Electromagnetic Theory with Quantum Principal Bundles

TL;DR

This paper develops a non-commutative geometric formulation of electromagnetic theory using quantum principal bundles on Moyal--Weyl space, deriving Maxwell equations with electric and magnetic charges, and exploring instanton solutions unrelated to Yang--Mills equations.

Contribution

It introduces a novel non-commutative geometric framework for electromagnetism with quantum principal bundles, extending classical theory to quantum space--time.

Findings

Derived Maxwell equations with electric and magnetic charges in quantum space

Presented a mathematical model with instantons not solving Yang--Mills equations

Formulated a non-commutative electromagnetic theory using quantum principal bundles

Abstract

The aim of this paper is to formulate a {\it non--commutative geometrical} version of the classical electromagnetic field theory in the vacuum with the Moyal--Weyl algebra as the space--time by using the theory of quantum principal bundles and quantum principal connections. As a result we will present the correct Maxwell equations in the vacuum of the model, in which we can appreciate the existence of electric and magnetic charges and currents. Finally, in the fourth section we are going to present a {\it mathematical model} for which there are instantons that are not solutions of the corresponding Yang--Mills equation.