Cubic root of Klein-Gordon equation

TL;DR

This paper introduces a new relativistic linear differential equation in multiple dimensions that generalizes the Dirac equation by using Clifford algebra of a cubic polynomial related to the Klein-Gordon operator, revealing novel spin and covariance properties.

Contribution

It constructs a new class of relativistic equations based on cubic Clifford algebra, extending Dirac's framework and exploring their spin, covariance, and gauge coupling features.

Findings

Derived a new relativistic equation generalizing Dirac's equation.

Identified conditions for spin and Lorentz covariance based on algebra representations.

Discussed coupling to gauge fields and connections to anyon-like fields in 1+1 dimensions.

Abstract

We construct new relativistic linear differential equation in $d$ dimensions generalizing Dirac equation by employing the Clifford algebra of the cubic polynomial associated to Klein-Gordon operator multiplied by the mass parameter. Unlike the Dirac case where the spin content is unique and Lorentz covariance is manifest, here the spin as well as Lorentz covariance of the theory are related to the choice of representation of the Clifford algebra. One of the considered explicit matrix representations gives rise to anyon-like fields in $d=1+1$. Coupling to a U(1) gauge field is discussed and compared with Dirac theory.