D. Eelbode
University of Antwerp, Mathematics and Computer Science, Faculty Member
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... For the hyperbolic unit ball or the hyperbolic half plane we refer to the work of Heinz Leutwiler and his students (see [15],[9],[12]) and to the one of Cnops (see [5],[6]). Where the above attempts generalize the Dirac operator for... more
... For the hyperbolic unit ball or the hyperbolic half plane we refer to the work of Heinz Leutwiler and his students (see [15],[9],[12]) and to the one of Cnops (see [5],[6]). Where the above attempts generalize the Dirac operator for Spinl fields, in this paper we work with a ...
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... For the hyperbolic unit ball or the hyperbolic half plane we refer to the work of Heinz Leutwiler and his students (see [15],[9],[12]) and to the one of Cnops (see [5],[6]). Where the above attempts generalize the Dirac operator for... more
... For the hyperbolic unit ball or the hyperbolic half plane we refer to the work of Heinz Leutwiler and his students (see [15],[9],[12]) and to the one of Cnops (see [5],[6]). Where the above attempts generalize the Dirac operator for Spinl fields, in this paper we work with a ...
Project Euclid. ...
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In this contribution, we introduce higher spin Dirac operators, i.e. a specific class of differential operators in Clifford analysis of several vector variables, motivated by equations from theoretical physics. In particular, the higher... more
In this contribution, we introduce higher spin Dirac operators, i.e. a specific class of differential operators in Clifford analysis of several vector variables, motivated by equations from theoretical physics. In particular, the higher spin Dirac operator in three vector variables will be explicitly constructed, starting from a description of the so-called twisted Rarita-Schwinger operator.
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In this paper, the fundamental solution of the Dirac equation on hyperbolic space will be calculated by means of the fundamental solution for the wave-operator in the $(m+1)$-dimensional Minkowski space-time of signature $(1,m)$. This... more
In this paper, the fundamental solution of the Dirac equation on hyperbolic space will be calculated by means of the fundamental solution for the wave-operator in the $(m+1)$-dimensional Minkowski space-time of signature $(1,m)$. This leads to addition formulas for the fundamental solution in terms of the solution in a lower-dimensional Minkowski space-time. Certain identities between hypergeometric functions can then be used to obtain a closed form for the fundamental solution of the Dirac equation.
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... ELEMENTARY PARTICLES AND FIELDS Theory A Toy Model for Higher Spin Dirac Operators* ... Abstract—This paper deals with the higher spin Dirac operator Q2,1 acting on functions taking values in an irreducible representation space for... more
... ELEMENTARY PARTICLES AND FIELDS Theory A Toy Model for Higher Spin Dirac Operators* ... Abstract—This paper deals with the higher spin Dirac operator Q2,1 acting on functions taking values in an irreducible representation space for so(m) with highest weight (5 ...
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ABSTRACT This paper is devoted to the algebraic analysis of the system of differential equations described by the Hermitian Dirac operators, which are two linear first order operators invariant with respect to the action of the unitary... more
ABSTRACT This paper is devoted to the algebraic analysis of the system of differential equations described by the Hermitian Dirac operators, which are two linear first order operators invariant with respect to the action of the unitary group. In the one variable case, we show that it is possible to give explicit formulae for all the maps of the resolution associated to the system. Moreover, we compute the minimal generators for the first syzygies also in the case of the Hermitian system in several vector variables. Finally, we study the removability of compact singularities. We also show a major difference with the orthogonal case: in the odd dimensional case it is possible to perform a reduction of the system which does not affect the behavior of the free resolution, while this is not always true for the case of even dimension.
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... The inverse Radon transform and the fundamental solution of the hyperbolic Dirac equation D. Eelbode⋆, F. Sommen ... Abstract In this paper the fundamental solution of the Dirac equation in an m-dimensional hyperbolic space is... more
... The inverse Radon transform and the fundamental solution of the hyperbolic Dirac equation D. Eelbode⋆, F. Sommen ... Abstract In this paper the fundamental solution of the Dirac equation in an m-dimensional hyperbolic space is constructed. ...
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In this paper an explicit expression is determined for the elliptic higher spin Dirac operator, acting on functions f(\underline{x}) taking values in an arbitrary irreducible finite-dimensional module for the group Spin(m) characterized... more
In this paper an explicit expression is determined for the elliptic higher spin Dirac operator, acting on functions f(\underline{x}) taking values in an arbitrary irreducible finite-dimensional module for the group Spin(m) characterized by a half-integer highest weight. Also a special class of solutions of these operators is constructed, and the connection between these solutions and transvector algebras is explained.
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ABSTRACT
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ABSTRACT The Fueter theorem states that regular (resp. monogenic) functions in quaternionic (resp. Clifford) analysis can be constructed from holomorphic functions [Inline formula] in the complex plane, hereby using a combination of a... more
ABSTRACT The Fueter theorem states that regular (resp. monogenic) functions in quaternionic (resp. Clifford) analysis can be constructed from holomorphic functions [Inline formula] in the complex plane, hereby using a combination of a formal substitution and the action of an appropriate power of the Laplace operator. In this paper we interpret this theorem on the level of representation theory, as an intertwining map between certain [Inline formula]-modules.
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... One might argue that function theoretical results for the hyperbolic Dirac operator follow from results obtained by for example Van Lancker and Sommen (see references [1617. Van Lancker P 1997 Clifford Analysis on the Unit Sphere,... more
... One might argue that function theoretical results for the hyperbolic Dirac operator follow from results obtained by for example Van Lancker and Sommen (see references [1617. Van Lancker P 1997 Clifford Analysis on the Unit Sphere, PhD-thesis Ghent Belgium View all ...
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Project Euclid. ...
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In this article, we realize a projective model for the equivalent of the unit sphere S ⊂ℝ in the orthogonal space ℝ of ultrahyperbolic signature (p, q), and we define a Dirac operator on this surface within the context of the Clifford... more
In this article, we realize a projective model for the equivalent of the unit sphere S ⊂ℝ in the orthogonal space ℝ of ultrahyperbolic signature (p, q), and we define a Dirac operator on this surface within the context of the Clifford analysis. We then construct null solutions of a special type for this operator, hence obtaining a relationship between hypergeometric functions and spin-invariant systems.
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ABSTRACT
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ABSTRACT 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... more
ABSTRACT 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 f(x0, x)f(x_0, \underline{x}) on \mathbbRm+1{\mathbb{R}}^{m+1} , i.e. a solution for the generalized Cauchy-Riemann operator D on \mathbbRm+1{\mathbb{R}}^{m+1} , construct a monogenic function g(x0, x)g(x_0, \underline{x}) such that [`(D)]g = f\overline{D}g = f . In view of the fact that, for monogenic functions g, this can be written as ¶x0\partial_{x_0} g = f, a straightforward generalization consists in replacing the scalar generator ¶x0\partial_{x_0} of translations in the x 0-direction by a generator of another transformation group. In this paper we consider translations in more dimensions.