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Rafai Ablamowicz William E. Baylis Thomas Branson Pertti Lounesto Lan Porteous Johnryan J.M RafaI Ablamowicz William E. Baylis Thomas Branson Pertti Lounesto lan Porteous JohnRyan J.M. Selig Garret Sobczyk Lectures on Clifford (Geometric) Algebras and Applications RafaI Ablamowicz Garret Sobczyk Editors Springer Science+Business Media, LLC Rafal Ablamowicz Garret Sobczyk Department of Mathematics Dept. de Frsica y Matemâticas Tennessee Technological University Universidad de las Am~ricas-Puebla Cookeville, TN 38505 Santa Catarina Martir U.S.A. Cholula, Puebla, 72820 M~xico Library of Congress Cataloging-in-Publication Data Lectures on Clifford (geometric) algebras and applications / [edited by] Rafai Ablamowicz and Garret Sobczyk. p. cm. Includes bibliograpbical references and index. ISBN 978-0-8176-3257-1 ISBN 978-0-8176-8190-6 (eBook) DOI 10.1007/978-0-8176-8190-6 1. Clifford algebras. 1. Ablamowicz, Rafal. II. Sobczyk, Garret, 1943- QAI99.lA32003 512'.57-dc21 2003052205 CIP AMS SubjectClassifications: llE88, 15-75,15A66 Printed on acid-free paper <02004 Springer Science+Business Media New York Originally published by Birkhliuser Boston in 2004 AlI rights reserved. Tbis work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN 978-0-8176-3257-1 Cover design by Joseph Sherman, Camden, CT. 98765 4 3 2 1 Professor Pertti Lounesto presenting Lecture 1 on May 18, 2002, during the "6th Conference on Clifford Algebras and their Applications in Mathematical Physics," May 20-25, 2002, Tennessee Technological University, Cookeville, TN. Dedicated to Professor Pertti Lounesto Mathematician, Teacher, Friend, and Colleague Contents Preface Rafal Ablamowicz and Garret Sobczyk xiii Lecture 1: Introduction to Clifford Algebras Pertti Loullestot 1 1.1 Introduction 1 1.2 Clifford algebra of the Euclidean plane 2 1.3 Quaternions 7 3 1.4 Clifford algebra of the Euclidean space IR ... .. .. .. ...... ...... .. •.. 11 1.5 The electorn spin in a magnetic field 14 1.6 From column spinors to spinor operators 18 4 1.7 In 4D: Clifford algebra Cf 4 of 1R 23 1.8 Clifford algebra of Minkowski spacetime 24 1.9 The exterior algebra and contractions 25 1.10 The Grassmann-Cayley algebra and shuffle products 26 1.11 Alternative definitions of the Clifford algebra 26 1.12 References 29 Lecture 2: Mathematical Structure of Clifford Algebras Ian Porteous 31 2.1 Clifford algebras 31 2.2 Conjugation 40 2.3 References 51 x Contents Lecture 3: Clifford Analysis John Ryan 53 3.1 Introduction 53 3.2 Foundations of Clifford analysis 54 3.3 Other types of Clifford holomorphic functions 62 3.4 The equation D k f = 0 65 3.5 Conformal groups and Clifford analysis 71 3.6 Conformally flat spin manifolds 74 3.7 Boundary behavior and Hardy spaces 75 3.8 More on Clifford analysis on the sphere 78 3.9 The Fourier transform and Clifford analysis 82 3.10 Complex Clifford analysis 86 3.11 References 87 Lecture 4: Applications of Clifford Algebras in Physics William E. Baylis 91 4.1 Introduction 91 4.2 Three Clifford algebras 92 4.3 Paravectors and relativity 102 4.4 Eigenspinors 113 4.5 Maxwell 's equation 116 4.6 Quantum theory 130 4.7 Conclusions 132 4.8 References 132 Lecture 5: Clifford Algebras in Engineering J.M. Selig 135 5.1 Introduction 135 5.2 Quatemions 137 5.3 Biquatemions 141 Contents xi 5.4 Points, lines, and planes 144 5.5 Computer vision example 150 5.6 Robot kinematics 151 5.7 Concluding remarks 154 5.8 References 155 Lecture 6: Clifford Bundles and Clifford Algebras Thomas Branson 157 6.1 Spin geometry 157 6.2 Conformal structure '" 164 6.3 Tractor constructions 176 6.4 References 186 Appendix Rafol Ablamowicz and Garret Sobczyk 189 7.1 Software for Clifford algebras 189 7.2 References 206 Index 211 Preface Advances in technology over the last 25 years have created a situation in which workers in diverse areas of computer science and engineering have found it neces­ sary to increase their knowledge of related fields in order to make further progress. Clifford (geometric) algebra offers a unified algebraic framework for the direct expression of the geometric ideas underlying the great mathematical theories of linear and multilinear algebra, projective and affine geometries, and differential geometry. Indeed, for many people working in this area, geometric algebra is the natural extension of the real number system to include the concept of direction. The familiar complex numbers of the plane and the quaternions of four dimen­ sions are examples of lower-dimensional geometric algebras . During "The 6th International Conference on Clifford Algebras and their Ap­ plications in Mathematical Physics" held May 20--25, 2002, at Tennessee Tech­ nological University in Cookeville, Tennessee, a Lecture Series on Clifford Ge­ ometric Algebras was presented. Its goal was to to provide beginning graduate students in mathematics and physics and other newcomers to the field with no prior knowledge of Clifford algebras with a bird's eye view of Clifford geometric algebras and their applications. The lectures were given by some of the field's most recognized experts. The enthusiastic response of the more than 80 partici­ pants in the Lecture Series, many of whom were graduate students or postdocs, encouraged us to publish the expanded lectures as chapters in book form. The book, which contains many up-to-date references to the rapidly evolving litera­ ture, should also serve as a handy reference to professionals who want to keep abreast with the latest developments and applications in the field. In Chapter I, Pertti Lounesto states that "Clifford algebra is by definition the minimal construction designed to control the geometry in question ," He then gives a concise but coherent introduction to geometric algebras by thoroughly examin­ ing the Clifford product of two vectors, the definition of a bivector, and how these concepts are used to represent reflections and rotations in the plane and in higher dimensional spaces. The geometric significance of quaternions is explained in three and four dimensions. He carefully shows us how the more advanced con­ cepts of spinors, exterior algebra and contraction, the Grassmann-Caley algebra and shuffle product, naturally evolve from the more basic concepts. An important feature is that he shows how the lower dimensional Clifford algebras can be rep­ resented in terms of the familiar algebra of square matrices. His lecture ends with xiv Preface several categorical definitions of Clifford algebras of a quadratic form, and their deformations to Clifford algebras of an arbitrary bilinear form. In Chapter 2, Ian Porteous shows us that the "control of the geometry" that each Clifford algebra has follows directly from its close relationship to the correspond­ ing classical group. He begins with a systematic study of Clifford algebras by showing how they can be constructed from matrix algebras over the real numbers, complex numbers, or quaternions with the famous periodicity of eight. He cata­ logs all this information into useful tables of their most important features, such as tables of the spinor groups, groups of motion, conjugation types, the general linear groups, and the corresponding dimensions of the associated Lie algebras. He gives a brief history and discussion of Vahlen matrices and conformal trans­ formations, which have recently found diverse applications in pattern recognition and image processing. John Ryan, in Chapter 3, introduces the basic concepts of Clifford analysis which extends the well-known complex analysis of the plane to three and higher dimensions.A generalized Cauchy-Riemann operator, called the Dirac operator, makes it possible to introduce the analogues of holomorphic functions . Cauchy's theorem and Cauchy's integral formula all generalize nicely to the higher dimen­ sions of Clifford analysis. He shows that holomorphic functions are preserved under the action of conformal Mobius transformations. One of the most impor­ tant topics in classical complex analysis is the study of boundary value problems. These problems generalize nicely to the study of boundary behavior in Hardy spaces, which is closely related to the study of monogenic functions on the n-ball and its boundary, the (n - I)-sphere. By introducing the Fourier transform, and complex Clifford analysis, monogen ic functions defined on the n-ball can be holo­ morphically extended to larger "tube" domains. The last three chapters are built upon the core material set down in the first three chapters. Much of the recent interest in Clifford (geometric) algebras can be traced back to the work of David Hestenes and coworkers in late 1960's and early 1970's, who viewed Clifford 's geometric algebra as a unified language for mathematics and physics . In Chapter 4, William Baylis explores some of the extensive appli­ cations that have been made to physics. One of the earliest applications was to electromagnetism and the efficient formulation of Maxwell's equation as a single equation when expressed in the Pauli algebra, which is the geometric algebra of the physical space, or in the even more powerful Dirac algebra, which is the al­ gebra of Minkowski spacetime. Both of these geometric algebras can be applied to directly to the study of special relativity because of the ease of representing not only the rotations of 3-dimensional space, but the more general Lorentz trans­ formations of space and time.
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