Gauge Invariant Fractional Electromagnetic Fields

TL;DR

This paper introduces a novel, gauge-invariant fractional electromagnetic field theory with symmetric, causal operators derived from a Lagrangian, addressing previous issues of non-causality and asymmetry in fractional electromagnetism.

Contribution

It develops a spatially symmetric, causal fractional electromagnetic field theory with a new fractional vector calculus from a Lagrangian framework.

Findings

Defined fractional gradient, divergence, and curl operators.

Formulated fractional Maxwell's equations.

Ensured gauge invariance and causality.

Abstract

Fractional derivatives and integrations of non-integers orders was introduced more than three centuries ago but only recently gained more attention due to its application on nonlocal phenomenas. In this context, several formulations of fractional electromagnetic fields was proposed, but all these theories suffer from the absence of an effective fractional vector calculus, and in general are non-causal or spatially asymmetric. In order to deal with these difficulties, we propose a spatially symmetric and causal gauge invariant fractional electromagnetic field from a Lagrangian formulation. From our fractional Maxwell's fields arose a definition for the fractional gradient, divergent and curl operators.