Table Of ContentRafaI 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
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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
