Microscopic Amp\`ere current-current interaction

With the rapid development of modern measurement techniques, the energy resolution of $1 \, meV$ can now be easily obtained. Generally, the driving mechanisms of the physical, chemical or biological processes of the matters or the living organisms on Earth at about $1 \, meV$ energy scale are assumed to stem from the fundamental microscopic Coulomb interaction, its various reduced ones and the relativistic corrections. In this article, by using a path integral approach on a non-relativistic quantum electrodynamics theory, we show that there is another fundamental microscopic electromagnetic interaction at this energy scale, the microscopic Amp\`ere current-current interaction. It has time-dependent dynamical feature and can be the driving interaction of the physical, chemical or biological processes at about $1\, meV$ energy scale. A new Amp\`ere-type exchange spin interaction is also found with a magnitude about $10^{-4}$ of the well-known Heisenberg exchange spin interaction.