Quantum mechanics of the electric charge

The Coulomb field of the proton in a hydrogen atom is a completely classical object. We know it from the success of the Dirac equation in which the classical Coulomb field is put in. However, the proton's charge, which gives the scale of the Coulomb field, is quantized. Thus the Coulomb field behaves like Bohr's orbits in the old quantum theory: its spatio-temporal shape is classical but its magnitude is quantized. The Author explains this curious state of affairs. There are two distinct regimes of the electromagnetic field: the regime described by the standard quantum electrodynamics and zero-frequency regime, which is translationally invariant and has only the Lorentz group as its symmetry group. The electric charge is a part of the translationally invariant zero-frequency regime and as such can indeed be quantized. (author)