Introduction to Clifford analysis

Abstract

The aim of this plenary lecture is to give an introduction to the topics of the spe-cial session on " Applicable Hypercomplex Analysis " . These topics deal with extensions of the complex numbers to " more complex numbers " , called hypercomplex numbers, involving an ar-bitrary number of imaginary units, and with the function theories that one can build on algebras of hypercomplex numbers. The motivation for studying this subject is the long-standing desire to create a mathematical framework that is capable of modelling the geometric and analytic content of higher dimensional physical phenomena. After a basic general introduction to hypercomplex numbers, a more detailed overview of a practical type of hypercomplex analysis, namely Clifford Analysis (CA), will be presented. Clif-ford Analysis is a part of mathematical analysis where one studies a chosen subset of functions, which take values in a particular hypercomplex algebra, called a Clifford algebra. We will first see what a Clifford algebra is and then the geometrical importance of such algebras will be explained. Thereafter, we will recall the key elements of the familiar Complex Analysis and show how it can be generalized to higher dimensions. The definition of a typical Clifford Anal-ysis, involving a first order vector derivation operator called Dirac operator, will be stated and the physical relevant distinction between Euclidean and pseudo-Euclidean Clifford analyses is discussed. Along the way, the mathematical formulations will be supplemented with interesting physical interpretations, revealing the naturalness of Clifford Analysis and its potential for use in physical applications. Especially, the particular Clifford Analysis based on the Clifford alge-bra of signature (1, 3) will emerge as a tailor-made function theory describing electromagnetic and quantum fields in Minkowski space.