentials.

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