Clifford Algebras: Applications to Mathematics, Physics, and Engineering

626 Pages·2004·23.36 MB·English

by  Carlos A. Berenstein

Progress in Mathematical Physics
Volume 34
Editors-in-Chief
Anne Boutet de Monvel, Universite Paris VII Denis Diderot
Gerald Kaiser, The Virginia Center for Signals and Waves
Editorial Board
D. Bao, University of Houston
C. Berenstein, University ofM aryland, College Park
P. Blanchard, Universitat Bielefeld
A.S. Fokas, Imperial College of Science, Technology and Medicine
C. Tracy, University of California, Davis
H. van den Berg, Wageningen University
Clifford Algebras
Applications to Mathematics,
Physics, and Engineering
Rafal Ablamowicz
Editor
Birkhtiuser
Boston • Basel· Berlin
Rafal Ablamowicz
Department of Mathematics, Box 5054
Tennessee Technological University
Cookeville. TN 38505
U.s.A.
rablamoyicz~tntech.edu
http://www.math.tntech.eduirafaV
Library of Congress Cataloging-In-Publication Data
A CIP ~atalogue record for this book is available from the Library of Congress,
Washinglon D.C., USA.
AMS SubjcclClassifications: Primary: 30035. ISA66, ISA69, I SA7S. 53C27, 5SB32. 5SB34,6SUIO, SI R2S,
SIRSO; S«,ondary: 05EIO. 05EIS, IIESl, I lESS, IIF03, 11F41, 13ElO, 14D21, 15·04, ISA33, 16W30,
16WSO, I1A3S, 17B, 17B10. I1B3S, 17B37. I1B4S, 19D5S, 2OC0S,2OC30,2OGIS,2OO42,22E2S,3OG25,
31CI2, 31C20, 32A36, 32A4O, 32A50, 32W05, 35F15, 35HlO, 35K15, 35040. 35S0S, 42B30, 42835,
43ASO. 46H30. 46LS7, 47A13, 47A60, 51N2S, 53A30, 53820, 53C07, S3C15. S3(2S, 53C30, 53C50.
5SC35, SSJ4O,58J60, 65D17,65KOS,65M60, 65RlO, 78A45, 78A46, 81PIS, 81Q05, 8ITIO, 81T13, 81nS,
81 V22, 83AOS, 83COS, 83(40, 8307, 83C6O
ISBN-13: 978-1-4612-7393-6 c-1SBN·13: 918-1-4612-2044-2
DOl: 10.1007/918-1-4612-2044-2
a»"
02004 Birkhltuser Boston Birkhiiuser lLIJl.l
Soflcovcr reprint of the hardcover in lSI edition 2004
All rights reserved. This work may 1101 be translated or copied in whole or in pan withoulthe written
permission of the publisher (BirkMuser Boslon, c/o Springer-Verlag New York, Inc., 175 Fifth Avenue,
New York, NY 10010, USA), c;I(cept for brief C;l(cerpts in connection with reviews or scholarly analysis.
Use in conneclion with any form of information storage and retrieval, electronic adaptatioo, compuler
software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks and similar tenns, even if they are
not identified as such, is not to be taken as an expressinn of opinion as to whether or 001 they are subject
to propeny rights.
Typeset by !be editor in AMS-IH&X.
9 8 7 6 S 4 3 2 I SPIN 10955864
BirWuser is pan of Springer Science+Business Media
www.birkhauser.com
Contents
PREFACE
Rafal Ablamowicz ...................................................... IX
PART I. CLIFFORD ANALYSIS ......................................... 1
1. The Morera Problem in Clifford Algebras and the Heisenberg Group
Carlos A. Berenstein, Der-Chen Chang, and Wayne M. Eby .................. 3
2. Multidimensional Inverse Scattering Associated
with the SchrOdinger Equation
Swanhild Bernstein ..................................................... 23
3. On Discrete Stokes and Navier-Stokes Equations in the Plane
Klaus Giirlebeck and Angela Hommel .................................... 35
4. A Symmetric Functional Calculus for Systems of Operators of Type w
Brian Jefferies .......................................................... 59
5. Poincare Series in Clifford Analysis
Rolf Soren Krausshar ................................................... 75
6. Harmonic Analysis for General First Order Differential Operators
in Lipschitz Domains
Emilio Marmolejo-Olea and Marius Mitrea ............................... 91
7. Paley-Wiener Theorems and Shannon Sampling
in the Clifford Analysis Setting
Tao Qian ............................................................. 115
8. Bergman Projection in Clifford Analysis
Guangbin Ren and Helmuth R. Malonek ................................. 125
9. Quaternionic Calculus for a Class of Initial Boundary Value Problems
Wolfgang Sprossig ........................ '" .......................... 141
vi Contents
PART II. GEOMETRY ................................................ 153
lO. A Nahm Transform for Instantons over ALE Spaces
Claudio Bartocci and Marcos Jardim .................................... 155
11. Hyper-Hermitian Manifolds and Connections
with Skew-Symmetric Torsion
Gueo Grantcharov ..................................................... 167
12. Casimir Elements and Bochner Identities on Riemannian Manifolds
Yasushi Homma ....................................................... 185
13. Eigenvalues of Dirac and Rarita-Schwinger Operators
Doojin Hong .......................................................... 201
14. Differential Forms Canonically Associated to Even-Dimensional
Compact Conformal Manifolds
William J. Ugalde ..................................................... 211
15. The Interface of Noncommutative Geometry and Physics
Joseph C. Vdrilly ...................................................... 227
PART III. MATHEMATICAL STRUCTURES ........................... 243
16. The Method of Virtual Variables and Representations of Lie
Superalgebras
Andrea Brini, Francesco Regonati, and Antonio Teolis .................... 245
17. Algebras Like Clifford Algebras
Michael Eastwood ..................................................... 265
18. Grade Free Product Formula: from Grassmann-Hopf Gebras
Bertfried Fauser ....................................................... 279
19. The Clifford Algebra in the Theory of Algebras, Quadratic Forms,
and Classical Groups
Alexander Hahn ....................................................... 305
20. Lipschitz's Methods of 1886 Applied
to Symplectic Clifford Algebras
Jacques Helmstetter ................................................... 323
Contents vii
21. The Group of Classes ofInvolutions
of Graded Central Simple Algebras
Jacques Helmstetter ................................................... 335
22. A Binary Index Notation for Clifford Algebras
Dennis W. Marks ...................................................... 343
23. Transposition in Clifford Algebra: SU (3)
from Reorientation Invariance
Bernd Schmeikal ...................................................... 351
PART IV. PHYSICS ................................................... 373
24. The QuantumlClassical Interface: Insights
from Clifford's (Geometric) Algebra
William E. Baylis ...................................................... 375
25. Standard Quantum Spheres
Francesco Bonechi, Nicola Ciccoli, and Marco Tarlini .................... 393
26. Clifford Algebras, Pure Spinors and the Physics of Fermions
Paolo Budinich ........................................................ 401
27. Spinor Formulations for Gravitational Energy-Momentum
Chiang-Mei Chen, James M. Nester, and Roh-Suan Tung .................. 417
28. Chiral Dirac Equations
Claude Daviau ........................................................ 431
29. Using Octonions to Describe Fundamental Particles
Tevian Dray and Corinne A. Manogue ................................... 451
30. Applications of Geometric Algebra in Electromagnetism,
Quantum Theory and Gravity
Anthony Lasenby, Chris Doran, and Elsa A rca ute ......................... 467
31. Noncommutative Physics on Lie Algebras,
(Z2t Lattices and Clifford Algebras
Shahn Majid .......................................................... 491
32. Dirac Operator on Quantum Homogeneous Spaces and
Noncommutative Geometry
Robert M. Owe zarek ................................................... 519
viii Contents
33. r-Fold Multivectors and Superenergy
Jose M. Pozo and Josep M. Parra ....................................... 531
34. The CC Approach to the Standard Model
7
Greg Trayling and William E. Baylis ..................................... 547
PART V. APPLICATIONS IN ENGINEERING .......................... 559
35. Implementation of a Clifford Algebra Co-Processor Design
on a Field Programmable Gate Array
Christian Perwass, Christian Gebken, and Gerald Sommer ................ 561
36. Image Space
Jan 1. Koenderink ..................................................... 577
37. Pose Estimation of Cycloidal Curves
by using Twist Representations
Bodo Rosenhahn and Gerald Sommer ................................... 597
INDEX .............................................................. 613
Preface
This volume contains a selection of invited papers, often in expanded form, that
are based on presentations made at the 6th Conference on Clifford Algebras and
their Applications in Mathematical Physics, May 20-25, 2002, in Cookeville,
Tennessee [1]. The organizers of the conference, Rafal Ablamowicz (Tennessee
Technological University), and John Ryan (University of Arkansas) grouped all
conference presentations into five sessions in Clifford analysis, geometry, mathe
matical structures, physics, and applications in engineering. Thus it seemed natu
ral to organize the book into five parts under the same titles.
The 6th Conference on Clifford Algebras continued a 16-year sequence of in
ternational conferences devoted to the mathematical aspects of Clifford algebras
and their varied applications in mathematical physics, and, more recently, in cy
bernetics, robotics, image processing and engineering. Previous meetings took
place at the University of Kent, Canterbury, U.K., 1985; University of Montpel
lier, Montpellier, France, 1989; University of Gent, Gent, Belgium, 1993; Univer
sity of Aachen, Germany, 1996; and Ixtapa, Mexico, 1999. Three edited volumes
appeared after the Ixtapa conference, published by Birkhauser, Boston [2--4].
Chapter I, in this volume, is devoted exclusively to topics from Clifford anal
ysis and range from the Morera problem, inverse scattering associated with the
SchrOdinger equation, through discrete Stokes equations in the plane, a symmetric
functional calculus, Poincare series, to differential operators in Lipschitz domains,
Paley-Wiener theorems and Shannon sampling, Bergman projections, and quater
nionic calculus for a class of boundary value problems. Among geometry topics,
not so visibly present at previous conferences that gave rise to Chapter 2, are
spin structures and Clifford bundles, eigenvalue problems for Dirac and Rarita
Schwinger operators, differential forms on conformal manifolds, connection and
torsion, Bochner identities on Riemannian manifolds, and noncom mutative ge
ometry in physics. Chapter 3 is devoted to mathematical structures such as Grass
mann algebras, Lie superalgebras, Grassmann-Hopf algebras, symplectic Clifford
algebras and graded central algebras. Applications in physics, collected in Chap
ter 4, cover a wide range of topics from classical mechanics to general relativity,
twistor and octonionic methods, electromagnetism and gravity, elementary parti
cle physics, noncommutative physics, Dirac's equation, quantum spheres, and the
Standard Model. Chapter 5 includes papers on Clifford geometric algebras and
designing co-processors in computer engineering, applications in the description
of an image space using Cayley-Klein geometry, and pose estimation. Below the
reader will find brief introductions to all chapters in this volume.
While the papers collected in this volume require that the reader possess a solid
x Preface
knowledge of appropriate background material, as they lead to the most current
research topics, the fundamentals of this background were presented prior to the
6th Conference in the form of six lectures. The lectures were delivered by promi
nent specialists in the field (including Pertti Lounesto to whom this volume is
dedicated). They were aimed at graduate students and newcomers to the field.
These lectures will appear in a separate volume from Birkhauser in 2003 [6].
Papers from regular sessions at the 6th Conference included here were selected
by the conference session organizers who oversaw the refereeing process:
o Clifford Analysis: Marius Mitrea (University of Missouri-Columbia, Co
lumbia, Missouri), and Mircea Martin (Baker University, Baldwin City,
Kansas);
o Geometry: Tom Branson (University of Iowa, Iowa City, Iowa), and Ugo
Bruzzo (International School for Advanced Studies, Trieste, Italy);
o Mathematical Structures: Ludwik Dabrowski (SISSA, Trieste, Italy), and
Bertfried Fauser (Universitat Konstanz, Konstanz, Germany);
o Physics: William Baylis (University of Windsor, Windsor, Canada), and
Giovanni Landi (Universita di Trieste, Trieste, Italy);
o Applications: Jon Selig (South Bank University, London, England), and
Gerald Sommer (Christian-Albrechts-Universitat Kiel, Kiel, Germany).
Papers presented as plenary talks were selected by the Editor. The main aim of
this volume, which guided the selection process, was to collect the best papers
in each of the five representative areas ranging from pure mathematical theory
to a wide array of applications in physics and computer engineering. Following
the spirit of the conference, this volume unlike previously published volumes, in
cludes a separate chapter on geometry.
Dedication
I am dedicating this book to the memory of Pertti Lounesto, friend and colleague,
whose plenary lecture at the 6th Conference proved to be his last major contribu
tion to the field of Clifford algebras - the field he loved so much.
PART I. CLIFFORD ANALYSIS
Berenstein, Chang and Eby: In this paper the authors discuss the Morera prob
lem for functions f taking values in a Clifford algebra Cin. As a replacement
for the Morera theorem from the complex plane C, which gives necessary and
sufficient conditions for a continuous function f in an open subset n of C to be
n,
holomorphic in they formulate integral conditions whose vanishing on a Lip
schitz boundary of every bounded domain secures that f be left regular. They
establish a connection, similar to the one from the Euclidean space, between the