Fractional Clifford Analysis

Presenter: Uwe Kaehler, University of Aveiro, Portugal

Joint work with P. Cerejeiras, N. Vieira

In recent years one can observe a growing interest in fractional calculus. This theory allows a modeling of problems – be it in optics or quantum mechanics – which provides a wider degree of freedom since it can be used for a more complete characterization of the object under study or as an additional relevant parameter. Particularly, the factorization of the Laplace operator by fractional Dirac operators is quite interesting since it allows to arrive at Dirac operators which are not limited to the usual $SU(2)$-symmetry. In fact this setting provides relativistic covariant equations generated by taking the $n$-th root of the d'Alembert operator are fractional wave equations with an inherent $SU(n)$ symmetry. One of the principal points from the view of Clifford analysis is the establishment of a corresponding function theory, This will be the principal point of this talk. We introduce fractional Dirac operators and we present the framework of a higher-dimensional function theory adapting the method of F. Sommen based on fractional Weyl relations, which allows to circumvent problems coming from the standard $\mathfrak{osp}(1|2)$-approach.