Scalar higher spin operators in Clifford analysis

Presenter: Matthias Roels, University of Antwerp, Belgium

Joint work with David Eelbode, Hendrik De Bie

Clifford analysis has become a branch of multi-dimensional analysis in which far-reaching generalisations of the classical Cauchy-Riemann operator in complex analysis are studied from a function theoretical point of view. Without claiming completeness, one could say that the theory focuses on first-order conformally invariant operators acting on functions taking values in irreducible representations for the spin group. This then leads to function theories refining (poly-)harmonic analysis on $\mathbb{R}^m$. In this talk we will explain how to extend these results to a certain class of second-order conformally invariant operators, leading to analogues of the Rarita-Schwinger function theory for functions taking values in the space of harmonics (at the same time generalising classical harmonic analysis for the Laplace operator). After a brief introduction to some notions of differential geometry, we will give a detailed description on how to define and construct this class of operators. As some equations of motion that often appear in physics are expressed in terms of these operators, we will explain how to translate our notation to the tensor notation used by physicists. Finally, some function theoretical results that were already obtained or are under construction will be discussed.