Relativistic Equations for Fractional-Spin particles

TL;DR

This paper develops a new family of relativistic equations for particles with fractional spin by generalizing Dirac's equation, demonstrating invariance under Lorentz transformations, and identifying specific fractional spins for different cases.

Contribution

It introduces a novel method to derive equations for fractional-spin particles by taking roots of the energy-momentum relation, extending Dirac's approach.

Findings

Equations are invariant under rotations and boosts.

Particles with fractional spins of 3/8, 1/8, 1/6, and 0 are described.

Explicit construction for N=3 and N=4 cases.

Abstract

This paper generalizes the method of deducing Dirac's equation to constructing a family of equations that represent the $N$-th root of the basic Energy-Momentum relation for a free particle \cite{Dattoli1, Dattoli2, Dattoli3, Dattoli4}. Then these equations are recast in a form that allows one to interpret them as the fundamental dynamical equations for particles with fractional spin, which we study in detail for the case of $N=4$ and $N=3$. We explicitly prove that the equation is invariant to rotations and boosts and indeed represents spin-$\frac{3}{8}$ and spin-$\frac{1}{8}$ particles for $N=4$ and spin-$\frac{1}{6}$ and $0$ for $N=3$.