  The theory of scale relativity provides a new insight into the origin of
fundamental laws in physics. Its application to microphysics allows us to
recover quantum mechanics as mechanics on a non-differentiable (fractal)
spacetime. The Schrodinger and Klein-Gordon equations are demonstrated as
geodesic equations in this framework. A development of the intrinsic properties
of this theory, using the mathematical tool of Hamilton's bi-quaternions, leads
us to a derivation of the Dirac equation within the scale-relativity paradigm.
The complex form of the wavefunction in the Schrodinger and Klein-Gordon
equations follows from the non-differentiability of the geometry, since it
involves a breaking of the invariance under the reflection symmetry on the
(proper) time differential element (ds <-> - ds). This mechanism is generalized
for obtaining the bi-quaternionic nature of the Dirac spinor by adding a
further symmetry breaking due to non-differentiability, namely the differential
coordinate reflection symmetry (dx^mu <-> - dx^mu) and by requiring invariance
under parity and time inversion. The Pauli equation is recovered as a
non-relativistic-motion approximation of the Dirac equation.
