  In the last decades, the blossoming of experimental breakthroughs in the
domain of electron energy loss spectroscopy (EELS) has triggered a variety of
theoretical developments. Those have to deal with completely different
situations, from atomically resolved phonon mapping to electron circular
dichroism passing by surface plasmon mapping. All of them rely on very
different physical approximations and have not yet been reconciled, despite
early attempts to do so. As an effort in that direction, we report on the
development of a scalar relativistic quantum electrodynamic (QED) approach of
the inelastic scattering of fast electrons. This theory can be adapted to
describe all modern EELS experiments, and under the relevant approximations,
can be reduced to any of the last EELS theories. In that aim, we present in
this paper the state of the art and the basics of scalar relativistic QED
relevant to the electron inelastic scattering. We then give a clear relation
between the two once antagonist descriptions of the EELS, the retarded green
Dyadic, usually applied to describe photonic excitations and the quasi-static
mixed dynamic form factor (MDFF), more adapted to describe core electronic
excitations of material. We then use this theory to establish two important
EELS-related equations. The first one relates the spatially resolved EELS to
the imaginary part of the photon propagator and the incoming and outgoing
electron beam wavefunction, synthesizing the most common theories developed for
analyzing spatially resolved EELS experiments. The second one shows that the
evolution of the electron beam density matrix is proportional to the mutual
coherence tensor, proving that quite universally, the electromagnetic
correlations in the target are imprinted in the coherence properties of the
probing electron beam.
