# A scale-relativistic derivation of the Dirac Equation

Abstract : 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 Schrödinger 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 $D^2x/ds^2=0$, 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 Schrödinger 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.
Document type :
Journal articles

https://hal.archives-ouvertes.fr/hal-00134534
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Submitted on : Friday, March 2, 2007 - 2:33:03 PM
Last modification on : Wednesday, November 17, 2021 - 12:27:09 PM

### Citation

Marie-Noelle Celerier, Laurent Nottale. A scale-relativistic derivation of the Dirac Equation. Electromagn. Phenom., 2003, 3, pp.70-80. ⟨hal-00134534⟩

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