Interaction theories are usually based on a elativistically invarieant Lagrange function. This function is generally known and accepted for the electromagnetic interaction. The variation of that Lagrangian leads to the system of the coupled Maxwell-Dirac equations. It contains a non-linear term. If you neglect this term, you obtain the well-known linear Dirac equation and rules for determining the correct values of the spectral lines of atoms. However, one cannot describe the radiation process and has to introduce the quantum hypothesis. But, if the non-linear term is also taken into account, there are solutions of the system what describe the emission of "quantum jumps" in space and time with correct frequencies. This is demonstrated in the presented work for hydrogen and helium atoms. It explains the entangled eigenfunctions in the context of a classical mear-field theory. Further problems like diffraction effects, photo effects and relativistic transformation of the field tensor are discussed. Aim of the work is a proposal of an alternative to the statistical interpretation of the quantum theory in context of a classical near-field theory.