REVOLUTION DE L'OPERATEUR DE DIRAC DANS L'ANALYSE DE CLIFFORD

Abstract

Given (Rn; q) the quadratic space of signature (r,s) with r + s = n. The problem we are concerned with is the identification of the Cliffordien distributions and spinoriel distributions defined on the group spin(r,s) at values in the Clifford algebra CL(IRn); (resp the spinors space) biinvariant by its maximal subgroup compact = spin(r) x spin(s)); the own solutions of Dirac Operator.

To solve this problem, we profit from the existence of the close relation between the Dirac D operator and the Laplace operator realized by the equality D2 =□.

On The other hand, we can not forget the basic role of the coordinator that the universal morphism plays between the spin(r,s) groups and SO(r,s) and its relevement.

Thus, these two powerful tools contribute to the complete identification of Cliffordien distributions from spheric distributions and vise versa.

I can be said that the Clifford analysis is a refinement of the harmonic analysis. Her techniques are used to construct a fundamental solution for first order homogeneous differential operators on the n-dimensional hyperbolic space.

In most of the literature on Clifford analysis the Dirac operator considered is the elliptic one acting on a flat euclidean space.

For these reasons, the Dirac operator is currently making a revolution to realise the unification of the spinoriel geometry and the Clifford analysis by using the calculation techniques of the Clifford algebra.