Complex Clifford Algebra Research Articles

Calculation of Elements of Spin Groups Using Generalized Pauli’s Theorem

We formulate generalizations of Pauli's theorem on the cases of real and complex Clifford algebras of even and odd dimensions. We give analogues of these theorems in matrix formalism. Using these theorems we present an algorithm for computing elements of spin groups that correspond to elements of orthogonal groups as double cover.

Cyclic Structures of Cliffordian Supergroups and Particle Representations of Spin+(1, 3)

Supergroups are defined in the framework of \({\mathbb{Z}_2}\) 2-graded Clifford algebras over the fields of real and complex numbers, respectively. It is shown that cyclic structures of complex and real supergroups are defined by Brauer-Wall groups related with the modulo 2 and modulo 8 periodicities of the complex and real Clifford algebras. Particle (fermionic and bosonic) representations of a universal covering (spinor group Spin +(1, 3)) of the proper orthochronous Lorentz group are constructed via the Clifford algebra formalism. Complex and real supergroups are defined on the representation system of Spin +(1, 3). It is shown that a cyclic (modulo 2) structure of the complex supergroup is equivalent to a supersymmetric action, that is, it converts fermionic representations into bosonic representations and vice versa. The cyclic action of the real supergroup leads to a much more high-graded symmetry related with the modulo 8 periodicity of the real Clifford algebras. This symmetry acts on the system of real representations of Spin +(1, 3).

Far‐field patterns of solutions of the perturbed Dirac equation

The purpose of the paper is to study the asymptotic behavior at infinity of solutions of a perturbed Dirac equation in called k‐monogenic. Every such solution is a solution of the Helmholtz equation with values in a complex Clifford algebra. The main goal is to use the far‐field pattern to characterize the radiating (outgoing) k‐monogenic functions among the radiating solutions of the Helmholtz equation. It will be shown that an algebraic condition characterizes these far‐field patterns. Copyright © 2012 John Wiley & Sons, Ltd.

CLIFFORD MODULES AND SYMMETRIES OF TOPOLOGICAL INSULATORS

We complete the classification of symmetry constraints on gapped quadratic fermion hamiltonians proposed by Kitaev. The symmetry group is supposed compact and can include arbitrary unitary or antiunitary operators in the Fock space that conserve the algebra of quadratic observables. We analyze the multiplicity spaces of real irreducible representations of unitary symmetries in the Nambu space. The joint action of intertwining operators and antiunitary symmetries provides these spaces with the structure of Clifford module: we prove a one-to-one correspondence between the ten Altland–Zirnbauer symmetry classes of fermion systems and the ten Morita equivalence classes of real and complex Clifford algebras. The antiunitary operators, which occur in seven classes, are projectively represented in the Nambu space by unitary "chiral symmetries". The space of gapped symmetric hamiltonians is homotopically equivalent to the product of classifying spaces indexed by the dual object of the group of unitary symmetries.

Hyperholomorphic theory on kaehler manifolds

First of all, using the relations (2.3), (2.4), and (2.5), we define a complex Clifford algebra ▪ and the Witt basis. Secondly, we utilize the Witt basis to define the operators ▪ on Kaehler manifolds which act on ▪-valued functions. In addition, the relation between above operators and Hodge-Laplace operator is argued. Then, the Borel-Pompeiu formulas for ▪-valued functions are derived through designing a matrix Dirac operator ▪ and a 2 × 2 matrix–valued invariant integral kernel with the Witt basis.

On self-adjoint operators in Krein spaces constructed by Clifford algebra Cl_2

Let \(J\) and \(R\) be anti-commuting fundamental symmetries in a Hilbert space \(\mathfrak{H}\). The operators \(J\) and \(R\) can be interpreted as basis (generating) elements of the complex Clifford algebra \(Cl_2(J,R):=\text{span}\{I,J,R,iJR\}\). An arbitrary non-trivial fundamental symmetry from \(Cl_2(J,R)\) is determined by the formula \(J_{\vec{\alpha}}=\alpha_1 J +\alpha_2 R+\alpha_3 iJR\), where \(\vec{\alpha} \in \mathbb{S}^2\). Let \(S\) be a symmetric operator that commutes with \(Cl_2(J,R)\). The purpose of this paper is to study the sets \(\Sigma_{J_{\vec{\alpha}}}\) (\(\forall \vec{\alpha} \in \mathbb{S}^2\)) of self-adjoint extensions of \(S\) in Krein spaces generated by fundamental symmetries \(J_{\vec{\alpha}}\) (\(J_{\vec{\alpha}}\)-self-adjoint extensions). We show that the sets \(\Sigma_{J_{\vec{\alpha}}}\) and \(\Sigma_{J_{\vec{\beta}}}\) are unitarily equivalent for different \(\vec{\alpha}, \vec{\beta} \in \mathbb{S}^2\) and describe in detail the structure of operators \(A \in \Sigma_{J_{\vec{\alpha}}}\) with empty resolvent set.

$${\overline{\partial}}$$ -problem for an overdetermined system with two higher dimensional variables

Our main purpose is to describe necessary and sufficient conditions for the solvability of the \({{\overline\partial}}\) -problem for biregular complex Clifford algebra valued functions of two higher dimensional variables in even dimensional Euclidean spaces. Since the biregular function theory falls naturally from the so-called isotonic Clifford analysis, the main idea of the proof is based on the notion of biregular-conjugate harmonic functions to be introduced here. Besides, whenever the \({{\overline\partial}}\) -problem is solvable, we give the general solution of it in a quite explicit form.

A representation of sl(2;C) on clifford algebra of even dimension

Under the foundation of Hermitean Clifford setting, we define the fundamental operators for complex Clifford algebra valued functions, obtain some properties of these operators, and discuss a representation of sl(2;C) on Clifford algebra of even dimension.

A Fractal Representation of the Complex Clifford Algebra Equivalent to the Fock Representation

We construct a representation of the infinite dimensional complex Clifford algebra on the Hilbert space of square-integrable complex-valued functions on the Cantor set, which we show to be equivalent to the classical Fock representation.

On the Structure of Complex Clifford Algebra

The structure of a complex Clifford algebra is studied by direct sum decompositions into eigenspaces of specific linear operators.

Representations of Clifford Algebras on Function Spaces on the Cantor Set

We give a representation of the (infinite-dimensional) complex Clifford algebra on the Hilbert space of square-integrable complexvalued functions on the Cantor set.

Some properties of holomorphic Cliffordian functions in complex Clifford analysis

In this article, we mainly develop the foundation of a new function theory of several complex variables with values in a complex Clifford algebra defined on some subdomains of C n+1 , so-called complex holomorphic Cliffordian functions. We define the complex holomorphic Cliffordian functions, study polynomial and singular solutions of the equation DΔ m f = 0, obtain the integral representation formula for the complex holomorphic Cliffordian functions with values in a complex Clifford algebra defined on some submanifolds of C n+1 , deduce the Taylor expansion and the Laurent expansion for them and prove an invariance under an action of Lie group for them.

A Symplectic Subgroup of a Pseudounitary Group as a Subset of Clifford Algebra

Let Cl1(1,3) and Cl2(1,3) be the subsets of elements of the Clifford algebra Cl(1,3) of ranks 1 and 2 respectively. Recently it was proved that the subset Cl2(p,q)+iCl1(p,q) of the complex Clifford algebra can be considered as a Lie algebra. In this paper we prove that for p=1, q=3 the Lie algebra Cl2(p,q)+iCl1(p,q) is isomorphic to the well known matrix Lie algebra sp(4,R) of the symplectic Lie group Sp(4,R). Also we define the so called symplectic group of Clifford algebra and prove that this Lie group is isomprphic to the symplectic matrix group Sp(4,R).

The Hermitian Clifford Analysis Toolbox

Hermitian Clifford analysis focusses on monogenic functions taking values in a complex Clifford algebra or in a complex spinor space. Here monogenicity is expressed by means of two complex mutually adjoint Dirac operators, which are invariant under the action of a representation of the unitary group. In this paper we have properly developed the Hermitian Clifford analysis framework and established a Hermitian Fischer decomposition, which, as is the case in traditional Clifford analysis, is a corner stone of the function theory.

Unitary Spaces on Clifford Algebras

For the complex Clifford algebra (p, q) of dimension n = p + q we define a Hermitian scalar product. This scalar product depends on the signature (p, q) of Clifford algebra. So, we arrive at unitary spaces on Clifford algebras. With the aid of Hermitian idempotents we suggest a new construction of, so called, normal matrix representations of Clifford algebra elements. These representations take into account the structure of unitary space on Clifford algebra.

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 \({\mathbb{R}}^m\).

Fundaments of Hermitean Clifford analysis part II: splitting of h -monogenic equations§

Hermitean Clifford analysis focuses on h-monogenic functions taking values in a complex Clifford algebra or in a complex spinor space, where h-monogenicity is expressed by means of two complex and mutually adjoint Dirac operators, which are invariant under the action of a Clifford realization of the unitary group. In part 1 of the article the fundamental elements of the Hermitean setting have been introduced in a natural way, i.e., by introducing a complex structure on the underlying vector space, eventually extended to the whole complex Clifford algebra . The two Hermitean Dirac operators are then shown to originate as generalized gradients when projecting the gradient on invariant subspaces. In this part of the article, the aim is to further unravel the conceptual meaning of h-monogenicity, by studying possible splittings of the corresponding first-order system into independent parts without changing the properties of the solutions. In this way further connections with holomorphic functions of several complex variables are established. As an illustration, we give a full characterization of h-monogenic functions for the case n = 2. During the final redaction of this article, we received the sad news that our friend, colleague and co-author Jarolím Bureš died on 1 October 2006.

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 \(\underline \partial = \sum\nolimits_{j = 1}^m {} e_j \partial _{x_j }\), where (\(e_{1},\ldots, e_m\)) forms an orthogonal basis for the quadratic space \(\mathbb{R}^m\) underlying the construction of the Clifford algebra \(\mathbb{R}_{0,m}\). When allowing for complex constants and taking the dimension to be even: m = 2n, the same set of generators produces the complex Clifford algebra \(\mathbb{C}_{2n}\), 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].

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;\({\mathbb{R}}\)), 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 (eα, α = 1, . . . , 2n) onto the Witt basis and moreover give rise to a direct sum decomposition of Open image in new window into two components, where the SO(2n;\({\mathbb{R}}\))-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;\({\mathbb{R}}\)), denoted Spin J (2n;\({\mathbb{R}}\)), being isomorphic with the unitary group U(n;\({\mathbb{C}}\)). 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;\({\mathbb{R}}\)). The eventual goal is to extend the complex structure J to the whole Clifford algebra Open image in new window , in order to conceptually unravel the true meaning of Hermitean monogenicity and its connections to orthogonal monogenicity.

A theoretical framework for wavelet analysis in a Hermitean Clifford setting

Hermitean Clifford analysis focusses on monogenic functions taking values in a complex Clifford algebra or in a complex spinor space. Here monogenicity is expressed by means of two complex mutually adjoint Dirac operators, which are invariant under the action of a representation of the unitary group. In this paper we have further developed the Hermitean theory by introducing so-called zonal functions and by studying plane wave null solutions of the Hermitean Dirac operators. Moreover we have defined new Hermite polynomials in this Hermitean setting and expressed them in terms of the former Clifford-Hermite polynomials and of the one-dimensional Laguerre polynomials. These Hermitean Hermite polynomials are good candidates for mother wavelets in a Hermitean continuous wavelet transformation theory yet to be developed.