Motion equations for relativistic particles in an external electromagnetic field in scale relativity

TL;DR

This paper explores the derivation of relativistic particle motion equations in electromagnetic fields within scale relativity, revealing how to recover the standard Dirac equation without extra terms by modifying covariance implementation.

Contribution

It introduces a novel method to derive the electromagnetic Dirac equation in scale relativity, accounting for spin-charge coupling and resolving additional terms in the presence of external fields.

Findings

Standard Dirac equation can be recovered without extra terms

Modified covariance approach encompasses spin-charge coupling

Method is rooted in deeper scale relativity principles

Abstract

Klein-Gordon and Dirac equations are the motion equations for relativistic particles with spin 0 (so-called scalar particles) and 1/2 (electron/positron) respectively. For a free particle, the Dirac equation is derived from the Klein-Gordon equation by taking its square root in a bi-quaternionic formalism fully justified by the first principles of the scale relativity theory. This is no more true when an external electro-magnetic field comes into play. If one tries to derive the electro-magnetic Dirac equation in a manner analogous to the one used when this field is absent, one obtains an additional term which is the relativistic analogue of the spin-magnetic field coupling term encountered in the Pauli equation, valid for a non-relativistic particle with spin 1/2. There is however a method to recover the standard form of the electro-magnetic Dirac equation, with no additional term, which amounts modifying the way both covariances involved here, quantum and scale, are implemented. Without going into technical details, it will be shown how these results suggest this last method is based on more profound roots of the scale relativity theory since it encompasses naturally the spin-charge coupling.