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A scale-relativistic derivation of the Dirac Equation

By Marie-Noelle Celerier and Laurent Nottale

Abstract

13 pages, accepted for publication in Electromagnetic Phenomena, Special issue dedicated to the 75th anniversary of the discovery of the Dirac equationThe 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

Topics: [PHYS.HTHE] Physics [physics]/High Energy Physics - Theory [hep-th]
Publisher: HAL CCSD
Year: 2003
OAI identifier: oai:HAL:hal-00134534v1
Provided by: Hal-Diderot
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