  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.
