Holographic coordinates

TL;DR

This paper explores the conformal symmetry of the Laplace equation in 2D, connecting differential geometry, quantum systems, and physics through the Lie algebra of the conformal group and holographic coordinates.

Contribution

It introduces a novel representation of the conformal Lie algebra using solutions of the Laplace equation, linking diverse mathematical and physical concepts in a unified framework.

Findings

Representation of conformal Lie algebra via Laplace solutions

Connection between differential geometry and quantum systems

Potential applications in physics discussed

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.