Relativistic wave equations with fractional derivatives and pseudo-differential operators

TL;DR

This paper explores relativistic wave equations derived from fractional powers of the D'Alambertian operator, revealing their non-local nature for n>2 and connecting their algebraic structures to SU(n) groups and Poincaré symmetry.

Contribution

It introduces a new class of covariant relativistic equations using fractional derivatives, extending the algebraic framework of Dirac and Pauli matrices to arbitrary n.

Findings

Equations are local for n=1,2 but non-local for n>2.

Representation of generalized algebra of matrices related to SU(n).

Discussion of symmetry transformations and Green functions construction.

Abstract

The class of the free relativistic covariant equations generated by the fractional powers of the D'Alambertian operator $(\square^{1/n})$ is studied. Meanwhile the equations corresponding to n=1 and 2 (Klein-Gordon and Dirac equations) are local in their nature, the multicomponent equations for arbitrary n>2 are non-local. It is shown, how the representation of generalized algebra of Pauli and Dirac matrices looks like and how these matrices are related to the algebra of SU(n) group. The corresponding representations of the Poincar\'e group and further symmetry transformations on the obtained equations are discussed. The construction of the related Green functions is suggested.