Clifford analysis versus its quaternionic counterparts

Abstract

For a positive integer n let Cl0,n be the universal Clifford algebra with the signature (0,n). The name Clifford analysis is usually referred to the function theories for functions in the kernels of the two operators: the (Cliffordian) Cauchy–Riemann operator and the Dirac operator. For n=2, Cl0,2 becomes the skew-field of Hamilton's quaternions for which the two operators are widely known: the Moisil–Theodoresco and the Fueter operators. We establish the precise relations between the Moisil–Theodoresco operator and the Dirac operator for Cl0,3. It turns out that the case of the Cauchy–Riemann operator for Cl0,3 and the Fueter operator is more sophisticated, and we describe the peculiarities emerging here. Copyright © 2009 John Wiley & Sons, Ltd.