  Klein-Gordon and Dirac equations are the motion equations for relativistic
particles with spin 0 (so-called scalar particles) and 1/2 (electron/positron)
respectively. For a free particle, the Dirac equation is derived from the
Klein-Gordon equation by taking its square root in a bi-quaternionic formalism
fully justified by the first principles of the scale relativity theory. This is
no more true when an external electro-magnetic field comes into play. If one
tries to derive the electro-magnetic Dirac equation in a manner analogous to
the one used when this field is absent, one obtains an additional term which is
the relativistic analogue of the spin-magnetic field coupling term encountered
in the Pauli equation, valid for a non-relativistic particle with spin 1/2.
There is however a method to recover the standard form of the electro-magnetic
Dirac equation, with no additional term, which amounts modifying the way both
covariances involved here, quantum and scale, are implemented. Without going
into technical details, it will be shown how these results suggest this last
method is based on more profound roots of the scale relativity theory since it
encompasses naturally the spin-charge coupling.
