Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 079, 29 pages      arXiv:0707.2869

Clifford Algebras and Possible Kinematics

Alan S. McRae
Department of Mathematics, Washington and Lee University, Lexington, VA 24450-0303, USA

Received April 30, 2007, in final form July 03, 2007; Published online July 19, 2007

We review Bacry and Lévy-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.

Key words: Cayley-Klein geometries; Clifford algebras; kinematics.

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