Cover design Pictures on the cover were created by Jordi Vives Nebot (Dept. de Ffsica Fonamental, Universitat de Barcelona, Diagonal 647, E-08028, Barcelona, Spain). They represent sections x + ye12 + zet of two Julia-type 3-dimensional Clifford fractals viewed from an isometric perspective. Each image is a result of 10 iterations of the map c ~ cZ + b, where c and b are elements of the Clifford algebra Cl p,q' P + q = 2. Convergence of the resulting sequences of Clifford elements was determined by using a spinorial norm (the scalar part of the Clifford number times its Clifford-conjugate). The following metrics and base points b have been used to create these images: • Plane Fractal (larger image): b = -0.152815 + 0.656528e12 E Clz,o; • Ouaternionic Fractal (smaller image): b = -0.152815 + 0.5et + 0.656528e12 E CiO,2' The graphics have been produced on a 486 DX-l00 at a resolution of 1024 x 768 pixels. It took 24 hours of computing time to create the Plane Fractal and 15 hours to create the Ouaternionic Fractal. Clifford Algebras with Numeric and Symbolic Computations Rafal Ablamowicz Pertti Lounesto Josep M. Parra Editors 1996 Birkhauser Boston • Basel· Berlin Rafal Abramowicz Pertti Louesto Department of Mathematics Helsinki University of Technology Gannon University 02150 Espoo 15, Finland Erie, PA 16541 USA Josep M. Parra Department Ffsica Fonamental Facultat de Ffsica Universitat de Barcelona E-08028 Barcelona, Spain i Printed on acid-free paper Birkhiiuser lIP © 1996 Birkhauser Boston Softcover reprint of the hardcover 1s t edition 1996 Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhauser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of$6.00 per copy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests should be addressed directly to Birkhauser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. ISBN 978-1-4615-8159-8 ISBN 978-1-4615-8157-4 (eBook) DOI 10.1007/978-1-4615-8157-4 Typeset by the Editors in TEX. 987654321 TABLE OF CONTENTS Preface Rafal Ablamowicz, Pertti Lounesto, and Josep M. Parra .............. Vll List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . XVI 1. VERIFYING AND FALSIFYING CONJECTURES ................ 1 Counterexamples in Clifford algebras with CLICAL Pertti Lounesto ........................................ 3 2. DIFFERENTIAL GEOMETRY, QUANTUM MECHANICS, SPINORS AND CONFORMAL GROUP ............................. 31 The use of computer algebra and Clifford algebra in teaching mathematical physics hyme ~~ h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 33 General Clifford algebra and related differential geometry calculations with MATHEMATICA Josep M. Parra and Llorent; Rosello. . . . . . . . . . . . . . . . . . . . . . 57 Pauli-algebra calculations in MAPLE V W. E. Baylis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 The generative process of space-time and strong interaction quantum numbers of orientation Bernd Schmeikal ...................................... 83 On a new basis for a generalized Clifford algebra and its application to quantum mechanics A. Granik and M. Ross . .......................... . 101 Vector continued fraction algorithms D. E. Roberts . ............. . 111 LUCY: A Clifford algebra approach to spin or calculus lorg Schray, Robin W. Tucker and Charles H.-T. Wang . ............. 121 Computer algebra in spinor calculations Franco Piazzese .............. . 145 Vahlen matrices for non-definite metrics 1. Cnops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 155 3. GENERALIZED CLIFFORD ALGEBRAS AND NUMBER SYSTEMS, PROJECTIVE GEOMETRY AND CRYSTALLOGRAPHY. . . . . 165 On Clifford algebras of a bilinear form with an antisymmetric part Rafal Ablamowicz and Pertti Lounesto . . . . . . . . . . . . . . . . . . . . 167 V A unipodal algebra package for MATHEMATICA Garret Sobczyk . .................... . · 189 Octonion X-product orbits Geoffrey Dixon . . . . . . . . · 201 A commutative hypercomplex algebra with associated function theory Clyde M. Davenport. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 On generalized Clifford algebras - recent applications W. Bajguz and A. K Kwasniewski . ........ . · 229 Oriented projective geometry with Clifford algebra Richard C. Pappas . ......................... . · 233 The applications of Clifford algebras to crystallography using MATHEMATICA A. Gomez, J. L. Aragon, O. Caballero, and F. Davila ..... 251 4. NUMERICAL METHODS IN CLIFFORD ALGEBRAS .. 267 Orthonormal basis sets in Clifford algebras G. Bergdolt ................. . 269 Complex conjugation - relative to what? Alexander Soiguine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 285 Object-oriented implementations of Clifford algebras in C++: a prototype Arvind Raja. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 295 INDEX ............................................ 317 VI PREFACE The human mind is seldom satisfied, and is certainly never exercising its highest functions, when it is doing the work of a calculating machine. What the man of science, whether he is a mathematician or a physical inquirer, aims at is, to acquire and develope clear ideas of the things he deals with. For this purpose he is willing to enter on long calculations, and to be for a season a calculating machine, if he can only at last make his ideas clearer. - James Clerk Maxwell, 1870.1 These words of James Clerk Maxwell, who in September of 1870 gave a Presi dential Address at a meeting of Mathematical and Physical Sections of the British Association, fully express and justify our goals in planning and assembling this con tributed volume. Although we often indeed become "for a season" (or more) these "calculating machines", we look more and more into those marvelous time-saving machines we call computers as our new allies capable of symbolic, geometric, and algebraic methods. It is the intelligent use of computers that has given scientists more freedom, more time and more insight in their highest task of developing a scientific knowledge of Nature. Computer algebra systems such as AXIOM, CAYLEY, CLICAL, DERIVE, MAC SYMA, MAPLE, MATHEMATICA, MATLAB, and REDUCE have contributed considerably to mathematical research in the past few years. In addition to encour aging experimentation, their greatest significance has been in providing a common language and approach to a variety of mathematical problems. This way, they have contributed enormously to mutual understanding of people with different scientific backgrounds. We need to realize that we live in an "exceptional era" in which the new com puter tools not previously known to mathematicians and physicists are now available. Cartan's classification of real simple Lie algebras derived at the beginning of this century required an enormous amount of computation and time. Today, computa tions of that complexity would only take a few minutes of the computer time. Thus, it is important to develop confidence with computers doing numeric and symbolic computations in lieu of our becoming "calculating machines." Computers can be used for the following tasks: interactive and creative experimentation in creating new knowledge; verification of one's own and of others' hypotheses; falsification of conjectures, theorems, theories, paradigms, etc., by counter examples; 1 James Clerk Maxwell, 'Address to the Mathematical and Physical Sections of the British Association,' Liverpool, September 15, 1870, in: The Scientific Papers of James Clerk Maxwell, Vol. 2, ed. W. D. Niven, Dover Publications, Inc., New York, 1965, p. 219, lines 5 - 10. Vll verification and selection of right alternatives by elimination of implausible al ternatives. However, it is very important to know the limitations of the given computer program being used. Programs that have been used, some successfully and some unsuccessfully, to perform computations with Clifford algebras can be generally di vided into three groups: numeric, such as FORTRAN, C++, PASCAL (internal language of CLICAL), muSIMP (a function oriented language derived from LISP, an internal language of muM AT H); semi-symbolic, such as CLICAL; symbolic, such as MACSYMA, MAPLE, MATHEMATICA, DERIVE (previ ously known as muMATH) , AXIOM (previously known as SCRATCHPAD), MATLAB, REDUCE. For example, while MATLAB (Marcus, 1993) is used by physicists in quantum me chanics for matrix computations, the multivector approach offered by CLICAL (until recently the only computer program capable of such an approach) is much better precisely because instead of matrices it uses the much more efficient formalism of Clifford algebras. CLICAL has been used successfully by several of our contribu tors, namely Lounesto, Schmeikal, Pappas, and Soguine. Anthony Hearn's REDUCE (Hearn, 1968) was created in an unsuccessful attempt to find symbolic solutions to the Dirac equation with different potentials and for scattering computations with contracted Dirac matrices (Hearn, 1971). Later REDUCE was used in Gent, Bel gium, for symbolic computations with Clifford algebras. Use of REDUCE in teaching mathematical physics is discussed in our volume by J ayme Vaz. For historical ac curacy, we mention that the first computer algebra system which could actually run on a PC was muMATH created by Albert Rich and David Stoutemyer (Wooff and Hodgkinson, 1987; Freese et al., 1986), a precursor of DERIVE (Rich et al., 1989). Another group in Namur, Belgium, has used MACSYMA for symbolic com putations with Clifford algebras. None of these programs, muMATH, DERIVE and MACSYMA, is featured in this volume. A Grabner basis is a basis for an ideal generated by a given set of so called "distributed multivariate polynomials" (Geddes et al., 1993). The basis is generally not unique since it may depend on the order in which variables are specified. Grabner bases facilitate computations with multivariate polynomials, such as deciding if the given algebraic system of polynomial equations has a solution and, if so, solving it. Their introduction was very important for the development of symbolic programs such as MACSYMA (see first chapter of Davenport et al., 1988), AXIOM (with its precursor SCRATCHPAD), MATHEMATICA (with its precursor SMP computer algebra system), and MAPLE, capable of handling polynomials, solving systems of linear and non-linear equations, solving differential equations, etc. SCRATCHPAD, created by IBM researchers in the mid-70's, has developed into AXIOM, which offers a categorical approach to computer mathematics and contains a Clifford algebra domain for symbolic computations with these algebras (Jenks and Sutor, 1992). For a review of computer algorithms used in MACSYMA, muMATH, REDUCE, Vlll and SCRATCHPAD see (Davenport et al., 1988). For a brief history of computer algebra systems see (Geddes et al., 1993, pages 1 - 10). The two symbolic computer algebra systems used by most of our contributors are MAPLE and MATHEMATICA. The development of MAPLE started in 1980. An important property of the system is that most of its algebraic facilities are implemented in its high-level language which can also be used as a programming language by the user (Char et al., 1992; Geddes et al., 1993). Several of our con tributors, namely Baylis, Schray et al., and Ablamowicz, have benefited from that feature by creating extensive packages for Clifford algebra symbolic computations with MAPLE. MATHEMATICA, announced by Stephen Wolfram in 1988, also offers a high-level programming language (Wolfram, 1991). Parra and Rosello, Sobczyk, Gomez et al. have used MATHEMATICA in such diverse areas as mathematical physics and its teaching, unipodal number systems, and crystallography. Finally, two of our contributors, Bergdolt and Raja, opted for the numerical languages FOR TRAN and C++, respectively. Clifford algebras are at a crossing point in a variety of research areas, including abstract algebra, crystallography, projective geometry, quantum mechanics, differ ential geometry and complex analysis. For many researchers working with these algebras, the computer algebra systems have become an indispensable tool in a dis covery of new knowledge, in gaining a better understanding of the existing theory and its applications, and in the classroom teaching mathematical physics in the formidable language of Clifford algebras. In organizing our volume we have given priority to: manuscripts describing results on Clifford algebras which have been obtained with one or more of the above computer systems, including original packages written by our contributors; manuscripts summarizing contributors' own experiences in using one or more of these systems in Clifford algebra teaching; articles of high scientific quality which would be of interest to Clifford alge bra researchers and those wishing to learn about Clifford algebras and their applications in various fields. In the following section, we briefly review each contribution in an effort to guide the reader through the theory of Clifford algebras and some of its applications, and some new developments in both areas afforded by the use of computers. Chapter 1 contains a contribution from Pertti Lounesto, and it accomplishes several goals. First, it teaches us the need for critical reading of scientific literature not only because authors and referees make mistakes but mainly as a path to deeper understanding and creative thinking. Second, it proves beyond any doubt that well designed computer programs are most useful - we dare to say vital, due to the present-day complexity of mathematical science - for that critical learning task. They make possible a continuous exchange between general abstract theory and specific models or examples which is so essential in keeping science alive. James Clerk Maxwell publicly said it in an unmistakable way: IX There are men who, when any relation or law, however complex, is put before them in a symbolical form, can grasp its full meaning as a relation among abstract quantities. Such men sometimes treat with indifference the further statement that quantities actually exist in nature which fulfil this relation. The mental image of the concrete reality seems rather to disturb than to assist their contemplations. But the great majority of mankind are utterly unable, without long training, to retain in their minds the unembodied symbols of the pure mathematician, so that, if science is ever to become popular, and yet remain scientific, it must be by a profound study and copious application of those principles of the mathematical classification of quantities which, as we have seen, lie at the root of every truly scientific illustration. - James Clerk Maxwell, 1870.2 Third, it vindicates the falsifying strategy whenever there is any doubt about the validity of any proposition (and, most interestingly, when there seems to be no doubt at all in the minds of most). As a large sample of counter-examples shows, this is not at all a destructive task. Exactly the opposite is true: by recognizing where our present knowledge fails us, we are given an opportunity to correct it, extend it, and to select a better strategy for advancing it. Fourth, it will qualify the tenacious if not advanced reader as a "Clifford algebra critical expert" thanks to its wide spectrum of topics, such as spinor norm, intricacies of the covering groups and general scalar spinor products to the Dirac theory of electron, a relationship between Clifford and exterior algebras, as well as the method of exposition that requires a direct confrontation with well-documented sources. This accomplished, the following chapters offer a wide arena for critical learning, and, due to the open nature of this contributed monograph, also for a public debate. A newcomer to the field should not mistake Chapter 1 for an elementary introduction to Clifford algebras and should be prepared to postpone reading until tools offered in the remaining chapters or in Crumeyrolle (1990) are mastered. Chapter 2 presents those aspects of Clifford algebras that have established them as an essential part of what has been called physical mathematics. The use of Clifford geometric algebra facilitates formulation of fundamental physical laws and brings better understanding to physics, at least for those of us who prefer geometrical images or "illustrations" when connecting mathematical abstraction with physical reality. J ayme Vaz presents rotations and Lorentz transformations, electromagnetism, the Dirac equation, etc., in the language of Clifford algebras which has become a well-established practice in physics. Through a use of REDUCE packages, we are also led to interesting applications of Clifford algebras in differential geometry and gauge theories. We note here that REDUCE allows an extremely simple definition of (and manipulation with) Clifford algebras. Parra and Rosello contribute a MATHEMATICA package for algebraic and dif ferential geometry. Although specially devised for physicists' use, the package allows for arbitrary dimension, signature, and orthogonal curvilinear coordinates. Flexibil ity of its input/output facilities and simplicity of expression have made it an ideal 2 James Clerk Maxwell, 'Address to the Mathematical and Physical Sections of the British Association,' Liverpool, September 15, 1870, ibid., pp. 219-220, lines -4 - 8. x