Hermitian Clifford analysis and its connections with representation theory (original) (raw)

Fundaments of Hermitean Clifford Analysis Part I: Complex Structure

Complex Analysis and Operator Theory, 2007

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 realisation of the unitary group. In Part I of the paper the fundamental elements of the Hermitean setting have been introduced in a natural way, i.e. by introducing a complex structure J ∈ SO(2n; R) on the underlying vector space, eventually extended to the whole complex Clifford algebra Cℓ C 2n . 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 paper, 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.

The Hermitean Hilbert–Dirac Connection

Advances in Applied Clifford Algebras, 2009

Hermitean Clifford analysis is a recent branch of Clifford analysis, refining the Euclidean case; it focusses on the simultaneous null solutions, called Hermitean monogenic functions, of two complex Dirac operators which are invariant under the action of the unitary group. The specificity of the framework, introduced by means of a complex structure creating a Hermitean space, forces the underlying vector space to be even dimensional. Thus, any Hilbert convolution kernel in R 2n should originate from the non-tangential boundary limits of a corresponding Cauchy kernel in R 2n+2 . In this paper we show that the difficulties posed by this inevitable dimensional jump can be overcome by following a matrix approach. The resulting matrix Hermitean Hilbert transform also gives rise, through composition with the matrix Dirac operator, to a Hermitean Hilbert-Dirac convolution operator "factorizing" the Laplacian and being closely related to Riesz potentials.

On the definition of geometric Dirac operators

For the definition of a spin c structure and its associated Dirac operators there can be found two different approaches in the literature. One of them uses lifts of the orthonormal frame bundle to principal spin c bundles (cf.

The Clifford algebra of physical space and Dirac theory

European Journal of Physics, 2016

The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term by in the usual Dirac factorization of the Klein-Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation.

The Hermitian Clifford Analysis Toolbox

Advances in Applied Clifford Algebras, 2008

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 realization of the unitary group. In this paper we have properly developed the Hermitean Clifford analysis framework and established the structure for a Hermitean Fischer decomposition, which, as is the case in traditional Clifford analysis, is a corner stone of the function theory. Keywords: Hermitean Clifford analysis, Hermitean Fischer decomposition.

Complex geometry and Dirac equation

International journal of theoretical …, 1998

Complex geometry represents a fundamentalingredient in the formulation of the Dirac equation bythe Clifford algebra. The choice of appropriate complexgeometries is strictly related to the geometricinterpretation of the complex imaginary unit i=-1. We discuss two ...

On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces

Journal of Geometry and Physics, 1996

The eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces of arbitrary dimension are computed by separating variables in geodesic polar coordinates. These eigenfunctions are then used to derive the heat kernel of the iterated Dirac operator on these spaces. They are then studied as cross sections of homogeneous vector bundles, and a group-theoretic derivation of the spinor spherical functions and heat kernel is given based on Harish-Chandra's formula for the radial part of the Casimir operator.

Discrete Dirac Operators in Clifford Analysis

Advances in Applied Clifford Algebras, 2007

We develop a constructive framework to define difference approximations of Dirac operators which factorize the discrete Laplacian. This resulting notion of discrete monogenic functions is compared with the notion of discrete holomorphic functions on quad-graphs. In the end Dirac operators on quad-graphs are constructed. . Primary 30G35, 30G25; Secondary 05C78.

On Fundamental Solutions in Clifford Analysis

Complex Analysis and Operator Theory, 2012

Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centred around the concept of monogenic functions, which constitute the kernel of a first order vector valued, rotation invariant, differential operator ∂ called the Dirac operator, which factorizes the Laplacian. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focusses on a subclass of monogenic functions, i.e. the simultaneous null solutions, called Hermitean (or h−) monogenic functions, of two Hermitean Dirac operators ∂ z and ∂ z † which are invariant under the action of the unitary group, and constitute a splitting of the original Euclidean Dirac operator. In Euclidean Clifford analysis, the Clifford-Cauchy integral formula has proven to be a corner stone of the function theory, as is the case for the traditional Cauchy formula for holomorphic functions in the complex plane. Also a Hermitean Clifford-Cauchy integral formula has been established by means of a matrix approach. Naturally Cauchy integral formulae rely upon the existence of fundamental solutions of the Dirac operators under consideration. The aim of this paper is twofold. We want to reveal the underlying structure of these fundamental solutions and to show the particular results hidden behind a formula such as, e.g. ∂ E = δ. Moreover we will refine these relations by constructing fundamental solutions for the differential operators issuing from the Euclidean and Communicated by