Topological interpretation of Dirac equation and geometrisation of electromagnetic field

TL;DR

This paper proposes a novel approach to geometrize electromagnetic fields by modeling interacting fields as a unified nonmetrized space-time manifold, linking Dirac equations to topological and geometric properties.

Contribution

It introduces a topological and geometric interpretation of Dirac and electromagnetic fields within a unified nonmetrized space-time framework, extending geometrization beyond gravity.

Findings

Dirac equation as a group-theoretic relation

Electromagnetic field components as curvature tensor parts

Potential extension to weak interaction geometrization

Abstract

A new concept of geometrization of electromagnetic field is proposed. Instead of the concept of extended field and its point sources, the interacting Maxwellian and Dirac electron--positron fields are considered as a microscopic unified closed connected nonmetrized space--time 4-manifold. Within this approach, the Dirac equation proves to be a group-theoretic relation that accounts for the topological and metric properties of this manifold. The Dirac spinors serve as basis functions of its fundamental group representation, while the tensor components of electromagnetic field prove to be the components of a curvature tensor of the relevant covering space. A basic distinction of the suggested approach from the geometrization of gravitational field in general relativity is that, first, not only the field is geometrized but also are its microscopic sources and, second, the field and its sources are treated not as a metrized Riemannian space--time but as a nonmetrized space-- time manifold. A possibility to geometrize weak interaction is also discussed.