Fractional relativity

TL;DR

This paper introduces fractional relativity, a theoretical framework extending special relativity with a fractional dispersion relation, exploring its implications on quantum equations and classical electrodynamics.

Contribution

It establishes a fractional relativity framework with a curved energy-momentum space and derives a fractional Schrödinger equation as a nonrelativistic limit.

Findings

Fractional Schrödinger equation derived from fractional Klein-Gordon equation.

Relative locality does not affect classical position uncertainty.

Faraday's law is modified by fractional derivatives.

Abstract

By fractional relativity we mean a theoretical framework to study physics with the dispersion relation $E^{\alpha}=m^{\alpha}c^{2\alpha}+p^{\alpha}c^{\alpha}$, which recovers special relativity at $\alpha=2$. One such framework is established in a particular curved energy-momentum space. It is shown that the fractional Schr\"{o}dinger equation arises as a nonrelativistic limit of the Klein-Gordon equation in fractional relativity. In this framework, the relative locality makes no contribution to the position uncertainty at the classical level, and the Faraday's law in classical electrodynamics is modified by fractional derivatives.