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Dirac Equation in Scale Relativity

Marie-Noelle Celerier·Laurent Nottale
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Abstract

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 Schrödinger 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 Schrödinger and Klein-Gordon equations, the breaking of further symmetries, namely, the differential coordinate symmetry ($dx^μ \leftrightarrow - dx^μ$) and the parity and time reversal symmetries.

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Marie-Noelle Celerier, Laurent Nottale. 2001-12-21. Dirac Equation in Scale Relativity. https://arxiv.org/abs/hep-th/0112213

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