Clifford algebra and spinor-valued functions. A function theory for the Dirac operator. Related REDUCE software by F. Brackx and D. Constales. (English) Zbl 0747.53001

Delanghe, R.; Sommen, F.; Souček, V.

Mathematics and its Applications (Dordrecht). 53. Dordrecht etc.: Kluwer Academic Publishers. xvii, 485 p. (1992).

This book is a significant contribution in unravelling the fertile interaction between Clifford algebras, the Dirac operator, and the Spin group. It is noteworthy in that it merges these topics into a new theory of functions called Clifford analysis. The book consists of six chapters: Chapter 0 — Clifford algebras and lower dimensional Euclidean spaces, Chapter I – Clifford algebras and spinor spaces (both of these primarily contain review material with few proofs, but the presentation is very lucid); Chapter II – Monogenic functions (dealing with the fundamental concepts of the function theory arising from null solutions of the Dirac operator); Chapter III – Special functions and methods (development of special monogenic functions intended to play a role analogous to that of classical orthogonal polynomials, including an exposition of the Cauchy- Kovalevska method, Cauchy-type and plane wave integrals); Chapter IV – Monogenic differential forms and residues (a generalized theory of holomorphic differential forms and residues based on incorporating invariance with respect to a Spin group); Chapter V – Clifford analysis and the Penrose transform (elliptic and hyperbolic integral formulas, isotropic flat manifolds, twistor correspondence, the Penrose transform for the Dirac equations). The book concludes with three appendices with a REDUCE software package, and a ninety-four item bibliography. The authors are to be warmly congratulated for having produced an exciting and fascinating book.

Reviewer: J.D.Zund (Las Cruces)

MSC:

53-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53C27	Spin and Spin\({}^c\) geometry
53-04	Software, source code, etc. for problems pertaining to differential geometry
53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
15A66	Clifford algebras, spinors
30G99	Generalized function theory

Keywords:

spin group; Clifford analysis; monogenic functions; Penrose transform

Software:

REDUCE