The question of the physical nature of the electron, of the electromagnetic field and of gauge invariance is revisited in the framework of the theory of scale relativity. Space-time is described as a non-derivable "manifold", which implies that its geometry is fractal (i.e., it is explicitly dependent on the resolution scale). The electromagnetic field can then be re-interpreted as a field of dilations induced by displacements, while the electric charge is the conservative quantity that originates from the scale symmetry. This means that gauge transformations are identified with scale transformations on the resolution variables (which are internal to the electron, i.e., relevant at scales smaller than its Compton length). In this general framework, the Lorentz force and the Maxwell equations are derived from first principles, and the QED-covariant derivative naturally emerges instead of being merely postulated. Now, in the framework of special scalerelativity, the Planck length-scale becomes a minimal, impassable scale, invariant under dilations (that replaces the zero point). As a consequence scale ratios are limited and therefore one demonstrates charge quantization and the existence of a relation between the mass and the charge of the electron. From this relation, the mass of the electron explicitly depends on the number of Higgs doublets: one finds that the theoretical expectation agrees with the experimental mass for one Higgs doublet.