The theory of scale relativity provides a new insight into the origin of
fundamental laws in physics. Its application to microphysics allows to recover
quantum mechanics as mechanics on a non-differentiable (fractal) space-time.
The Schr\"odinger and Klein-Gordon equations have already been demonstrated as
geodesic equations in this framework. We propose here a new development of the
intrinsic properties of this theory to obtain, using the mathematical tool of
Hamilton's bi-quaternions, a derivation of the Dirac equation, which, in
standard physics, is merely postulated. The bi-quaternionic nature of the Dirac
spinor is obtained by adding to the differential (proper) time symmetry
breaking, which yields the complex form of the wave-function in the
Schr\"odinger and Klein-Gordon equations, the breaking of further symmetries,
namely, the differential coordinate symmetry (dxμ↔−dxμ) and the parity and time reversal symmetries.Comment: 33 pages, 4 figures, latex. Submitted to Phys. Rev.
