A key to the projective model of homogeneous metric spaces

TL;DR

This paper develops a vector algebra-based translation of the projective model for Cayley-Klein geometries, enabling easier application to Minkowski, de-Sitter, and anti-de-Sitter space-times, and unifies the treatment of isometries.

Contribution

It translates the projective model of Cayley-Klein geometries into vector algebra, making it more accessible for physicists and mathematicians, and applies it to various space-times.

Findings

Unified algebraic description of isometries in Minkowski, de-Sitter, and anti-de-Sitter spaces

Representation of Poincare group actions using Clifford algebra

Facilitation of the projective model's adoption in physics and applied mathematics

Abstract

A metric introduced on a projective space yields a homogeneous metric space known as a Cayley-Klein geometry. This construction is applicable not only to Euclidean and non-Euclidean spaces but also to kinematic spaces (space-times). A convenient algebraic framework for Cayley-Klein geometries called the projective model is developed in [1, 2]. It is based on Grassmann and Clifford algebras and provides a set of algebraic tools for modeling points, lines, planes and their geometric transformations such as projections and isometries. Isometry groups and their Lie algebras find a natural and intuitive expression in the projective model. The aim of this paper is to translate the foundational concepts of the projective model from the language of projective geometry to a more familiar language of vector algebra and thereby facilitate its spread and adoption among physicists and applied mathematicians. I apply the projective model to Minkowski, de-Sitter, and anti-de-Sitter space-times in two dimensions. In particular, I show how the action of the Poincare group can be captured by the Clifford algebra in a uniform fashion with respect to rotations (boosts) and translations.