314 research outputs found
A Clifford analysis approach to superspace
A new framework for studying superspace is given, based on methods from
Clifford analysis. This leads to the introduction of both orthogonal and
symplectic Clifford algebra generators, allowing for an easy and canonical
introduction of a super-Dirac operator, a super-Laplace operator and the like.
This framework is then used to define a super-Hodge coderivative, which,
together with the exterior derivative, factorizes the Laplace operator. Finally
both the cohomology of the exterior derivative and the homology of the Hodge
operator on the level of polynomial-valued super-differential forms are
studied. This leads to some interesting graphical representations and provides
a better insight in the definition of the Berezin-integral.Comment: 15 pages, accepted for publication in Annals of Physic
Rarita-Schwinger Type Operators on Spheres and Real Projective Space
In this paper we deal with Rarita-Schwinger type operators on spheres and
real projective space. First we define the spherical Rarita-Schwinger type
operators and construct their fundamental solutions. Then we establish that the
projection operators appearing in the spherical Rarita-Schwinger type operators
and the spherical Rarita-Schwinger type equations are conformally invariant
under the Cayley transformation. Further, we obtain some basic integral
formulas related to the spherical Rarita-Schwinger type operators. Second, we
define the Rarita-Schwinger type operators on the real projective space and
construct their kernels and Cauchy integral formulas.Comment: 21 pages. arXiv admin note: text overlap with arXiv:1106.358
Fundamental solutions for the super Laplace and Dirac operators and all their natural powers
The fundamental solutions of the super Dirac and Laplace operators and their
natural powers are determined within the framework of Clifford analysis.Comment: 12 pages, accepted for publication in J. Math. Anal. App
Spherical harmonics and integration in superspace
In this paper the classical theory of spherical harmonics in R^m is extended
to superspace using techniques from Clifford analysis. After defining a
super-Laplace operator and studying some basic properties of polynomial
null-solutions of this operator, a new type of integration over the supersphere
is introduced by exploiting the formal equivalence with an old result of
Pizzetti. This integral is then used to prove orthogonality of spherical
harmonics of different degree, Green-like theorems and also an extension of the
important Funk-Hecke theorem to superspace. Finally, this integration over the
supersphere is used to define an integral over the whole superspace and it is
proven that this is equivalent with the Berezin integral, thus providing a more
sound definition of the Berezin integral.Comment: 22 pages, accepted for publication in J. Phys.
The class of Clifford-Fourier transforms
Recently, there has been an increasing interest in the study of hypercomplex
signals and their Fourier transforms. This paper aims to study such integral
transforms from general principles, using 4 different yet equivalent
definitions of the classical Fourier transform. This is applied to the
so-called Clifford-Fourier transform (see [F. Brackx et al., The
Clifford-Fourier transform. J. Fourier Anal. Appl. 11 (2005), 669--681]). The
integral kernel of this transform is a particular solution of a system of PDEs
in a Clifford algebra, but is, contrary to the classical Fourier transform, not
the unique solution. Here we determine an entire class of solutions of this
system of PDEs, under certain constraints. For each solution, series
expressions in terms of Gegenbauer polynomials and Bessel functions are
obtained. This allows to compute explicitly the eigenvalues of the associated
integral transforms. In the even-dimensional case, this also yields the inverse
transform for each of the solutions. Finally, several properties of the entire
class of solutions are proven.Comment: 30 pages, accepted for publication in J. Fourier Anal. App
Hermite and Gegenbauer polynomials in superspace using Clifford analysis
The Clifford-Hermite and the Clifford-Gegenbauer polynomials of standard
Clifford analysis are generalized to the new framework of Clifford analysis in
superspace in a merely symbolic way. This means that one does not a priori need
an integration theory in superspace. Furthermore a lot of basic properties,
such as orthogonality relations, differential equations and recursion formulae
are proven. Finally, an interesting physical application of the super
Clifford-Hermite polynomials is discussed, thus giving an interpretation to the
super-dimension.Comment: 18 pages, accepted for publication in J. Phys.
A Cauchy integral formula in superspace
In previous work the framework for a hypercomplex function theory in
superspace was established and amply investigated. In this paper a Cauchy
integral formula is obtained in this new framework by exploiting techniques
from orthogonal Clifford analysis. After introducing Clifford algebra valued
surface- and volume-elements first a purely fermionic Cauchy formula is proven.
Combining this formula with the already well-known bosonic Cauchy formula
yields the general case. Here the integration over the boundary of a
supermanifold is an integration over as well the even as the odd boundary (in a
formal way). Finally, some additional results such as a Cauchy-Pompeiu formula
and a representation formula for monogenic functions are proven.Comment: 14 pages, accepted for publication in the Bulletin of the LM
Spherical harmonics and integration in superspace II
The study of spherical harmonics in superspace, introduced in [J. Phys. A:
Math. Theor. 40 (2007) 7193-7212], is further elaborated. A detailed
description of spherical harmonics of degree k is given in terms of bosonic and
fermionic pieces, which also determines the irreducible pieces under the action
of SO(m) x Sp(2n). In the second part of the paper, this decomposition is used
to describe all possible integrations over the supersphere. It is then shown
that only one possibility yields the orthogonality of spherical harmonics of
different degree. This is the so-called Pizzetti-integral of which it was shown
in [J. Phys. A: Math. Theor. 40 (2007) 7193-7212] that it leads to the Berezin
integral.Comment: 18 pages, accepted for publication in J. Phys.
A General Geometric Fourier Transform
The increasing demand for Fourier transforms on geometric algebras has
resulted in a large variety. Here we introduce one single straight forward
definition of a general geometric Fourier transform covering most versions in
the literature. We show which constraints are additionally necessary to obtain
certain features like linearity or a shift theorem. As a result, we provide
guidelines for the target-oriented design of yet unconsidered transforms that
fulfill requirements in a specific application context. Furthermore, the
standard theorems do not need to be shown in a slightly different form every
time a new geometric Fourier transform is developed since they are proved here
once and for all.Comment: First presented in Proc. of The 9th Int. Conf. on Clifford Algebras
and their Applications, (2011
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