Lectures on Clifford (Geometric) Algebras and Applications PDF
Preview Lectures on Clifford (Geometric) Algebras and Applications
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 ProfessorPertti Lounesto presentingLecture 1onMay 18,2002, during the "6thConferenceonCliffordAlgebrasand their Applications inMathematicalPhysics," May 20-25,2002, TennesseeTechnologicalUniversity,Cookeville,TN. DedicatedtoProfessorPertti Lounesto Mathematician,Teacher,Friend, andColleague Contents Preface Rafal AblamowiczandGarret Sobczyk xiii Lecture1:IntroductiontoCliffordAlgebras t PerttiLoullesto 1 1.1 Introduction 1 1.2 Clifford algebra of theEuclidean plane 2 1.3 Quaternions 7 1.4 Clifford algebra of theEuclidean space IR3 .......................•.. 11 1.5 The electorn spin inamagnetic field 14 1.6 From column spinors tospinoroperators 18 1.7 In4D:Clifford algebra Cf of1R4 23 4 1.8 Clifford algebra ofMinkowski spacetime 24 1.9 The exterior algebra andcontractions 25 1.10 The Grassmann-Cayleyalgebra andshuffleproducts 26 1.11 Alternative definitions oftheClifford algebra 26 1.12 References 29 Lecture2:MathematicalStructureofClifford Algebras IanPorteous 31 2.1 Clifford algebras 31 2.2 Conjugation 40 2.3 References 51 x Contents Lecture3: CliffordAnalysis John Ryan 53 3.1 Introduction 53 3.2 Foundations ofCliffordanalysis 54 3.3 OthertypesofCliffordholomorphicfunctions 62 3.4 TheequationDkf = 0 65 3.5 Conformal groupsandCliffordanalysis 71 3.6 Conformally flatspinmanifolds 74 3.7 Boundary behaviorandHardyspaces 75 3.8 MoreonCliffordanalysisonthesphere 78 3.9 TheFourier transformandCliffordanalysis 82 3.10 Complex Cliffordanalysis 86 3.11 References 87 Lecture4:ApplicationsofCliffordAlgebrasin Physics William E.Baylis 91 4.1 Introduction 91 4.2 Three Cliffordalgebras 92 4.3 Paravectors andrelativity 102 4.4 Eigenspinors 113 4.5 Maxwell'sequation 116 4.6 Quantum theory 130 4.7 Conclusions 132 4.8 References 132 Lecture5:CliffordAlgebrasin Engineering J.M. Selig 135 5.1 Introduction 135 5.2 Quatemions 137 5.3 Biquatemions 141 Contents xi 5.4 Points,lines,andplanes 144 5.5 Computer visionexample 150 5.6 Robotkinematics 151 5.7 Concluding remarks 154 5.8 References 155 Lecture6: CliffordBundlesandCliffordAlgebras Thomas Branson 157 6.1 Spin geometry 157 6.2 Conformal structure '" 164 6.3 Tractorconstructions 176 6.4 References 186 Appendix RafolAblamowiczandGarretSobczyk 189 7.1 SoftwareforCliffordalgebras 189 7.2 References 206 Index 211 Preface Advances in technologyover the last 25 years havecreated a situation in which workersindiverseareasofcomputerscienceandengineeringhavefounditneces sarytoincreasetheirknowledgeofrelatedfieldsinordertomakefurtherprogress. Clifford (geometric) algebra offers a unified algebraic framework for the direct expression of the geometric ideas underlyingthe great mathematical theories of linear and multilinear algebra, projective and affine geometries, and differential geometry. Indeed, for manypeople working inthis area, geometricalgebra isthe natural extension of the real number system to include the conceptof direction. The familiar complex numbers of the plane and the quaternions of four dimen sions areexamples oflower-dimensionalgeometricalgebras. During"The6th InternationalConferenceon 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 priorknowledgeofCliffordalgebras withabird'seyeviewofClifford 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 graduatestudents 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 thelatest developmentsandapplicationsinthefield. InChapter I, Pertti Lounesto states that "Clifford algebra isby definition the minimalconstructiondesigned tocontrol thegeometryinquestion," Hethengives aconcisebutcoherentintroductiontogeometricalgebras bythoroughlyexamin ingtheClifford productoftwovectors,thedefinitionofabivector,andhowthese concepts are used torepresentreflections androtations intheplane and inhigher 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, exterioralgebra and contraction,the Grassmann-Caleyalgebra andshuffleproduct, naturally evolvefrom themore basicconcepts. Animportant feature isthat heshows howthelowerdimensionalClifford algebras canbe rep resented intermsofthefamiliar algebraofsquare matrices.Hislectureends with xiv Preface several categoricaldefinitions of Clifford algebras ofaquadraticform, and their deformationstoClifford algebras ofanarbitrarybilinearform. InChapter2,IanPorteousshowsusthatthe"controlofthegeometry"thateach Clifford algebra hasfollowsdirectly fromitscloserelationshiptothecorrespond ing classical group. He begins with a systematic study of Clifford algebras by showing howtheycanbeconstructedfrommatrixalgebrasovertherealnumbers, complexnumbers, or quaternions with the famous periodicityof eight. He cata logs all this information into useful tables of theirmost importantfeatures,such as tables of the spinor groups, groups of motion, conjugationtypes, the general linear groups, and the correspondingdimensions of the associated Lie algebras. He gives a briefhistory and discussion of Vahlenmatrices and conformaltrans formations,whichhaverecently founddiverseapplicationsinpattern recognition and image processing. John Ryan, in Chapter 3, introduces the basic concepts of Clifford analysis which extends the well-known complex analysis of theplane to three and higher dimensions.AgeneralizedCauchy-Riemannoperator, called theDirac operator, makes itpossible tointroduce theanalogues ofholomorphicfunctions.Cauchy's theorem andCauchy'sintegralformula allgeneralizenicely tothehigherdimen sions of Clifford analysis. He shows that holomorphic functions are preserved under the action of conformal Mobius transformations. One of the most impor tanttopics inclassical complex analysis isthestudyofboundaryvalueproblems. These problems generalize nicely to the study of boundary behavior in Hardy spaces, whichisclosely related tothestudyofmonogenic functions onthen-ball and its boundary, the (n - I)-sphere. By introducingthe Fouriertransform, and complexCliffordanalysis,monogenicfunctionsdefinedonthen-ballcanbeholo morphicallyextended tolarger"tube"domains. The last three chapters are built upon the core material set down in the first threechapters. Much oftherecent interest inClifford (geometric) algebras canbetracedback 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.InChapter 4, William Baylis explores some of theextensive appli cations that have been made to physics. One of the earliest applications was to electromagnetismandtheefficientformulation ofMaxwell'sequation asasingle equationwhen expressed in the Pauli algebra, which isthe 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 geometricalgebras can be applied to directly to the study of special relativity because of the ease of representing notonlytherotations of3-dimensionalspace,butthemoregeneralLorentz trans formationsof space andtime. Therelationshipsand interplay between these two important algebras are clearly explained,and a set of fully integrated exercises provides many opportunities for the reader to get a hands-on knowledge of the basic ideas. Some of the many important topics explored are charge dynamics in uniform fields,directed plane waves,polarizationand phase shifters,standing waves,andthepotentialofamovingcharge.Thespinorialformulationofclassical
The list of books you might like
