共查询到20条相似文献,搜索用时 250 毫秒
1.
2.
3.
4.
Lin ZHU Xiu Hong FENG Yan Lin YU 《数学学报(英文版)》2007,23(9):1629-1638
In this paper, we give identifications of bispinor space with Grassmann algebra, and with Clifford algebra. The multiplication in Clifford algebra provides an action on them. Lastly we have researched on the geometry of bispinor space, and define Dirac operators to get a Pythagoras equality. 相似文献
5.
In this paper we establish a general principle which may be used to construct many explicit solutions to special inhomogeneous
Dirac equations with distributional right-hand side. These solutions are presented as series of products of Clifford algebra
valued functions which themselves satisfy Dirac equations in a lower dimension. We also present several special examples,
including plane waves, zonal functions, Cauchy kernels and electromagnetic fields. 相似文献
6.
V. V. Monakhov 《Theoretical and Mathematical Physics》2016,186(1):70-82
We consider a Clifford extension of the Grassmann algebra in which operators are constructed from products of Grassmann variables and derivatives with respect to them. We show that this algebra contains a subalgebra isomorphic to a matrix algebra and that it additionally contains operators of a generalized matrix algebra that mix states with different numbers of Grassmann variables. We show that these operators are extensions of spin-tensors to the case of superspace. We construct a representation of Dirac matrices in the form of operators of a generalized matrix algebra. 相似文献
7.
Applications of Clifford analysis to three-dimensional elasticity are addressed in the present paper. The governing equation for the displacement field is formulated in terms of the Dirac operator and Clifford algebra valued functions so that a general solution is obtained analytically in terms of one monogenic function and one multiple-component spatial harmonic function together with its derivative. In order to solve numerically the three-dimensional problems of elasticity for an arbitrary domain with complicated boundary conditions, Clifford algebra valued boundary integral equations (BIEs) for multiple-component spatial harmonic functions at an observation point, either inside the domain, on the boundary, or outside the domain, are constructed. Both smooth and non-smooth boundaries are considered in the construction. Moreover, the singularities of the integrals are evaluated exactly so that in the end singularity-free BIEs for the observation point on the boundary taking values on Clifford numbers can be obtained. A Clifford algebra valued boundary element method (BEM) based on the singularity-free BIEs is then developed for solving three-dimensional problems of elasticity. The accuracy of the Clifford algebra valued BEM is demonstrated numerically. 相似文献
8.
Ricardo Abreu-Blaya Juan Bory-Reyes Fred Brackx Hennie De Schepper Frank Sommen 《Complex Analysis and Operator Theory》2012,6(5):971-985
The theory of complex Hermitean Clifford analysis was developed recently as a refinement of Euclidean Clifford analysis; it focusses on the simultaneous null solutions, called Hermitean monogenic functions, of two Hermitean Dirac operators constituting a splitting of the traditional Dirac operator. In this function theory, the fundamental integral representation formulae, such as the Borel?CPompeiu and the Clifford?CCauchy formula have been obtained by using a (2 ×?2) circulant matrix formulation. In the meantime, the basic setting has been established for so-called quaternionic Hermitean Clifford analysis, a theory centred around the simultaneous null solutions, called q-Hermitean monogenic functions, of four Hermitean Dirac operators in a quaternionic Clifford algebra setting. In this paper we address the problem of establishing a quaternionic Hermitean Clifford?CCauchy integral formula, by following a (4?× 4) circulant matrix approach. 相似文献
9.
For a positive integer n let Cl0,n be the universal Clifford algebra with the signature (0,n). The name Clifford analysis is usually referred to the function theories for functions in the kernels of the two operators: the (Cliffordian) Cauchy–Riemann operator and the Dirac operator. For n=2, Cl0,2 becomes the skew‐field of Hamilton's quaternions for which the two operators are widely known: the Moisil–Théodoresco and the Fueter operators. We establish the precise relations between the Moisil–Théodoresco operator and the Dirac operator for Cl0,3. It turns out that the case of the Cauchy–Riemann operator for Cl0,3 and the Fueter operator is more sophisticated, and we describe the peculiarities emerging here. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
10.
11.
Fractional Dirac equations are constructed and fractional Dirac operators on Clifford algebra in four dimensional are introduced within the framework of the fractional calculus of variations recently introduced by the author. Many interesting consequences are revealed and discussed in some details. 相似文献
12.
13.
In this paper we study Clifford Fourier transforms (CFT) of multivector functions taking values in Clifford’s geometric algebra, hereby using techniques coming from Clifford analysis (the multivariate function theory for the Dirac operator). In these CFTs on multivector signals, the complex unit \({i \in \mathbb{C}}\) is replaced by a multivector square root of ?1, which may be a pseudoscalar in the simplest case. For these integral transforms we derive an operator representation expressed as the Hamilton operator of a harmonic oscillator. 相似文献
14.
John Ryan 《Journal of Functional Analysis》1987,70(2)
A number of Runge approximation theorems are proved for complex Clifford algebra valued holomorphic functions which either satisfy the holomorphic, homogeneous Dirac equation, or complex Laplacian. The results are applied to establish analogues of the homological version of the Mittag-Leffler theorem. 相似文献
15.
Jayme Vaz 《Advances in Applied Clifford Algebras》1997,7(1):37-70
We introduce a calculus over a lattice based on a lattice generalization of the Clifford algebras. We show that Clifford algebras,
in contrast to the continuum, are not an adequated algebraic structure for lattice problems. Then we introduce a new algebraic
structure, that reduces to a Clifford algebra in the continuum limit, in terms of which we can develop a formalism analogous
to the differential geometry of the continuum, also in the sense that we have intrinsic expressions. The differential operator
is given by the graded commutator of an operator that generalizes the Dirac operator. We also discuss the applications of
this formalism in lattice gauge theories, with particular attention to the fermion doubling problem.
On leave of absence from Department of Applied Mathematics, State University at Campinas (UNICAMP), 13081-970 Campinas, SP,
Brazil — vaz@ime.unicamp.br 相似文献
16.
Jacques Helmstetter 《Advances in Applied Clifford Algebras》2008,18(2):153-196
A meson algebra is involved in the Duffin wave equation for mesons in the same way as a Clifford algebra is involved in the
Dirac wave equation for electrons. Therefore meson algebras too should have geometrical properties after the manner of Grassmann.
Actually it is possible to define interior multiplications with similar properties, and deformations too. Every meson algebra
is a deformation of a neutral meson algebra, in the same way as (almost) every Clifford algebra is a deformation of an exterior
algebra. Some applications follow: the PBW-property is proved for all meson algebras, the injectiveness of Jacobson’s diagonal
morphism is proved with the minimal hypothesis, and the existence of Lipschitz monoids is established at least for meson algebras
over fields.
相似文献
17.
F. Brackx H. De Schepper M. E. Luna-Elizarrarás M. Shapiro 《Complex Analysis and Operator Theory》2012,6(2):325-339
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 ?{\underline{\partial}} 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{\partial_{\underline{z}}} and ?zf{\partial_{\underline{z}^\dagger}} 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 = d{\underline{\partial}E = \delta}. Moreover we will refine these relations by constructing fundamental solutions for the differential operators issuing from
the Euclidean and Hermitean Dirac operators by splitting the Clifford algebra product into its dot and wedge parts. 相似文献
18.
Discrete Clifford analysis is a higher dimensional discrete function theory, based on skew Weyl relations. The basic notions are discrete monogenic functions, i.e. Clifford algebra valued functions in the kernel of a discrete Dirac operator. In this paper, we introduce the discrete Fueter polynomials, which form a basis of the space of discrete spherical monogenics, i.e. discrete monogenic, homogeneous polynomials. Their definition is based on a Cauchy–Kovalevskaya extension principle. We present the explicit construction for this discrete Fueter basis, in arbitrary dimension m and for arbitrary homogeneity degree k. 相似文献
19.
Gaston Casanova 《Advances in Applied Clifford Algebras》2000,10(1):39-44
Using a formula derived from the Dirac equation, written in Clifford algebra, we prove that either in relativistic or in corpuscular
(quantum) mechanics (Bohm, Vigier) free electrons cannot be localized. This makes the corpuscular hypothesis unacceptable,
disproving the possibility of a subquantum field. 相似文献
20.
A particular orthogonal map on a finite-dimensional real quadratic vector space (V,?Q) with a non-degenerate quadratic form Q of any signature (p,?q) is considered. It can be viewed as a correlation of the vector space that leads to a dual Clifford algebra C?(V*,?Q) of linear functionals (multiforms) acting on the universal Clifford algebra C?(V,?Q). The map results in a unique involutive automorphism and a unique involutive anti-automorphism of C?(V,?Q). The anti-involution reduces to reversion (resp. conjugation) for any Euclidean (resp. anti-Euclidean) signature. When applied to a general element of the algebra, it results in transposition of the element matrix in the left regular representation of C?(V,?Q). We also give an example for real spinor spaces. The general setting for spinor representations will be treated in part II of this work [R. Ab?amowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras II: Stabilizer groups of primitive idempotents, Linear Multilinear Algebra, to appear]. 相似文献