The Teodorescu Operator in Clifford Analysis

Euclidean Clifford analysis is a higher dimensional function theory centred around monogenic functions,i.e.,null solutions to a first order vector valued rotation invariant differential operator (θ) ca...     

Clifford Algebra-Valued Wavelet Transform on Multivector Fields

This paper presents a construction of the n = 2 (mod 4) Clifford algebra Cl _n,0-valued admissible wavelet transform using the admissible similitude group SIM(n), a subgroup of the affine group of \mathbbRⁿ{\mathbb{R}^{n}} . We express the admissibility condition in terms of the Cl _n,0 Clifford Fourier transform (CFT). We show that its fundamental properties such as inner product, norm relation, and inversion formula can be established whenever the Clifford admissible wavelet satisfies a particular admissibility condition. As an application we derive a Heisenberg type uncertainty principle for the Clifford algebra Cl _n,0-valued admissible wavelet transform. Finally, we provide some basic examples of these extended wavelets such as Clifford Morlet wavelets and Clifford Hermite wavelets.     

Universal Similarity Factorization Equalities over Generalized Clifford Algebras

For any element a in a generalized 2^n-dimensional Clifford algebra Lln （F） over an arbitrary field F of characteristic not equal to two, it is shown that there exits a universal invertible matrix Pn over Lln（F） such that Pn^-1DnPn= φ（α）∈F^2n×2n, where φ（a） is a matrix representation of α over and Dα is a diagonal matrix consisting of a or its conjugate.     

The Invariants of the Clifford Groups

The automorphism group of the Barnes-Wall lattice L _m in dimension 2 ^m (m ; 3) is a subgroup of index 2 in a certain Clifford group of structure 2 ₊ ^1+2m . O ⁺(2m,2). This group and its complex analogue of structure .Sp(2m, 2) have arisen in recent years in connection with the construction of orthogonal spreads, Kerdock sets, packings in Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs. In this paper we give a simpler proof of Runge@apos;s 1996 result that the space of invariants for of degree 2k is spanned by the complete weight enumerators of the codes , where C ranges over all binary self-dual codes of length 2k; these are a basis if m k - 1. We also give new constructions for L _m and : let M be the -lattice with Gram matrix . Then L _m is the rational part of M ^m, and = Aut(M^m). Also, if C is a binary self-dual code not generated by vectors of weight 2, then is precisely the automorphism group of the complete weight enumerator of . There are analogues of all these results for the complex group , with doubly-even self-dual code instead of self-dual code.     

Between quantum Virasoro algebra $$\mathcal{L}_c $$ and generalized Clifford algebras

In this paper we construct the quantum Virasoro algebra generators in terms of operators of the generalized Clifford algebras C_n^k. Precisely, we show that can be embedded into generalized Clifford algebras. Junior Associate at The Abdus Salam ICTP, Trieste, Italy.     

Partial Primitives for Clifford Algebra-Valued Functions

In this paper we generalize the concept of primitivation of monogenic functions taking values in a Clifford algebra, which is on its own a generalization to higher dimension of the primitivation problem for holomorphic functions in the complex plane. This problem can be stated as follows: given a monogenic function on , i.e. a solution for the generalized Cauchy-Riemann operator D on , construct a monogenic function such that . In view of the fact that, for monogenic functions g, this can be written as g = f, a straightforward generalization consists in replacing the scalar generator of translations in the x ₀-direction by a generator of another transformation group. In this paper we consider translations in more dimensions.     

Hyperbolic Function Theory in the Clifford Algebra {\mathcal {C}}\ell_{n+1, 0}

The aim of this paper is to give the basic principles of hyperbolic function theory on the Clifford algebra . The structure of the theory is quite similar to the case of Clifford algebras with negative generators, but the proofs are not obvious. The (real) Clifford algebra is generated by unit vectors with positive squares e²_i = + 1. The hyperbolic Dirac operator is of the form where Q₀f is represented by the composition . If is a solution of H_kf = 0, then f is called k-hypergenic in Ω, where is an open set. We introduce some basic results of hyperbolic function theory and give some representation theorems on . Received: October, 2007. Accepted: February, 2008.     

Hermitean Clifford-Hermite Polynomials

Orthogonal Clifford analysis in flat m–dimensional Euclidean space focusses on monogenic functions, i.e. null solutions of the rotation invariant vector valued Dirac operator , where ( ) forms an orthogonal basis for the quadratic space underlying the construction of the Clifford algebra . When allowing for complex constants and taking the dimension to be even: m = 2n, the same set of generators produces the complex Clifford algebra , which we equip with a Hermitean Clifford conjugation and a Hermitean inner product. Hermitean Clifford analysis then focusses on the simultaneous null solutions of two mutually conjugate Hermitean Dirac operators, naturally arising in the present context and being invariant under the action of a realization of the unitary group U (n). In this so–called Hermitean setting Clifford–Hermite polynomials are constructed, starting from a Rodrigues formula involving both Dirac operators mentioned. Due to the specific features of the Hermitean setting, four different types of polynomials are obtained, two types of even degree and two types of odd degree. We investigate their properties: recurrence relations, structure, explicit form and orthogonality w.r.t. a deliberately chosen weight function. They also give rise to the definition of the Hermitean Clifford–Hermite functions, and may be used to develop a Hermitean continuous wavelet transform, see [4].     

Hartogs Extension Theorem for Functions with Values in Complex Clifford Algebras

A regular extension phenomenon of functions defined on Euclidean space with values in a Clifford algebra was studied by Le Hung Son in the 90’s using methods of Clifford analysis, a function theory which, is centred around the notion of a monogenic function, i.e. a null solution of the firstorder, vector-valued Dirac operator in . The isotonic Clifford analysis is a refinement of the latter, which arises for even dimension. As such it also may be regarded as an elegant generalization to complex Clifford algebra-valued functions of both holomorphic functions of several complex variables and two-sided biregular function theories. The aim of this article is to present a Hartogs theorem on isotonic extendability of functions on a suitable domain of . As an application, the extension problem for holomorphic functions and so for the two-sided biregular ones is discussed.      

Normal generation and Clifford index on algebraic curves

For a smooth curve C it is known that a very ample line bundle on C is normally generated if Cliff() < Cliff(C) and there exist extremal line bundles (:non-normally generated very ample line bundle with Cliff() = Cliff(C)) with . However it has been unknown whether there exists an extremal line bundle with . In this paper, we prove that for any positive integers (g, c) with g = 2c + 5 and (mod 2) there exists a smooth curve of genus g and Clifford index c carrying an extremal line bundle with . In fact, a smooth quadric hypersurface section C of a general projective K3 surface always has an extremal line bundle with . More generally, if C has a line bundle computing the Clifford index c of C with , then C has such an extremal line bundle . For all authors, this work was supported by Korea Research Foundation Grant funded by Korea Government (MOEHRD, Basic Reasearch Promotion Fund)(KRF-2005-070-C00005).     

A characterization of Clifford tori with constant scalar curvature one by the first stability eigenvalue

Let M be a compact hypersurface with constant scalar curvature one immersed into the unit Euclidean sphere . As is well-known, such hypersurfaces can be characterized variationally as critical points of the integral _M Hdv. In this paper we derive a sharp upper bound for the first eigenvalue of the corresponding Jacobi operator in terms of the mean curvature of the hypersurface. Moreover, we prove that this bound is achieved only for the Clifford tori in with scalar curvature one.—Dedicated to the memory of Prof. José F. Escobar, Chepe     

An Analogue of Beurling–Hörmander’s Theorem Associated with Partial Differential Operators

In this paper for a positive real number α we consider two partial differential operators D and D_α on the half–plane We define a generalized Fourier transform associated with the operators D and D_α. We establish an analogue of Beurling–H?rmander’s Theorem for this transform and we give some applications of this theorem.     

New construction of causal tensors

We study the problem of constructing tensors satisfying the dominant property, a generalization of the dominant energy condition T_ab u^a v^b ≥ 0 for all future directed causal vectors u, v. The construction is done on the paravector subspace of the r-fold Euclidean Clifford algebra and is a generalization of the representation of superenergy tensors with complex 2-spinors. Especially, as with 2-spinors, we are able to construct causal tensors of arbitrary rank, contrary to earlier constructions using tensors or the r-fold Lorentzian Clifford algebra that only produce causal tensors of even rank. An advantage of the construction in is that several algebraic properties become trivial due to the Euclidean norm on it.     

Adjoint Clifford Rings

Let > be a ring and define , which yields a monoid , called the {\it adjoint semigroup\/} or {\it circle semigroup\/} of . This paper continues the authors" investigations into the relationship between a ring and its adjoint semigroup, focusing here on the case where is a Clifford semigroup. Such rings are called {\it adjoint Clifford rings.\/} Structure theory for such rings is developed and examples are given to illustrate and delimit the theory.     

Clifford tetrads,null zig zags,and quantum gravity

Quantum gravity has been so elusive because we have tried to approach it by two paths which can never meet: standard quantum field theory and general relativity. These contradict each other, not only in superdense regimes, but also in the vacuum, where the divergent zero-point energy would roll up space to a point. The solution is to build in a regular, but topologically nontrivial distribution of vacuum spinor fields right from the start. This opens up a straight road to quantum gravity, which we map out here. The gateway is covariance under the complexified Clifford algebra of our space-time manifold and its spinor representations, which Sachs dubbed the Einstein group, E. The 16 generators of E transformations obey both the Lie algebra of Spin^c -4, and the Clifford (SUSY) algebra of We derive Einstein’s field equations from the simplest E-invariant Lagrangian density, contains effective electroweak and gravitostrong field actions, as well as Dirac actions for the matter spinors. On microscales the massive Dirac propagator resolves into a sum over null zig-zags. On macroscales, we see the energy-momentum current, *T, and the resulting Einstein curvature, G. For massive particles, *T flows in the “cosmic time” direction—centri-fugally in an expanding universe. Neighboring centrifugal currents of *T present opposite radiotemporal vorticities G_or to the boundaries of each others’ worldtubes, so they advect, i.e. attract, as we show here by integrating by parts in the spinfluid regime. This boundary integral not only explains why stress-energy is the source for gravitational curvature, but also gives a value for the gravitational constant, κ (T), that couples them. κ turns out to depend on the dilation factor T = y⁰, which enters kinematically as “imaginary time”: the logradius of our expanding Friedmann 3-brane. On the microscopic scale, quantum gravity appears as the statistical mechanics of the null zig-zag rays of spinor fields in imaginary time T. Our unified field/particle action also contains new couplings of gravitomagnetic fields to strong fields and weak potentials. These predict new physical phenomena: Axial jets of nuclear decay products emitted with left helicity along the axis of a massive, spinning body. This paper gives the derivations of the results I reported at the PIMS conference entitled “Brane World and Supersymmetry” in July, 2002 at Vancouver, B.C. It also contains new results on spin-gravity coupling, on how a topologically-nontrivial distribution of vacuum spinors removes singularities and divergences, and how the amplitude of the vacuum spinors determines the gravitational constant and the rate of cosmic expansion     

Fundaments of Hermitean Clifford Analysis Part I: Complex Structure

Hermitean Clifford analysis focusses on h–monogenic functions taking values in a complex Clifford algebra or in a complex spinor space. Monogenicity is expressed here by means of two complex mutually adjoint Dirac operators, which are invariant under the action of a Clifford realisation of the unitary group. In this contribution we present a deeper insight in the transition from the orthogonal setting to the Hermitean one. Starting from the orthogonal Clifford setting, by simply introducing a so-called complex structure J ∈ SO(2n; ), the fundamental elements of the Hermitean setting arise in a quite natural way. Indeed, the corresponding projection operators 1/2 (1 ± iJ) project the initial basis (e_α, α = 1, . . . , 2n) onto the Witt basis and moreover give rise to a direct sum decomposition of into two components, where the SO(2n; )-elements leaving those two subspaces invariant, commute with the complex structure J. They generate a subgroup which is doubly covered by a subgroup of Spin(2n; ), denoted Spin _J (2n; ), being isomorphic with the unitary group U(n; ). Finally the two Hermitean Dirac operators are shown to originate as generalized gradients when projecting the gradient on the invariant subspaces mentioned, which actually implies their invariance under the action of Spin _J (2n; ). The eventual goal is to extend the complex structure J to the whole Clifford algebra , in order to conceptually unravel the true meaning of Hermitean monogenicity and its connections to orthogonal monogenicity. During the final redaction of this paper, we received the sad news that our friend, colleague and co-author Jarolím Bureš died on October 1, 2006 Submitted: October 16, 2006. Accepted: December 29, 2006.     

On the Integral Representation of Spherical k-Regular Functions in Clifford Analysis

We study the null solutions of iterated applications of the spherical (Atiyah-Singer) Dirac operator on locally defined polynomial forms on the unit sphere of ; functions valued in the universal Clifford algebra , here called spherical k-regular functions. We construct the kernel functions, get the integral representation formula and Cauchy integral formula of spherical k-regular functions, and as applications, the weak solutions of higher order inhomogeneous spherical (Atiyah-Singer) Dirac equations . We obtain, in particular, the weak solution of an inhomogeneous spherical Poisson equation Δ _s g = f. This work was partially supported by NNSF of China (No.10471107) and RFDP of Higher Education (No.20060486001).     

Applications of Matrix Algebra to Clifford Groups

In this paper, we will show that all of nonzero vectors and nonzero bivectors in the Clifford algebra ${\mathcal{C} \ell_{0,3}}$ are invertible and we will find some conditions for those objects to be element of the Clifford group ??_0,3 using the corresponding properties in the subalgebra L ₈ of the matrix algebra ${M_8 \mathbb{(R)}}$ .     

Extended Grassmann and Clifford Algebras

This paper is intended to investigate Grassmann and Clifford algebras over Peano spaces, introducing their respective associated extended algebras, and to explore these concepts also from the counterspace viewpoint. The presented formalism explains how the concept of chirality stems from the bracket, as defined by Rota et all [1]. The exterior (regressive) algebra is shown to share the exterior (progressive) algebra in the direct sum of chiral and achiral subspaces. The duality between scalars and volume elements, respectively under the progressive and the regressive products is shown to have chirality, in the case when the dimension n of the Peano space is even. In other words, the counterspace volume element is shown to be a scalar or a pseudoscalar, depending on the dimension of the vector space to be respectively odd or even. The de Rham cochain associated with the differential operator is constituted by a sequence of exterior algebra homogeneous subspaces subsequently chiral and achiral. Thus we prove that the exterior algebra over the space and the exterior algebra constructed on the counterspace are only pseudoduals each other, if we introduce chirality. The extended Clifford algebra is introduced in the light of the periodicity theorem of Clifford algebras context, wherein the Clifford and extended Clifford algebras can be embedded in which is shown to be exactly the extended Clifford algebra. We present the essential character of the Rota’s bracket, relating it to the formalism exposed by Conradt [25], introducing the regressive product and subsequently the counterspace. Clifford algebras are constructed over the counterspace, and the duality between progressive and regressive products is presented using the dual Hodge star operator. The differential and codifferential operators are also defined for the extended exterior algebras from the regressive product viewpoint, and it is shown they uniquely tumble right out progressive and regressive exterior products of 1-forms. R. da Rocha is supported by CAPES     

Fueter's theorem in discrete Clifford analysis

In this paper, we discretize techniques for the construction of axially monogenic functions to the setting of discrete Clifford analysis. Wherefore, we work in the discrete Hermitian Clifford setting, where each basis vector e_j is split into a forward and backward basis vector: . We prove a discrete version of Fueter's theorem in odd dimension by showing that for a discrete monogenic function f(ξ₀,ξ₁) left‐monogenic in two variables ξ₀ and ξ₁ and for a left‐monogenic P_k(ξ), the m‐dimensional function is in itself left monogenic, that is, a discrete function in the kernel of the discrete Dirac operator. Closely related, we consider a Vekua‐type system for the construction of axially monogenic functions. We consider some explicit examples: the discrete axial‐exponential functions and the discrete Clifford–Hermite polynomials. Copyright © 2015 John Wiley & Sons, Ltd.