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Klein-Gordon and Dirac equations are the motion equations for relativisticparticles with spin 0 (so-called scalar particles) and 1/2 (electron/positron)respectively. For a free particle, the Dirac equation is derived from theKlein-Gordon equation by taking its square root in a bi-quaternionic formalismfully justified by the first principles of the scale relativity theory. This isno more true when an external electro-magnetic field comes into play. If onetries to derive the electro-magnetic Dirac equation in a manner analogous tothe one used when this field is absent, one obtains an additional term which isthe relativistic analogue of the spin-magnetic field coupling term encounteredin 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-magneticDirac equation, with no additional term, which amounts modifying the way bothcovariances involved here, quantum and scale, are implemented. Without goinginto technical details, it will be shown how these results suggest this lastmethod is based on more profound roots of the scale relativity theory since itencompasses naturally the spin-charge coupling.

Motion equations for relativistic particles in an external electromagnetic field in scale relativity