Origin of Complex and Quaternionic Wavefunctions in Quantum Mechanics: the Scale-Relativistic View

In the theory of scale relativity, one attempts to set the foundations of quantum physics on a geometric basis. Space-time is assumed to be a nondifferentiable continuum, that can be characterized by its fractality. Namely, the physical quantities, among which the coordinates themselves, become in this geometry explicit functions of the scale (i.e., these function are divergent when the scale interval tends to zero). We recall how the main tools of quantum mechanics (complex, then spinorial wave functions), and the equations they satisfy, which are obtained as geodesics equations of the fractal space-time (Schrödinger, Klein-Gordon then Pauli and Dirac equations) can be derived in such a framework. Indeed, the nondifferentiability manifests itself, because of discrete symmetry breakings of the scale variables, in terms of successive doublings of the velocity field, which are naturally accounted for by complex then biquaternionic numbers. In this contribution, new improvements of this construction are given, including the description of the metric of a fractal space-time, a full derivation of the Born postulate, a new and more complete derivation of the Compton relation, and a generalization of the Schrödinger equation also valid for non-differentiable wave functions.