Electromagnetic Klein-Gordon and Dirac equations in scale relativity

TL;DR

This paper advances quantum field theory by deriving electromagnetic Klein-Gordon and Dirac equations within the framework of scale relativity, revealing geometric interpretations of gauge fields and naturally incorporating spin-charge coupling.

Contribution

It introduces a novel derivation of electromagnetic Dirac and Klein-Gordon equations using scale relativity, unifying gauge fields and spin-charge interactions geometrically.

Findings

Electromagnetic Klein-Gordon equation derived with scale-covariant derivatives.

Electromagnetic Dirac equation obtained without extra terms, preserving its standard form.

Spin-charge coupling emerges naturally from the scale relativity framework.

Abstract

We present a new step in the foundation of quantum field theory with the tools of scale relativity. Previously, quantum motion equations (Schr\"odinger, Klein-Gordon, Dirac, Pauli) have been derived as geodesic equations written with a quantum-covariant derivative operator. Then, the nature of gauge transformations, of gauge fields and of conserved charges have been given a geometric meaning in terms of a scale-covariant derivative tool. Finally, the electromagnetic Klein-Gordon equation has been recovered with a covariant derivative constructed by combining the quantum-covariant velocity operator and the scale-covariant derivative. We show here that if one tries to derive the electromagnetic Dirac equation from the Klein-Gordon one as for the free particle motion, i.e. as a square root of the time part of the Klein-Gordon operator, one obtains an additional term which is the relativistic analog of the spin-magnetic field coupling term of the Pauli equation. However, if one first applies the quantum covariance, then implements the scale covariance through the scale-covariant derivative, one obtains the electromagnetic Dirac equation in its usual form. This method can also be applied successfully to the derivation of the electromagnetic Klein-Gordon equation. This suggests it rests on more profound roots of the theory, since it encompasses naturally the spin-charge coupling.