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Anonymous
Posted December 6, 1999
This is an excellent book whose title, unfortunately, reveals little of its important contents. The subject is a rigorous formulation of classical electrodynamics and relativistic quantum mechanics in terms of complex quaternions. The latter form a mutidimenional algebra extending the complex numbers and possessing powerful generalizations of Cauchy's integral formulae. Although there are several books on the market using Clifford algebras (including complex quaternions) in mathematical physics, this book is distinguished in several ways. Most such books stop with the albegra, while the book under review also contains much rigorous analysis, including the 'integral representations' of the title. Specialzing to complex quaternions (instead of considering general Clifford algebras) leaves ample freedom to describe the chosen physics while allowing the development of powerful new methods. The connections with classical results by Moisil, Theodorescu and others are emphasized, helping to motivate and orient the mathematical reader.
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More About This Textbook
Overview
This book provides a new mathematical theory for the treatment of an ample series of spatial problems of electrodynamics, particle physics, quantum mechanics and elasticity theory. This technique proves to be as powerful for solving the spatial problems of mathematical physics as complex analysis is for solving planar problems.
The main analytic tool of the book, a non-harmonic version of hypercomplex analysis recently developed by the authors, is presented in detail. There are given applications of this theory to the boundary value problems of electrodynamics and elasticity theory as well as to the problem of quark confinement. A new approach to the linearization of special classes of the self-duality equation is also considered. Detailed proofs are given throughout. The book contains an extensive bibliography on closely related topics.
This book will be of particular interest to academic and professional specialists and students in mathematics and physics who are interested in integral representations for partial differential equations. The book is self-contained and could be used as a main reference for special course seminars on the subject.
Product Details
Table of Contents
Introduction and some remarks on generalisations of complex analysis a-holomorphic function theory Electrodynamical models Massive spinor fields Hypercomplex factorization, systems of non-linear partial differential equations generated by Futer-type operators.