Relativistic wave equations with fractional derivatives and pseudodifferential operators

Abstract

We study the class of the free relativistic covariant equations generated by the fractional powers of the d′Alembertian operator $\left({\square }^{1/n}\right)$. The equations corresponding to $n=1$ and $2$ (Klein-Gordon and Dirac equations) are local in their nature, but the multicomponent equations for arbitrary $n>2$ are nonlocal. We show the representation of the generalized algebra of Pauli and Dirac matrices and how these matrices are related to the algebra of $\text{SU}\left(n\right)$ group. The corresponding representations of the Poincaré group and further symmetry transformations on the obtained equations are discussed. The construction of the related Green functions is suggested.