Rarita–Schwinger Type Operators in Clifford Analysis

Bureš, Jarolím; Sommen, Franciscus; Souček, Vladimír; Lancker, P. Van

doi:10.1006/jfan.2001.3781

Cited by 69 publications

(133 citation statements)

References 7 publications

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“…Let f (x; u) be a polynomial in C ∞ (R m , M k ), homogeneous of degree h ≥ k in the vector variable x. In [7], the following equivalence was proved:…”

Section: Solutions Of Type Bmentioning

confidence: 99%

“…The necessary and sufficient conditions (called compatibility conditions for short) under which an inhomogeneous system in several Dirac operators has a solution, were thoroughly studied in [9]. Referring to [7,9] for details, the compatibility conditions for the existence of a solution for the system above are ∆ u (ug) = 0 and ∂ u ∂ x (ug) = 0. The first condition is equivalent with the monogeneity of g in the variable u, i.e.…”

Section: Solutions Of Type Bmentioning

confidence: 99%

“…g ∈ C ∞ (R m , M k−1 ), whereas the second compatibility condition is equivalent to g ∈ Ker h−1 R k−1 . In other words: these compatibility conditions signify that the kernel space for the operator R k−1 can be embedded into the kernel space for R k , using certain inversion operators which were described in [7]. Type B solutions for R k are thus equivalent with elements of Ker h−1 R k−1 ; their structure may thus be described through an inductive procedure.…”

Section: Solutions Of Type Bmentioning

confidence: 99%

“…Just like for the Rarita-Schwinger case, see [7], this requires the study of compatibility conditions for an inhomogeneous system of equations involving three Dirac operators. The system above is not of the form considered in [9] due to the presence of the last equation.…”

Section: Solutions Of Type Bmentioning

confidence: 99%

“…, 1 2 ), see e.g. [7,8]. Also higher spin Dirac operators acting on spinor-valued forms have been studied in detail, see e.g.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Higher Spin Dirac Operators Between Spaces of Simplicial Monogenics in Two Vector Variables

Brackx

Eelbode

Voorde

2010

Math Phys Anal Geom

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“…Let f (x; u) be a polynomial in C ∞ (R m , M k ), homogeneous of degree h ≥ k in the vector variable x. In [7], the following equivalence was proved:…”

Section: Solutions Of Type Bmentioning

confidence: 99%

Section: Solutions Of Type Bmentioning

confidence: 99%

Section: Solutions Of Type Bmentioning

confidence: 99%

Section: Solutions Of Type Bmentioning

confidence: 99%

“…, 1 2 ), see e.g. [7,8]. Also higher spin Dirac operators acting on spinor-valued forms have been studied in detail, see e.g.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Higher Spin Dirac Operators Between Spaces of Simplicial Monogenics in Two Vector Variables

Brackx

Eelbode

Voorde

2010

Math Phys Anal Geom

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Introduction to Clifford analysis

Sommen¹,

Sprößig²

2002

Math Methods in App Sciences

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EDITORIAL INTRODUCTION TO CLIFFORD ANALYSISComplex numbers were invented to answer a purely mathematical question. But before he discovered the quaternions, W.K. Hamilton was looking for a possible algebra structure for the three-dimensional (3D) physical space. Behind it is the Platonic idea that the physical reality would be the o print of a 'more perfect mathematical reality', which also means that physical problems can be modelled using mathematics and conversely, the modelling of physical problems leads to the discovery of interesting new mathematics. But the Platonic world of mathematics is also di erent from the real world and since Einstein we know that the physical space (or space-time) tends to behave not like a vector space or algebra but like a manifold with a weaker and hence more general structure. In the same way, the mathematical description of physical phenomena often needs the more general non-linear mathematics. But in spite of this, also the more restrictive and special mathematical structures like algebras may be quite useful for the applied sciences. Hereby we ÿrst of all mention the wide scope of applications of complex algebra and analysis and also of the quaternion algebra H = {q 0 + q 1 I + q 2 J + q 3 K : q 0 ; q 1 ; q 2 ; q 3 ∈ R} with its deÿning relations IJ = − JI = K; JK = − KJ = I; KI = − IK = J It namely allows the deÿnition of the so-called Fueter operator (already discovered by Hamilton) @ q = @ q0 + @ q1 I + @ q2 J + @ q3 K which may not be deÿned on the 'physical 3D-space' but does contain as a restriction the Moisil-Teodorescu operatorwhich has applications in electrostatics and magnetostatics. More so, after replacing the real co-ordinate q 0 by the imaginary i t; t meaning the time, the Fueter operator transforms into a kind of quaternion Dirac operator which has as null-solutions special source free solutions of the Maxwell ÿeld. Later on, still led by the idea to mathematize physical space, people looked for algebras which did not coincide with e.g. the physical space but were at least generated by it, more, in general, they were looking for algebras generated by a general m-dimensional space. To this class belong ÿrst of all the Grassmann algebra with deÿning relations f j f k = − f k f j

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Almansi‐type theorems in Clifford analysis

Malonek

Ren

2002

Math Methods in App Sciences

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SUMMARYIn this paper, we consider functions deÿned in a star-like domain ⊂ R n with values in the Cli ord algebra C' 0; n which are polymonogenic with respect to the (left) Dirac operator D = n j=1 e j @=@x j , i.e. they belong to the kernel of D k . We prove that any polymonogenic function f has a decomposition of the formwhere x = x 1 e 1 + · · · + xnen and f j ; j = 1; : : : ; k; are monogenic functions. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition of polynomials. Similar results are obtained for the powers of weighted Dirac operators of the formD = |x| − xD; ∈ R\{0}.

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Rarita–Schwinger Type Operators in Clifford Analysis

Cited by 69 publications

References 7 publications

Higher Spin Dirac Operators Between Spaces of Simplicial Monogenics in Two Vector Variables

Higher Spin Dirac Operators Between Spaces of Simplicial Monogenics in Two Vector Variables

Introduction to Clifford analysis

Almansi‐type theorems in Clifford analysis

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