About non-relativistic quantum mechanics and electromagnetism

We describe here the coherent formulation of electromagnetism in the non-relativistic quantum-mechanical many-body theory of interacting charged particles. We use the mathematical frame of the field theory and its quantization in the spirit of the QED. This is necessary because a manifold of misinterpretations emerged especially regarding the magnetic field and gauge invariance. The situation was determined by the historical development of quantum mechanics, starting from the Schr\"odinger equation of a single particle in the presence of given electromagnetic fields, followed by the many-body theories of many charged identical particles having just Coulomb interactions. Our approach to the non-relativistic QED emphasizes the role of the gauge-invariance and of the external fields. We develop further the $1/c^2$ approximation of this theory allowing a closed description of the interacting charged particles without photons. The resulting Hamiltonian coincides with the quantized version of the Darwin Hamiltonian containing besides the Coulomb also a current-current diamagnetic interaction. We show on some examples the importance of this extension of the many-body theory.