Construction of Arbitrary Order Conformally Invariant Operators in Higher Spin Spaces

Ding, Chao; Walter, Raymond; Ryan, John

doi:10.1007/s12220-017-9766-7

Cited by 15 publications

(14 citation statements)

References 30 publications

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“…Some basic integral formulas for the higher spin Laplace operator were also found recently, see [9]. In [10], the authors studied the construction of arbitrary order conformally invariant differential operators in higher spin spaces. These are fermionic operators when the orders are odd and bosonic operators when the orders are even.…”

Section: Introductionmentioning

confidence: 93%

“…The (2j − 1)-th order (j > 1) conformally invariant differential operator in higher spin theory, called the (2j − 1)-th order fermionic operator [10], is given by…”

Section: Preliminariesmentioning

confidence: 99%

“…where λ 2j−1 is a nonzero constant given in [10], and Z k (u, v) is the reproducing kernel for M k , which satisfies…”

Section: Preliminariesmentioning

confidence: 99%

“…In the past few decades, many authors [6,5,8,10,11,13] have been working on generalizations of classical Clifford analysis to the so-called higher spin theory. This investigates higher spin operators acting on functions on R m taking values in irreducible representations of the Spin group.…”

Section: Introductionmentioning

confidence: 99%

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Integral Formulas for Higher Order Conformally Invariant Fermionic Operators

Ding

2019

Adv. Appl. Clifford Algebras

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In this paper, we establish higher order Borel-Pompeiu formulas for conformally invariant fermionic operators in higher spin theory, which is the theory of functions on m-dimensional Euclidean space taking values in arbitrary irreducible representations of the Spin group. As applications, we provide higher order Cauchy's integral formulas for those fermionic operators. This paper continues the work of building up basic integral formulas for conformally invariant differential operators in higher spin theory.Mathematics Subject Classification (2010). Primary 30Gxx, Secondary 42Bxx, 46F12, 58Jxx.

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Section: Introductionmentioning

confidence: 93%

“…The (2j − 1)-th order (j > 1) conformally invariant differential operator in higher spin theory, called the (2j − 1)-th order fermionic operator [10], is given by…”

Section: Preliminariesmentioning

confidence: 99%

“…where λ 2j−1 is a nonzero constant given in [10], and Z k (u, v) is the reproducing kernel for M k , which satisfies…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Integral Formulas for Higher Order Conformally Invariant Fermionic Operators

Ding

2019

Adv. Appl. Clifford Algebras

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“…Higher spin representations have been considered in recent years in the context of string theory, investigations of the matter spectrum, and Clifford analysis [58,59,60,61,62,63,64,65,66,67] and they are therefore of special interest. With respect to higher spin note that degree n-polynomials in z ∈ C form the carrier module for the n + 1-dimensional irreducible spin representation of SU (2, C) [39].…”

Section: Higher Spinmentioning

confidence: 99%

Holographic coordinates

Ulrych¹

2018

Journal of Mathematical Analysis and Applications

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The Laplace equation in the two-dimensional Euclidean plane is considered in the context of the inverse stereographic projection. The Lie algebra of the conformal group as the symmetry group of the Laplace equation can be represented solely in terms of the solutions and derivatives of the solutions of the Laplace equation. It is then possible to put contents from differential geometry and quantum systems, like the Hopf bundle, relativistic spin, bicomplex numbers, and the Fock space into a common context. The basis elements of the complex numbers, considered as a Clifford paravector algebra, are reinterpreted as differential tangent vectors referring to dilations and rotations. In relation to this a homogeneous space is defined with the Lie algebra of the conformal group, where dilations and rotations are the coset representatives. Potential applications in physics are discussed. et al. [25]. Furthermore, a hierarchy of projective geometric spaces and Möbius geometries can be introduced based on Vahlen matrices [26,27,28,29]. Maks [30] investigated explicitly the sequence of Möbius geometries represented in terms of Clifford algebras, which is considered also in [31] in a bicomplex matrix representation. In relation to particle physics one may assume that projections of a more general geometry can arise as part of the measurement process via electromagnetic forces. In accordance with the holographic principle Möbius geometries and Möbius transformations can provide here a new perspective on the experimental data.Atiyah, Manton, and Schroers [32,33,34,35] describe electrons, protons, neutrinos, and neutrons within a model that has been inspired by Skyrme's baryon theory [36]. The Skyrme model became popular when Witten came up with the idea that baryons arise as solitons of the classical meson fields [37]. The Skyrme theory turned out to be remarkably successful in describing baryons in the previous decades [38]. Thus matter is supposed to arise from a pure geometric foundation [39,40]. Solitons are usually considered in the context of gauge transformations, which relate in a mathematical context to fiber bundles and Cartan geometries [22].With regards to the topics discussed above research in two-dimensional conformal field theories [41,42,43,44,45] has turned out to be important, also because conceptual insights can be obtained more easily in a lower dimensional geometry. Motivation to proceed in this direction is also provided by [31]. In this article the two-dimensional Euclidean plane and the conformal symmetries, which refer to the Laplace equation in R 2 , are fundamental in a sequence of higher dimensional geometries and Clifford algebras. Therefore, one can ask the question how the Laplace equation can be embedded into higher dimensional geometries. Here concepts like the stereographic projection should play an important role. In this sense one can consider the Laplace equation in the base space S 2 instead of R 2 by means of what is denoted here as holographic coordinates. This conceptual change is adopt...

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Special bases for solutions of a generalized Maxwell system in 3‐dimensional space

Ding

Bock

Gürlebeck

2018

Math Methods in App Sciences

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The aim of this paper is to study complete polynomial systems in the kernel space of conformally invariant differential operators in higher spin theory. We investigate the kernel space of a generalized Maxwell operator in 3‐dimensional space. With the already known decomposition of its homogeneous kernel space into 2 subspaces, we investigate first projections from the homogeneous kernel space to each subspace. Then, we provide complete polynomial systems depending on the given inner product for each subspace in the decomposition. More specifically, the complete polynomial system for the homogenous kernel space is an orthogonal system wrt a given Fischer inner product. In the case of the standard inner product in L2 on the unit ball, the provided complete polynomial system for the homogeneous kernel space is a partially orthogonal system. Further, if the degree of homogeneity for the respective subspaces in the decomposed kernel spaces approaches infinity, then the angle between the 2 subspaces approaches π/2.

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Construction of Arbitrary Order Conformally Invariant Operators in Higher Spin Spaces

Cited by 15 publications

References 30 publications

Integral Formulas for Higher Order Conformally Invariant Fermionic Operators

Integral Formulas for Higher Order Conformally Invariant Fermionic Operators

Holographic coordinates

Special bases for solutions of a generalized Maxwell system in 3‐dimensional space

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