Some results in fractional Clifford analysis

Abstract

What is nowadays called (classic) Clifford analysis consists in the establishment of a function theory for functions belonging to the kernel of the Dirac operator. While such functions can very well describe problems of a particle with internal $SU(2)$-symmetries, higher order symmetries are beyond this theory. Although many modifications (such as Yang-Mills theory) were suggested over the years they could not address the principal problem, the need of a $n$-fold factorization of the d'Alembert operator.

In this paper we present the basic tools of a fractional function theory in higher dimensions, for the transport operator $(\alpha =\alfa)$, by means of a fractional correspondence to the Weyl relations via fractional Riemann-Liouville derivatives. A Fischer decomposition, fractional Euler and Gamma operators, monogenic projection, and basic fractional homogeneous powers are constructed.