Clifford algebras and geometric algebra

CLIFFORD ALGEBRAS AND GEOMETRIC ALGEBRA G. Arag£

J.L. Arag£ ~1 and M.A. RodrŸ167

Programa de Desarrollo Profesional en Automatizaci£ Universidad Autonoma Metropolitana, Azcapotzalco San Pablo, 108. Colonia Reynosa- Tamaulipas 02~00 M› Distrito Federal, M› $Instituto de FŸ Universidad Nacional Aut£ Apartado Postal ~0-36J 01000 M› Distrito Federal, M›

de M›

wDepartamento de Matem• Escuela Superior de FŸ y Matem• Instituto Polit› Nacional Unidad Pro]esional Adol]o L£ Mateos, Edificio 9 07300 M› Distrito Federal, M›

(Received: August 8, 1997, Accepted: October 3, 1997) A b s t r a c t . The geometric algebra as defined by D. Hestenes is compared with a constructive de¡ of Clifford algebras. Both 91 are discussed and the equivalence between a finite geometric algebra and the universal Clifford algebra Rp,q is shown. Also ah intermediate way to construct Clifford 91 is sketched. This attempt to conciliate two separ91 approaches may be useful taking into account the recognized importance of Clifford algebras in theoretical and 91 physics. PACS numbers: 02., 02.10.Tq, 03.65.Fd 1. I n t r o d u c t i o n Over the past 40 years the work of three central figures of the past century mathematics has received increased attention: R. Hamilton, H. Grassman and W.K. Clifford. In an attempt to generalize complex numbers to R 3, Hamilton established in 1843 (first published 1844) the theory of quaternions and was convinced that he had found a natural algebra of the three-dimensional Advances in Applied Ciifford Algebms 7 No. 2, 91-102 (1997)

92

Clifford Algebras and ...

G. Arag£ J.L. Arag£ and M.A. RodrŸ

space with many applieations in physics. In 1844 in the Die Lineale Ausdehnungslehre, Grassman expounded a systematic approach to multidimensional geometry and described what he called Eztensive Algebra which led to the development (among other areas) of differential forms. Finally, in 1876, Clifford proposed the geometric Algebra (now known as Clifford algebra) that included both extensive and quaternionic algebras. Clifford algebras where formalized in the fifties, between others, by C. Chevalley [2], E. Artin [3], M. Riesz [4], A. Crumeyrolle [5] and I. Porteous [6]. In all of these works a basically constructive approach was followed, with differences lying in very specific points such as the use of the quadratic form instead the bilinear form [6] of the use of the exterior algebra such that the Clifford algebra is viewed a s a subalgebra of the endomorphism algebra of the exterior algebra [2]. Almost at the same time, a seeond and different approach emerged. The physicist, David Hestenes (a Riesz' student) proposes the socalled geometrie algebra [7, 8] (as Clifford himself called it) which, as he points out, is an axiomatic approach very similar to those of the real numbers (but non-commutative) providing a unified approach to both metric and non-metric geometries. Hestenes claims that his geometric Algebra is a new language full of geometric information and constitutes the best available mathematical tool for theoretical physics. With this new language, he further developed the mathematical foundations of relativistic and non-relativistic classical and quantum mechanics [7, 9]. Most of the applications of Clifford algebras in physics ate based in the first approach but the use of geometric algebra is growing fast (see the special numbers 9, 10 and 11 of Foundations of Physics, volume 23, dedicate to D. Hestenes), and there ate claims that Hestenes' work has not had the impact it deserves to have [10]. We believe that there has been no attempts to conciliate both approaches (if it is possible) and consequently exploding to a maximum their respective possibilities. The purpose of this paper is double. First, with a didactic motivation, both approaches ate presented (Sections 2 and 3) with the aim of having a comparative view. Concerning the first approach, we shall use the one developed by Delanghe and coworkers in Ref.[ll]. Second, in Sec. 4, the equivalence between the finite geometric Algebra and the universal Clifford algebra of signature (p, q) is demonstrated. Fi¡ Sec. 5 is devoted to discussion and conelusions.

2. Real Clifford Algebras In this section we shall define a real Clifford algebra. Most definitions are taken from Delanghe, Sommen and Sou~ek's book [11], and the reader will be referred to this material when needed.

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Let us considera n-dimensional real vector space X a n d a non-degenerate symmetric bilinear form B on X. (X,B) is said to be a non-degenerate ndimensional real orthogonal vector space. DEFINITION 1. Let (2", B) be a non-degenerate n-dimensional real orthogonal rector space and let ,4 be a real associative algebra with unir element 1 such that: I. ,4 eontains isomorphic copies of R and X as linear subspaces. 2. For all v E X , v 2 = B (v, v). 3. A is aiso generaŸ asa ring by the copies of R and X of, eqnivalently, as a real associative algebra generated by {1) and X . Then ,4 is said to be a (real) Clifford algebra for (2d, B) and it is denoted by ,4 = C (X). As mentioned in the introduction, Clifford's term was geometric algebra but in order to avoid confusions, this name will be reserved to Hestenes' approach. 2.1. BASIS FOR THE CLIFFORD ALGEBRAS Let C (X) be a Clifford algebra of the real orthogonM space (X, B) of signature (p,q) and let {el, ...,en} be an orthonormal basis of ( X , B ) . The requirement 2 of definition 1 implies that: 2 f 1, ei -~ ~ - 1 , eiej + ejei = O,

i = 1,2,..,p. i=p+l,..,p+q.

(1)

i r j.

To make things easier, let us introduce the following notation. For each naturally ordered subset S of N = {1,2, ..., n}, es will stand for the product I-[lES ei, with e~ = 1. In particular, eN = I'LeN ei. The set S = {es I S C N} spans C(X) linearly and dimC(X) b = 0 . (iii) Bilinearity condition For elements a, b E 9:' and A E R we have: Aa. b = 89(Aab+ bAa) = ~ ( a b + ba) = A ( a . b) a . Ab = 89(aAb + Aba) : ~ (ab + ha) = A (a . b). Let us now consider the vector space X with a bilinear forro B (a, b) = a 9b on it, then since I = ele2...en is a n-blade it follows that {el, es, ..., en} is ah orthonormal basis of X as is inferred from: 1 ei. ej = ~ (eiej + ejei) = O, for i r j and i , j = 1,2,...,n. Thus the dimension of 291is n, and from all the previous results, we have that ~ (I) is a Clifford algebra. That G (I) is a universal Clifford algebra follows from the fact t h a t I%r the n-blade I = ele2...e,, there exists a vector a E ~, such that Ia is a n + 1 - b l a d e . Consequently I ~ P~ and dim ~ (I) = 2". Conversely, let us now proof that a universal Clifford algebra is a geometric algebra. As a consequence of the isomorphism between Clifford algebras (see Theorem 15.13 in Ref.[12] of Theorem 5 in Sec. 5), we can use the same vector space X without loosing generality. Let (X, B) be of type (n, 0) and let C (X, B) be its universal Clifford algebra 1~,0. In what follows, we show that this algebra satisfies the axioms of a geometric algebra (see Sect. 3.1). If R~, 0 denotes the k-space of l~,,0 then since C(X,B) = ~ Rn,0, k for k----0

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n

A ~ C ( X , B ) we have that A = ~ A k ,

where Ak E R~,0. The projection

k=l

()k :C (X, B) --+ R kn,o, de¡

(A+B)k

=

(A)k+(B)k

by (A)k = Ak, has the following properties:

,

(AA)k = A(A)k , ((A)k)k = (A) k, where A , B 6 C ( X , B ) , A = (A)0 6 R = R ~ 0 and/r = 1,2,..,n. We thus have two algebras defined in t~e vector space X, namely: rL

e(x,s)

=

G

R/c.,o

k=0

and n

~ (1) = (~~~ (I) k=0

where dim (R~,0) = d i m ( ~ k (I)) = (~), 0 < k < n. In each one of the above algebras inner and outer products, that depend on P~,0 and of Gk (I), respectively, are defined and share the same properties. But since B (a, b) = a. bit follows from the theorem of existence and uniqueness of Clifford algebras that both algebras coincide, that is ~ (I) ~ C (X, B). We have also that R~, 0 ~ ~k (I). T H E O R E M 3. Let ~ (I) be a subalgebm of the geometric algebra ~ generated by the n-blade I = ele2...en, where eieI = -ejei, for i,j = 1,...,n and i r j. Let X be the vector space spanned by the set {el,e~,...,en} and consider the Cligord atoebm e ( x , B), with B ( a, b) = 89(ab + be) a bitinear Ÿ of tVpe (n, 0). The finite geomelric al•ebra ~ (I) is a universal Clifford algebra l~.o, ~nd g ( Z) ~ C ( X , ~). We have demonstrated the equivalente between a finite geometric algebra a n d a universal Clifford algebra considering a positive definite bilinear forro. In case of a bilinear forro of type (p, q) we have to consider a n-blade I of signature (p, q) with p q- q = n, that is, I is composed of factors with p vectors with positive square and q vectors with negative square. In that case, the set

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G. Arag£

J.L. Arag£

and M.A. RodrŸ

of vectors a satisfying a A I = 0 constitutes a n-dimensional vector space with signature (p, q) and

a . b = ~ 1 (ab+ba), is a bilinear forro with signature (p, q). The axiomatic of the geometric algebra remains valid in this case with the exception of the axiom 7 which must be replaced by requirement that a 2 E R. Under this condition, we can state the following theorem. T H E O R E M 4. Let G (I) be a subalgebra of the geometric algebra ~ generated by the n-blade I of signature (p,q) wzth p + q = n, given by I = ele2...en with eiea = -ejei, i , j = 1,...,n an d i ~ j. Let X be the rector space of signature (p,q) spanned by the set {el,e=,...,en}, and consider the Clifford algebra g ( X, B), w~th 13 (a, b) = } (ab + ba) a bilinear form of type (p, q). The finite geometric algebra ~ (I) is a universal Clifford algebra Rp,q, and ~ (I)

c(x,t~). 5. D i s c u s s i o n Some discussion about the two approaches presented in this work shall be given here. We start by considering the tole that the bilinear form plays in a Clifford algebra, which is brought out by the following theorem. THEOREM 5. Let (X,B) and (X*,B*) two non-degenerate orthogonal spaces with the same signa•ure, respect to their respecŸ bilinear forms. Then, there exists ah isomorphism bdween the respective universal Clifford algebras C ( X , B ) and C (X*, B*). The proof follows from the Theorem 15.13 in Ref.[12]. An immediate corollary from this theorem includes that two 2'*-dimensional Clifford algebras of a real orthogonal non-degenerate n-dimensional space (n = p + q) ate isomorphic [12]. Consequently, since a universal Clifford algebra for 2" is uniquely defined up to an, essentially unique, isomorphism, one often speak of the universal Clifford algebra for X'. It only remains to prove the existence of such an algebra for any X. The existence theorem of a universal Clifford algebra can be found in Ref.[12], Theorem 15.14. A explained in See.2.1, if X is a real orthogonal non-degenerate space of dimension p+q, then k" "~ l~P,q. This lead us to the fac~ that it is the signature, not the bilinear form, what is important when considering Clifford algebras. From this point of view, the difference between the two approaches lies in the

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role ".hat the bilinear form plays. In the first construction, it is given a priori and remains fixed. In a geometric algebra, it is obtained directly from the geometric product, that is, a posteriori, thus in this case the bilinear form can be modified by changing the geometric product. In fact, let {el, e2, ..., en} be a linearly independent set of vectors of R p'q and let us define the geometric product between them by the relagionship (1), which can be extended by linea~ity to the whole R p,q. Prom here, we can define the inner product through a 9b = 89 + ba) a n d a universal Clifford algebra can be constructed from ttestenes' axioms. Thus, we have a bilinear form defined a posleriori, namely B(x, y) = x 9y. In the Clifford algebra so defined, the initial set {el, e2, ..., en ) is orthonormal and notice that we made use of neither the Gram-Schmidt process n o r a change of basis. Since it is well known that orthonormal basis make things easier, this approach allows us to define, from a linear independent set of vectors, the Clifford (or geometric) algebra best suited to our needs. This in principle may be of great advantage for the applications of Clifford algebras to, for instance, computer graphics [13] or crystallography [14]. From a concepgual point of view, the axiomatic approach is full of geometric significance and free of basis, requiring only basic concepts of linear algebra (vector space, basis, dimension, linear independence, etc) leaving aside many others well established concepts and techniques such as change of basis, GramSchmidt process, matrices, duality, etc. Ii may turn be important for teaching mathematics since, as ttestenes has pointed out, this approach reduces similarities in different algebraic systems to a common body of relations, definitions and theorems. Hesgenes' approach requires a new conceptualization of mathematics that we ghink is feasible provided that a definitive example showing the necessity of this approach can be found. Until now, several drawbacks exists. For in_ stance, i• has been pointed out [15] that Hestenes' construction yields a Dirac equation having a unnecessary complex form that should not be expected of a fundamental law. On the other side, using Clifford algebras instead the well known concepts and techniques of linear algebra requires to work in spaces with more dimensions ghat those really needed. Never•heless, the conceptual simplicity of the axiomatic construction, their unifying language and the stimulating work of D. Hestenes, deserves much more atgention and the search of future applications. A s a final consideration we would like to mention that ah intermediate way between both constructions presented here can be taken up. By starting with a set of n linearly independent vecgors in R p'q satisfying the anticommutativity law (1), the geometric product can be obtained by extending by linearity •his relationship. Now, as mengioned in the beginning of this Section, the inner

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J.L. Arag£ and M.A. RodrŸ

product can be considered a s a . b = 89 + ba) (a, b E RP'q). With some extra axioms to be established, we can construct the universal Clifford algebra Rv, q by demanding the pseudo-scalar not to be a real number. This work is under way.

Acknowledgment s The authors wish to acknowledge the very useful comments and suggestions of Alfredo G£ and Fabio D• This work was supported by C O N A C y T (grant 0088P-E) and DGAPA-UNAM (grants IN-104296 and IN-107296). References [1] To whom correspondence should be addressed. [2] Chev91 C.C., "The Algebraic Theory of Spinors', Columbia University Press (New York, 1954). [3] Artin E., ~Geometric Algebra", Interscience (New York, 1957, 1988). [4] Riesz M., "Clifford Number and Spinors", The Institute for Fluid Dynamics and Applied Mathematics, Lecture Series No. 38 (University of Maryland, 1958). [5] Crumeyrolle A., Ann. Inst. Henri Poincar› A l l , 19 (1969); A14, 309 (1971). [6] Porteous I.R., "Topological Geometry", Van Nostrand-Reinhold (London, 1969) also Cambridge University Press (•ambridge, 1981). [7] Hestenes D., "Space-Time Algebra", Gordon & Breach (New York, 1966, 1987). [8] Hestenes D. and G. Sobczyk, "Clifford Algebra to Geometric Calculus", Reidel (Dordrecht, 1984, 1987). [9] Hestenes D., "New Foundations for Classical Mechanics", Reidel (Dordrecht, 1985). [10] Gull S., A. L91 and C. Doran, Found. Phys., 23, 1175 (1993). [11] Delanghe R., F. Sommen and V. Sou~ek, "Clifford Algebra 91 Spinor Va]aed Functions: A Function Theory for the Dirac Operator ", Reidel (Dordrecht, 1992). [12] Porteous I.R., "Clifford Algebras a~d the Cl~sic 91 Groups", Cambridge University Press (Cambridge, 1995). [13] Bayro-Corrochano E. and J. Lasenby, in Proceedings of Europe-China Workshop on "Geometric Modeling and Invariants for Computer Vision", eds. R. Mohr and W. Ghengke. Xi'an, China. April (1995). [14] G£ A., J.L. Arag£ F. Ds and O. Caballero, in "Glifford Algebra~ with Numeric and Symbolic Computations", eds. R. Abtamowicz, P. Lounesto and J. M. Parra. Birkhauser (Boston, 1996) p.251. [15] Vrbik J., J. Math. Phys., 35, 2309 (1994).