Holographic coordinates

Abstract

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...