Guest Editor's Note: Clifford Algebras and Their Applications

THE SEVEN TOPICAL SESSIONS IN IXTAPA

Dedicated to Gian-Carlo Rota. Gian-Carlo Rota and Joel A. Stein in 1994 rai-sed the problem of accomodating Clifford algebra with the Hopf gebra or bigebra structure (this is the Bourbaki terminology (Bourbaki, 1989), instead of bial gebra, etc) and concluded that no such structure is possible (Rota & Stein, 1994, p. 13058). I considered Chevalley deformation of Woronowicz's braided generalization of an exterior algebra (and cogebra) (Woronowicz, 1989), leading to braided Clifford algebra. After Micho Durdevich arrived to México in 1993, we began stimulating discussions on this subject, which resulted in several separated and joint publications: on Clifford algebra for a Hecke braid (Oziewicz 1995), on Clifford algebra for arbitrary braid (Durdevich andOziewicz, 1994, 1996), and on Clifford quantum (i.e. Hopf) algebra (Durdevich, 1994(Durdevich, , 2001. However at this time I started to believe that the notion which must be studied first is Clifford cogebra alone, and that the question of how to define the Clifford bigebra in terms of a joint pair algebra and cogebra can be studied subsequently (Oziewicz, , 1998 in the spirit of a Lie bigebra (cf. Michaelis, 1980). In meantime, Durdevich defined Clifford Hopf gebra without Clifford cogebra (1994,2000).

Starting in 1995, I had many inspiring electronic discussions with Gian-Carlo Rota concerning how to accomodate the Hopf gebra or bigebra structure within a Clifford algebra. In 1997, we meet at the AMS Meeting in Oaxaca, México, where we also arranged for him to give the main invited plenary lecture at the Fifth Conference on Clifford Algebra in Ixtapa, México. Nonassociative Structures. Lev Sabinin organized the session on Moufang quasigroups and loops, which can be understand also as a triple of binary operations related by identities, generalizing a group operation to nonassociative structure.

Gravity and Elementary Particles. This session was organized by Leopold Halpern and Kurt Just.

sometimes it is necessary to read all the papers written by the same author(s) and scattered among different publications, and we sincerely suggest the reader do this. David Hestenes and many his followers are convinced that 'the much larger community of scientists and engineers is interested in the powerful tools of geometric algebra', as opposite to Clifford algebra. However, besides these irrelevant different names, there is also a deeper diversity of the philosophies that can not be discussed so easily (Hestenes, 1992).

I want to thank the authors for their contributions to the present volume and to the reviewers for their generous help. Most of these publications were reviewed by up to three independent referees. After referee reports, most of the contributions were expanded and several times revised.

We have tried to organize the present issue thematically, by grouping related contributions. This has not been completely successful as several authors span several areas.

DIRAC OPERATOR: CROSS RELATIONS

In the present volume most of the papers deal with the Dirac operators: from the minimal coupling, γ • ∇ ∼ γ µ (∂ µ − iA µ ), to the maximal coupling (Kähler, 1960(Kähler, , 1961(Kähler, , 1962, γ •(∇+c⊗−) with c in the Clifford algebra of the differential multiforms C (V, −) ∼ V ∧ , see the last Section for more details, c = Scalar + Vector + Bivector + Trivector + . . . ∈ C (−, −).

µν is said to be the Pauli term. There seems to be a general trend to generalize the Dirac equation to maximal coupling. In particular this is strongly advocated in this volume by Rolf Dahm through a phenomenology of the strong interactions, the model of the proton, etc, and also by José Gabriel Vargas and Douglas Graham Torr. Vargas and Torr's paper in this volume must be read together with the paper, Clifford-valued Clifforms: geometric language for Dirac equation, in Ab lamowicz & Fauser (2000). A Dirac operator with the maximal coupling, in particular with the Pauli coupling, is referred to as 'the generalized Dirac operator' by Jürgen Tolksdorf. Tolksdorf 's paper also must be read together with another paper of his, Geometry of generalized Dirac operators and the Standard Model of particle physics, in Ryan & Sprößig (2000).

One should expect that all other papers dealing here with the minimal coupling γ • ∇ will be extended soon to maximal coupling, ∇ → ∇ + c ⊗ −, or to γ µ ∂ µ − iγ µ A µ + 1 2 γ µν B µν +. . . . In particular, it would be very interesting to see the analysis made by Eduardo Piña extended to maximal coupling, including in particular the Pauli term. The same expectation holds for analysis of the Hestenes equation made here by Bertfried Fauser. Another paper related to Fauser's analysis is De Leo et al. (1999).

One can also expect that the spectral problems considered by Micho Durdevich and by Robert Owczarek for the minimal coupling can be extended to the maximal coupling (this seems to be not so easy).

Trautman et al., in Dietrich et al. (1998), study the spectral problem for the modified Dirac operator. This modification involve an orientation (a chirality) γ 5 and it seems that it can be included as a particular case in the maximal coupling problem.

Kurt Just et al. in the context of the quantized Dirac field, prove that a nonquantized maximal coupling Bose field is a functional of the Dirac field. It would be interesting to try to reiterate the same analysis in the framework of noncommutative algebra without Minkowski space, but with an additional structure of the Clifford C-algebra of a hermitian form. Kurt Just and James Thevenot, in the context of the quantized fields, raise the problem of the Pauli terms, and their paper, Pauli term must be absent in the Dirac equation, is published in Ab lamowicz & Fauser (2000).

Marcos Rosenbaum gives a fairly comprehensive overview of Alain Connes' noncommutative theory. Even more interesting subject, the Hopf algebra of Feynman diagrams, is reviewed in another paper by Marcos Rosenbaum jointly with J. David Vergara, Dirac operator, Hopf algebra of renormalization and structure of space-time, in Ab lamowicz & Fauser (2000).

Eckehard W. Mielke, Leopold Halpern and Norma Susana Mankoč Borštnik study the Dirac equation in the context of elementary particles and gravity.

It would be interesting also to consider the Clifford algebra and the Dirac operator on the lifted algebra in the spirit of Yano & Ishihara (1973) and Kainz and Michor (1987). In particular, if T denotes the tangent lift, then the Clifford algebra C (V, η) can be lifted to C (T V, T η). One can expect the Dirac operator for a lifted metric T η to be relevant to gravity.

Then Minkowski norm of p is the same as Pythagorean norm of the image,

However it seems that Piazzese's 'quasi-classical dynamics', as presented in Ixtapa, has nothing to do with the above 'change of signature' map and is just a consequence of the identity for the Lorentz's relativistic factor γ,

Miralles et al. explore that the sum of two diagonal metrics with different signatures, say the Minkowski g M and Euclidean g E metrics, is a degenerate metric 1 2 (g M + g E ) and such degenerate metric can be constructed from a splitting by selecting some unit vector(s).

Nonassociative algebraic structures are considered in several papers: by Artibano Micali, Jerzy Kociński, Lev Sabinin, Larissa V. Sbitneva and by Alexander I. Nesterov. Contribution by Micali will be published in one of the next issues for editorial reasons, see also his contribution to the volume edited by Dietrich et al. (1998). Micali deals with an associative Clifford k-algebra C (V, η ∈ V * ⊗V * ), for not necessarily associative nor necessarily unital k-algebra (V, m) with a non trivial weight ω ∈ alg(V, k). A scalar product η depends on this given weight ω in rather complicated way.

Lev Sabinin has long time been reformulating a Riemannian differential geometry in terms of the smooth quasigroups and loops (= unital quasigroups), introduced by Ruth Moufang around 1935, instead of the Lie groups. A quasigroup is a nonassociative generalization of a group, and can be understand, for example, as a triple of binary operations related by identities. This structure can be treated in the framework of Birkhoff's equational universal algebra and it would be desirable to study different axiomatics in the same way as in the group theory by trying to determine the minimal set of relations etc., as in the program presented by Tarski (1968). It also would be interesting to study the extension theory of quasigroups, i.e. short exact sequences of quasigroups, in the spirit of the Eilenberg (1948) program, in a similar way to extension of groups. One can expect that extensions of quasigroups should lead to the general theory of representations of quasigroups and in particular to general theory of odules (extending the family of modules, bimodules, etc.). Sbitnieva demonstrates how naturally the R-odule arises in special relativity. Sabinin introduces for the smooth loops ( Lie loops) the vector fields, Lie bracket (Lie algebra?), and (affine) connection, but without mentioning the Leibniz condition. I was expecting to see the (analogy of) the Leibniz condition for a derivation. In the differential geometry of the Lie groups, a vector field is by definition a derivation, so one can ask how this definition can, or cannot, be adopted to the smooth loop case?

Andreas Bette, and Julian Lawrynowicz and Osamu Suzuki, deal with the twistors invented by Roger Penrose and also studied independently, among many other, by Jan Rzewuski in Wroc law since 1970s (Kocik and Rzewuski, 1996). Bette's review paper on twistor approach to the Dirac equation is published in Ab lamowicz and Fauser (2000) and this introductory paper should be read first. Another approach to the Dirac operator in the framework of the twistor bundle was presented by Gusiew and Keller (1997).

Lawrynowicz and Suzuki decided to submit two papers on the same subject, first in Ab lamowicz and Fauser (2000) with a continuation is published in the present issue. They generalize the Penrose program for some (not any) other signatures, introducing pseudotwistors and bitwistors. However, the reader of both papers may find difficult to understand this terminology (among other) because all motivations was omitted and the relation with Penrose's twistors is not explained. Fortunately the papers contain the extensive list of references.

The name spinor have been introduced byÉlie Cartan in 1913. However, the Pauli σ matrices and the Dirac γ matrices with all defining Clifford algebra relations (without of these names) where already published by I. Schur in 1911. We add to this issue the first English translation of this historical paper by I. Schur. In Chapter VI (paragraph 21) of this paper Schur deals with a finite Clifford group and with a spinor representation of the Clifford algebra in terms of tensor product of essentialy Pauli matrices [Schur (1911), formulas (50) and ff.].

CLIFFORD COGEBRA AND CLIFFORD CONVOLUTION

My lecture in a session dedicated to the memory of Gian-Carlo Rota, was devoted to, among other things, Clifford cogebra. This was the only lecture devoted to this concept, besides a related paper on general cogebra by Borowiec and Vázques Coutiño (this volume). I hope that the next Conference in 2002 has a session on Clifford Cogebra and Applications. Here I would like give a brief review of its motivation (Oziewicz, , 1998Cruz and Oziewicz, 2000;Fauser and Oziewicz, 2000). I believe that a Clifford cogebra must play also an important role in all applications similar to a Clifford algebra.

In what follows, k is an associative and unital N-algebra k ⊗ N k → k (a semiring), or a Z-algebra k ⊗ Z k → k (a ring), not necessarily commutative and ⊗ means ⊗ k . Further, we will need also coscalars, i.e. N-cogebra k → k ⊗ N k (a co-ring). A binary k-algebra is a k-bimodule V, an extension of k, with a multiplication m V as a k-bimodule map m V : V ⊗ V → V. This looks like a Feynman tree graph with one vertex m describing an anihilation. A binary k-cogebra is like a creation process with a comultiplication V as a k-bimodule map V : V → V ⊗ V. It is hard to believe that an algebra structure may be sufficient to explain all problems in fundamental science, in applications, in engineering, in elementary particle physics, in logic, etc. The algebra and cogebra jointly give rise to convolution algebra and thus the name convolution. It was Heinz Hopf who discovered in 1941 that a convolution, m and , intertwine in algebraic topology, and thus the name Hopf gebra for an associative unital and antipodal convolution. The algebra and cogebra are like brother and sister, and it is unfortunate that present-day elementary textbooks on linear algebra do not mention this sister.

The reader not familar with semirings and bimodules can exchange a semiring for a field R and a k-bimodule for a vector R-space with almost no loss. The only difference is that in a case of a noncommutative k there are two different types of 'covectors', left and right covectors, because the right dual k-bimodule V * does not need to be the same as the left dual k-bimodule * V. In what follows,

If (V, V ) is a model of a k-cogebra , then a dual k-bimodule V * with * V is a k-algebra (and also a left dual k-bimodule * V with * V is an another k-algebra). In finite dimensions and in case of the graded dual in general, we have also thre converse statement. In what follows, V be a finite dimensional k-bimodule and a tensor η ∈ V * ⊗ V * be a k-valued 'arbitrary bilinear form' on V, where k needs not to be commutative. We do not need to assume that η T = η, where η T denotes transpose of η. Durdevich's noncommutative algebra Σ in this issue is a particular N-algebra or Z-algebra or R-algebra or C-algebra and can be understand as our k in what follows. Rota and Stein (1994) introduced a deformation of a convolution (and of a Hopf gebra), called Cliffordization, a graphical 'sausage', which can be applied to any convolution neither antipodal nor even unital. I showed in my lecture that Clifford cogebra alone can be obtained from an exterior Hopf gebra by anologous co-Clifordization, dual to Rota and Stein's Clifordization,

Here an exterior cogebra :

≡ ∧ * , and can be calculated explicitely as follows, for v, w ∈ V :

A tensor η ∈ V * ⊗ V * is a scalar product on V (coscalar on V * ), and ξ ∈ V ⊗ V is a coscalar product on V. These tensors lift to algebra maps η ∧ ∈ alg(V ∧ , V * ∧ ) & ξ ∧ ∈ alg(V * ∧ , V ∧ ). Let id|V * ∧n ∈ V * ∧n ⊗ V ∧n and id = id V * ∧ or appropriately id = id V ∧ Then for a basis {e i ∈ V } and a dual basis {ε i ∈ V * }, ε i e j = δ i j , we have the braid dependent expansions

An exponential of a tensor ξ, i.e. a lifted tensor ξ ∧ e ξ , in fact is braid dependent . A vertex ξ ∧ in a graphical sausage must be understand as a process v ⊗ w → v ⊗ ξ ∧ ⊗ w. The Graßmann R-cogebra (V ∧ , ) possess one group-like element only, namely 1 ∈ R, and therefore is a pointed irreducible cogebra. A Clifford cogebra possess a discrete number of group-like elements and this number is correlated with the signature.

In the same spirit one can treat Weyl ization or Heisenbergization of the symmetric exterior Weyl Hopf gebra. The quantization as the Moyal deformation of a symmetric multiplication (1949), in terms of the Poisson bivector field involves differential structure and most probably cannot be presented in as compact form as the above Rota & Stein's Cliffordization.

If C (V, η) = (V ∧ , ∧ η ) with ∧ η=0 ≡ ∧, is a Clifford k-algebra (or R-algebra) as the η-Cliffordization of an exterior Hopf k-gebra (or equivalently as the Chevalley deformation of an exterior k-algebra), then C (V * , η) = (V * ∧ , ∧ η * ) is a Clifford kcogebra (R-cogebra). One can define also a universal Clifford k-cogebra. In case a tensor η is invertible, we are dealing with a pair of mutually dual Clifford algebras of multivectors C (V, η) & of multicovectors C (V * , η −1 ) as was explained elsewhere (Oziewicz, , 1998. By duality, this gives that the k-bimodules (or the R-spaces) V ∧ & V * ∧ carry both structures: Clifford algebra & Clifford cogebra, and thus the name Clifford convolution.

Theorem 5.1 . The following unital and associative Clifford convolutions are antipode-less,

The above Theorem has been sharpened by Fauser and Oziewicz (2000): the Clifford convolution C (η, ξ) is antipode-less iff det(id − ξ • η) = 0.

No attempt has been made yet to find axioms for the Clifford convolution C (η, η −1 ). We believe that the set of such axioms may include the following