Table Of ContentCover 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
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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
