Relativistic Equations for Fractional-Spin particles

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 $N=3$.