Clifford Algebras and Possible Kinematics

TL;DR

This paper explores the use of Clifford algebras to unify the representation of various 2D kinematical groups and geometries, providing new insights into their structure and conformal models.

Contribution

It introduces a two-parameter family of Clifford algebras that unify Lie algebras and kinematical groups for 2D spacetimes, expanding the understanding of their geometric and algebraic properties.

Findings

Clifford algebras can represent all but one of the 2D kinematical groups.

Homogeneous spacetimes are associated with Cayley-Klein geometries.

Conformal models for these spacetimes are constructed.

Abstract

We review Bacry and Levy-Leblond's work on possible kinematics as applied to 2-dimensional spacetimes, as well as the nine types of 2-dimensional Cayley-Klein geometries, illustrating how the Cayley-Klein geometries give homogeneous spacetimes for all but one of the kinematical groups. We then construct a two-parameter family of Clifford algebras that give a unified framework for representing both the Lie algebras as well as the kinematical groups, showing that these groups are true rotation groups. In addition we give conformal models for these spacetimes.