Clifford Algebras

In an open subset Ω of the complex plane ℂ the Morera theorem gives a simple looking necessary and sufficient condition for a continuous function f to be holomorphic in Ω. Namely the vanishing of all the integrals ∫_γf(z) dz, where γ is an arbitrary Jordan curve in Ω whose interior also lies in Ω. The Morera problem consists in finding relatively small families Γ of Jordan curves such that the vanishing of the corresponding integrals still ensures that the conclusion of Morera’s theorem still holds. In this lecture we discuss a number of recent results obtained in the case one considers functions defined in two fairly different settings, the Clifford algebras and the Heisenberg group.

There is a deep connection between the theory of several complex variables and complex Clifford analysis. We will use a Borel—Pompeiu formula in ℂn and the representation of holomorphic functions obtained in the context of Clifford analysis to study the inverse scattering problem for an n-dimensional Schròdinger-type equation. Equations are found for reconstructing the potential from scattering data purely by quadratures. The solution also helps elucidate the problem of characterizing admissible scattering data. Especially we do not need a “miraculous condition”.

The main goal of the paper is to apply the theory of discrete analytic functions to the solution of Dirichlet problems for the Stokes and Navier—Stokes equations, respectively. The Cauchy—Riemann operator will be approximated by certain finite difference operators. Approximations of the classical T-operator as well as for the Bergman projections are constructed in such a way that the algebraic properties of the operators from complex function theory remain valid. This is used to approximate the solutions to the boundary value problems by adapted finite difference schemes.

For a system A = (A _i ,…, A _n ) of linear operators whose real linear combinations have spectra contained in a fixed sector in ℂ and satisfy resolvent bounds there, functions f(A) of the system A of operators can be formed for monogenic functions f having decay at zero and infinity in a corresponding sector in ℝn+1. In the case that the operators A _i ,…, A _n commute with each other and satisfy square function estimates in Hilbert space, the correspondence between bounded monogenic functions defined in a sector in ℝn+1 and bounded holomorphic functions defined in a sector in ℂn is used to define the functional calculus f→f(A) for bounded holomorphic functions f in a sector of ℂn. The treatment includes the Dirac operator on a Lipschitz surface in ℝn+1.

In this paper we deal with Clifford-valued generalizations of several families of classical complex-analytic Eisenstein series and Poincaré series for discrete subgroups of Vahlen’s group in the framework of Clifford analysis.

We develop a function theory associated with non-elliptic, variable co-efficient operators of Dirac type on Lipschitz domains. Boundary behavior, global regularity, integral representation formulas, are studied by means of tools originating in PDE and harmonic analysis.

This paper is concerned with the classical Paley—Wiener theorems in one and several complex variables, the generalization to Euclidean spaces in the Clifford analysis setting and their proofs. We prove a new Shannon sampling theorem in the Clifford analysis setting.

We study weighted Bergman projections in the monogenic Bergman spaces of the real unit ball \mathbb{B} in ℝ _n . We extend results of Forelli—Rudin, Coifman—Rochberg, and Djrbashian to Clifford analysis. The main result is as follows: Let P_α be the orthogonal projection from the Hilbert space L2( \mathbb{B} , Cl_0,n , dV_α) onto the subspace of monogenic functions A2( \mathbb{B} , Cl_0,n , dV_α. If p(α + 1) > β + 1 with 1 ≤ p < ∞ and α,β > - 1, then the operator P_α : Lp ( \mathbb{B} ,Cl_{0, n} , dV_β) → Ap( \mathbb{B} ,Cl_0,n , dV_β is bounded.

We study Galpern—Sobolev equations with the help of a quaternionic operator calculus. Previous work is extended to the case of a variable dispersive term. We approximate the time derivative by forward finite differences. Solving the resulting stationary problems by means of a quaternionic calculus, we obtain representation formulae.

We define a Nahm transform for instantons over hyperkàhler ALE 4-manifolds, and explore some of its basic properties

The aim of the present paper is to review general results and some constructions of the hyper-Kähler geometry with torsion. This is the geometry of a special type of hyper-Hermitian metrics on a hypercomplex manifold related to some questions in theoretical physics. In particular, we show that there is a local existence of such metrics based on an HKT-potential theory, a moment map and reduction theory, as well as a global non-existence property.

We show that the principal symbols of first order geometric differential operators on Riemannian manifolds are controlled by the enveloping algebra and higher Casimir elements of so(n). Then we give all the Bochner identities for the operators explicitly.

Let M=S1 × Sn-1 with metric Lorentzian or Riemannian and non-trivial spin structure on S1, Riemannian metric and standard spin structure on Sn-1, and n even. We give explicit formulas for the eigenvalues of Dirac and Rarita—Schwinger operators on M.

On a 6-dimensional, conformal, oriented and compact manifold M without boundary we compute a whole family of differential forms Ω₆(f, h) of order 6 with f, h ξ C∞(M). Each of these forms will be symmetric on f and h, conformally invariant, and such that ∫ _Mf₀Ω₆(f₁, f₂) defines a Hochschild 2-cocycle over the algebra C∞ (M). In the particular 6-dimensional conformally flat case, we compute a unique form satisfying Wres(f₀[F,f][F, h])=∫_Mf₀Ω₆(f, h) for the Fredholm module (H, F) associated by A. Connes [6] to the manifold M, and the Wodzicki residue Wres.

As a mathematical theory, noncommutative geometry (NCG) is by now well established. From the beginning, its progress has been crucially influenced by quantum physics: we briefly review this development in recent years.The standard model of fundamental interactions, with its central role for the Dirac operator, led to several formulations culminating in the concept of a real spectral triple. String theory then came into contact with NCG, leading to an emphasis on Moyal-like algebras and formulations of quantum field theory on noncommutative spaces. Hopf algebras have yielded an unexpected link between the noncommutative geometry of foliations and perturbative quantum field theory.The quest for a suitable foundation of quantum gravity continues to promote fruitful ideas, among them the spectral action principle and the search for a better understanding of “noncommutative spaces”.

We provide a brief account of Capelli’s method of virtual variables and of its relations with representations of general linear Lie superalgebras. More specifically, we study letterplace superalgebras regarded as bimodules under the action of superpolarization operators and exhibit complete decomposition theorems for these bimodules as well as for the operator algebras acting on them.

We compare various algebras associated with a representation of a semisimple Lie algebra. Their general construction is akin to that of the Clifford algebra (arising from the defining representation of the orthogonal algebra). These algebras arise from the symmetric product, skew product, or Cartan product of representations together with some additional data.

In the traditional approaches to Clifford algebras, the Clifford product is evaluated by recursive application of the product of a one-vector (span of the generators) on homogeneous, i.e., sums of decomposable (Grassmann), multi-vectors and later extended by bilinearity. The Hestenesian “dot” product, extending the one-vector scalar product, is even worse having exceptions for scalars and the need for applying grade operators at various times. Moreover, the multivector grade is not a generic Clifford algebra concept. The situation becomes even worse in geometric applications if a meet, join or contractions have to be calculated.Starting from a naturally graded Grassmann—Hopf gebra, we derive general formulæ for the products: meet and join, comeet and cojoin, left/right contraction, left/right cocontraction, Clifford and co-Clifford products. All these product formulæ are valid for any grade and any inhomogeneous multivector factors in Clifford algebras of any bilinear form, including non-symmetric and degenerated forms. We derive the three well-known Chevalley formulæ as a specialization of our approach and will display co-Chevalley formulæ. The Rota—Stein cliffordization is shown to be the generalization of Chevalley deformation. Our product formulæ are based on invariant theory and are not tied to representations/matrices and are highly computationally effective. The method is applicable to symplectic Clifford algebras too.

This article is an expanded version of my plenary lecture for the conference. It was the aim of the lecture to introduce the participants of the conference—their diverse realms of expertise ranged from theoretical physics, to computer science, to pure mathematics—to the algebraic matters of the title above. The basic texts listed in the references—note that this listing is by no means complete—serve as illustration of the rich and persistent interest in these topics and provide a reader with an opportunity to explore them in detail.

When Clifford algebras are studied in relation with exterior algebras, it is easy to undertake a parallel study of “symplectic Clifford algebras”, also called “Weyl algebras”: it suffices to replace exterior algebras with symmetric algebras, to remove all “twisting signs” from all places where they are present, and to intrude them into all places where they are absent. This enables one to imagine a symplectic counterpart of a theorem of Lipschitz about orthogonal transformations; unfortunately this counterpart needs an “enlargement” of the Weyl algebra, and leads to infinite sums and convergence problems. Some specific problems of the symplectic case, that result from this enlargement and that cannot be treated by purely algebraic means, are commented upon.

Some properties of Clifford algebras are actually common properties of all graded central simple algebras A provided with an involution ρi; with ρ is associated a “complex divided trace” (a complex number r such that r8 = 1), and thus all such involutions are classified by a cyclic group of order 8. Complex divided traces are also involved in the Brauer—Wall group of the field ℝ, and they bring efficiency and enlightenment in the study of bilinear forms on graded A-modules.

Hagmark and Lounesto’s binary labeling of the generators of a Clifford algebra can be extended by ordering the basis elements in either ascending order, denoted by a post-scripted binary index, or descending order, denoted by a pre-scripted binary index. Reversion is then represented by swapping prescripts and postscripts, grade involution by changing the sign of the index, and conjugation by both swapping and changing the sign. Bit inversion of the binary index is a duality operation that does not suffer the handedness problems of the Clifford dual or of the Hodge dual. Generators whose binary indices are bit inverses of each other commute (anti-commute) if the product of their grades is even (odd). If the number of generators is even (odd), the commutators (anti-commutators) of basis vectors labeled with binary indices and of covectors labeled by bit inversion of binary indices yield generalizations of the Heisenberg commutation relations.

Recoding base elements in a spacetime algebra is an act of cognition. But at the same time this act refers to the process of nature. That is, the internal interactions with their standard symmetries reconstruct the orientation of spacetime. This can best be represented in the Clifford algebra Cl_{3, 1} of the Minkowski spacetime. Recoding is carried out by the involutive automorphism of transposition. The set of transpositions of erzeugende Einheiten (primitive idempotents), as Hermann Weyl called them, generates a finite group: the reorientation group of the Clifford algebra. Invariance of physics laws with respect to recoding is not a mere matter of computing, but one of physics. One is able to derive multiplets of strong interacting matter from the recoding invariance of Cl_{3, 1} alone. So the SU(3) flavor symmetry essentially turns out to be a spacetime group. The original quark multiplet independently found by Gell-Mann and Zweig is reconstructed from Clifford algebraic eigenvalue equations of isospin, hypercharge, charge, baryon number and flavors treated as geometric operators. Proofs are given by constructing six possible commutative color spinor spaces ℂh_χ, or color tetrads, in the noncommutative geometry of the Clifford algebra Cl_{3, 1} Calculations are carried out with CLIFFORD, Maple V package for Clifford algebra computations. Color spinor spaces are isomorphic with the quaternary ring 4ℝ = ℝ ⊕ ℝ ⊕ ℝ ⊕ ℝ. Thus, the differential (Dirac) operator takes a very handsome form and equations of motion can be handled easily. Surprisingly, elements of Cl_{3, 1} representing generators of SU(3) bring forth (1) the well-known grade-preserving transformations of the Lorentz group together with (2) the heterodimensional Lorentz transformations, as Jose Vargas denoted them: Lorentz transformations of inhomogeneous differential forms. That is, trigonal tetrahedral rotations do not preserve the grade of a multivector but instead, they permute the base elements of the color tetrad having grades 0, 1, 2 and 3. In the present model the elements of each color space are exploited to reconstruct the flavor SU(3) such that each single commutative space contains three flavors and one color. Clearly, the six color spaces do not commute, and color rotations act in the noncommutative geometry. To give you a picture: Euclidean space with its reorientation group, i.e., the again embed the root spaces of the 6 flavor su(3). Color su(3) is exact because it does not involve relativistic effects. Flavor does and is therefore inexact. It seems that Cl_{3, 1} comprises enough structure for both the color- and the flavor-SU(3). In this very first approach both symmetries are reconstructed exact, whereas in reality the flavor SU(3) is only approximate. Here, the only way to make a difference between a color- and a flavor-rotation may be to distinguish between commutative and noncommutative geometry.We conclude that an extended heterodimensional Lorentz invariance of the Minkowski spacetime and the SU(3) of strongly interacting quarks result from each other. This involves a nongrade preserving-degree of freedom of motion taking spatial lines to spacetime areas and areas to spacetime-volumes and back to lines and areas. Although the form of the Dirac equation is preserved, a thorough study of the involved nonlinearities of equations of motion is still outstanding.

Classical relativistic physics in Clifford algebra has a spinorial formulation that is closely related to standard quantum formalism. The algebraic use of spinors and projectors, together with the bilinear relations of spinors to observed currents, gives quantum-mechanical form to many classical results, and the clear geometric content of the algebra makes it an illuminating probe of the quantum/classical interface. This paper extends past efforts to close the conceptual gap between quantum and classical phenomena while highlighting their essential differences. The paravector representation of spacetime in Cl₃ is used in particular to provide insight into spin-1/2 systems and their measurement.

We review the deformation of Podleś 2-sphere and of the 4-sphere introduced in [2], which display a quite similar behavior. Special attention is devoted to the development of the local picture of charts.

The equations defining pure spinors are interpreted as equations of motion formulated on the lightcone of momentum space P = ℝ1, 9. Most of the equations for fermion multiplets, usually adopted by particle physics, are then naturally obtained and their properties, such as internal symmetries, charges, and families, appear to be due to the correlation of the associated Clifford algebras with the three complex division algebras: complex numbers, quaternions and octonions. Pure spinors could be relevant not only because the underlying momentum space is compact, but also because they may throw some light on several problematic aspects of particle physics.

We first describe a class of spinor-curvature identities (SCI) which have gravitational applications. Then we sketch the topic of gravitational energy-momentum, its connection with Hamiltonian boundary terms and the issues of positivity and (quasi)localization. Using certain SCIs, several spinor expressions for the Hamiltonian have been constructed. One SCI leads to the celebrated Witten positive energy proof and the Dougan—Mason quasilocalization. We found two other SCIs which give alternate positive energy proofs and quasilocalizations. In each case the spinor field has a different role. These neat expressions for gravitational energy-momentum have much appeal. However it seems that such spinor formulations just have no room for angular momentum, which leads us to doubt that spinor formulations can really correctly capture the elusive gravitational energy-momentum.

A chiral relativistic wave equation is proposed for neutrinos. This wave equation allows a mass term. We study this equation in the Clifford algebra of space, and in the frame of the Clifford spacetime algebra, second-order equation, plane waves, Lagrangian formalism, conservative current. Next we extend the wave equation to the complete spacetime algebra and we obtain a chiral wave equation with a mass term and a charge term. We study the relativistic invariance, the Lagrangian formalism, the second-order equation, we solve the equation in the case of the hydrogen atom. We obtain the right number of energy levels and the right energy levels. We study the enlargement of the gauge invariance, with a real matrix formalism. The electric gauge group may be extended to SO(8), a subgroup of SO(16).

In previous work, the standard 4-dimensional Dirac equation was rewritten in terms of quaternionic 2-component spinors, leading to a formalism which unifies the treatment of massive and massless particles, and which describes the correct particle spectrum to be a generation of leptons, with the correct number of spin/helicity states. Furthermore, precisely three such generations naturally combine into an octonionic description of the 10-dimensional massless Dirac equation. We extend this formalism to 3-component octonionic “spinors”, which may lead to a description of fundamental particles in terms of the exceptional Jordan algebra, consisting of 3 × 3 octonionic Hermitian matrices.

We review the applications of geometric algebra in electromagnetism, gravitation and multiparticle quantum systems. We discuss a gauge theory formulation of gravity and its implementation in geometric algebra, and apply this to the fermion bound state problem in a black hole background. We show that a discrete energy spectrum arises in an analogous way to the hydrogen atom. A geometric algebra approach to multiparticle quantum systems is given in terms of the multiparticle spacetime algebra. This is applied to quantum information processing, multiparticle wave equations and to conformal geometry. The application to conformal geometry highlight some surprising links between relativistic quantum theory, twistor theory and de Sitter spaces.

We survey noncommutative spacetimes with coordinates being enveloping algebras of Lie algebras. We also explain how to do differential geometry on noncommutative spaces that are obtained from commutative ones via a Moyal-product type cocycle twist, such as the noncommutative torus, θ-spaces and Clifford algebras. The latter are noncommutative deformations of the finite lattice (ℤ₂)n and we compute their noncommutative de Rham cohomology and moduli of solutions of Maxwell’s equations. We exactly quantize noncommutative U(1)-Yang-Mills theory on ℤ₂ × ℤ₂ in a path integral approach.

Spectral triples constitute basic structures of Connes’ noncommutative geometry. In order to unify noncommutative geometry with quantum group theory, it is necessary to provide a proper description of spectral triples on quantum groups and, in a wider context, on their homogeneous spaces. Thus, a way to define Dirac-like operators on such quantum spaces has to be found. This paper is a brief summary on some problems we are facing in searching for such a way. We suggest that some modifications of either noncommutative geometry or quantum group theory, or both, are inevitable.Differential calculi and their relationship to the Dirac operators are the key concepts to understand for defining the spectral triples in question. Since there is no canonical way to construct such calculi on the quantum groups, and since there are two basic kinds, left-(right-) covariant and bicovariant, there are a number of issues to be resolved. We discuss these difficulties as well as difficulties implicit in the definition of the Dirac operator, which are present already at the classical manifolds level.Definition of quantum homogeneous spaces constitutes another problem. This is also briefly reviewed in the paper. Then a basic example associated with the quantum 2-sphere of Podleś, as a homogeneous space of the quantum SU(2) group of Woronowicz, is discussed. A naturally defined Dirac operator on such a 2-sphere does not satisfy one of Connes’ axioms.

A general structure combining Grassmann, Clifford and tensor products is presented. The r-fold multivectors provide the basis for the natural extension of Grassmann and Clifford algebras when several geometric entities are multilinearly related. Any tensor can be organized and understood as an r-fold multivector when its antisymmetries are taken into account. The r-fold Clifford algebra is contrasted with the multiparticle geometric algebra. The application of r-fold Clifford algebra to the study of superenergy tensors in physics is shown to provide their simplest definition. In addition, it constitutes a most efficient tool for obtaining and proving their essential properties, such as dominant positivity and conditions for their conservation.

A recent geometric approach to the standard model in terms of the Clifford algebra Cl₇ is summarized. The complete gauge group of the standard model is shown to arise naturally and uniquely by considering all rotations in seven-dimensional space that (1) conserve the spacetime components of the particle and antiparticle currents and (2) do not couple the right-chiral neutrino. The spinor mediates a physical coupling of Poincaré and isotopic symmetries within the restrictions of the Coleman-Mandula theorem. The four extra spacelike dimensions in the model form a basis for the Higgs isodoublet field. The charge assignments of both the fundamental fermions and the Higgs boson are produced exactly.

We present the design of a Clifford algebra co-processor and its implementation on a Field Programmable Gate Array (FPGA). To the best of our knowledge this is the first such design developed. The design is scalable in both the Clifford algebra dimension and the bit width of the numerical factors. Both aspects are only limited by the hardware resources. Furthermore, the signature of the underlying vector space can be changed without reconfiguring the FPGA. High calculation speeds are achieved through a pipeline architecture.

“Image processing” is a purely syntactical discipline that transforms images into other images. As such it is distinct from such fields as “image recognition,” “feature detection,” and so forth, which are semantically oriented. If semantics plays a role, the “meaning” is relative to a user’s world model. In the case of purely syntactical operations there is some hope to put the discipline on a foundation of first principles. Image processing in its present state is more of a bag of hat tricks though, and many of its methods are easily shown to be inconsistent. This contribution is an attempt to put image processing on a solid geometrical basis. In order to do so the (currently only vaguely defined though much used) concept of “image space” is formalized. Departing from a few commonly acknowledged properties one arrives at a three-dimensional Cayley-Klein space with a single isotropic dimension that can be understood as a limit of either Euclidean or Minkowskian three-space. In this geometry, planes containing an isotropic line are isomorphic with the dual number plane, whereas any other plane is isomorphic with the conventional complex number plane. The elements of the group of similarities are readily identified with image transformations that “don’t change the image,” thus an “image” can be defined as an invariant under similarities. The differential invariants are a substitute for the “feature operators” of current image processing. Their formal structure is rather simpler than the corresponding Euclidean differential invariants, but the differential geometry is just as rich as that of conventional Euclidean space.

This work concerns the 2D-3D pose estimation problem of cycloidal curves. Pose estimation means to estimate the relative position and orientation of a 3D object to a reference camera system. The 3D object features are in this work cycloidal curves, as extensions to classical 3D point or 3D line concepts. This means, we assume knowledge of a 3D cycloidal curve and observe it in an image of a calibrated camera. The aim is to estimate the rotation R and translation t to get a best fit of the transformed 3D object model to the observed 2D image data. Furthermore, other concepts such as 3D cycloidal surfaces and the numerical problems of estimating the pose parameters are discussed.