The application of the theory of scale relativity to microphysics aims at
recovering quantum mechanics as a new non-classical mechanics on a
non-derivable space-time. This program was already achieved as regards the
Schr\"odinger and Klein Gordon equations, which have been derived in terms of
geodesic equations in this framework: namely, they have been written according
to a generalized equivalence/strong covariance principle in the form of free
motion equations D2x/ds2=0, where D/ds are covariant derivatives built
from the description of the fractal/non-derivable geometry. Following the same
line of thought and using the mathematical tool of Hamilton's bi-quaternions,
we propose here a derivation of the Dirac equation also from a geodesic
equation (while it is still merely postulated in standard quantum physics). The
complex nature of the wave function in the Schr\"odinger and Klein-Gordon
equations was deduced from the necessity to introduce, because of the
non-derivability, a discrete symmetry breaking on the proper time differential
element. By extension, the bi-quaternionic nature of the Dirac bi-spinors
arises here from further discrete symmetry breakings on the space-time
variables, which also proceed from non-derivability.Comment: 13 pages, accepted for publication in Electromagnetic Phenomena,
Special issue dedicated to the 75th anniversary of the discovery of the Dirac
equatio