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

SIGMA 3 (2007), 079, 29 pages      arXiv:0707.2869      https://doi.org/10.3842/SIGMA.2007.079

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

Abstract
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.

pdf (571 kb)   ps (1412 kb)   tex (1366 kb)

References

  1. Bacry H., Lévy-Leblond J., Possible kinematics, J. Math. Phys. 9 (1968), 1605-1614.
  2. Ballesteros A., Herranz F.J., Superintegrability on three-dimensional Riemannian and relativistic spaces of constant curvature, SIGMA 2 (2006), 010, 22 pages, math-ph/0512084.
  3. Brannan D.A., Esplen M.F., Gray J.J., Geometry, Cambridge University Press, Cambridge, 1999.
  4. Cannata R., Catoni F., Catoni V., Zampetti P., Two-dimensional hypercomplex numbers and related trigonometries and geometries, Adv. Appl. Clifford Algebr. 14 (2004), 47-68.
  5. Gray J.J., Ideas of space, 2nd ed., Clarendon Press, Oxford, 1979.
  6. Gromov N., The Jordan-Schwinger representations of Cayley-Klein groups I: The orthogonal groups, J. Math. Phys. 31 (1990), 1047-1053.
  7. Gromov N., Transitions: contractions and analytic continuations of the Cayley-Klein groups, Internat. J. Theoret. Phys. 29 (1990), 607-620.
  8. Gromov N., The Gelfand-Tsetlin representations of the orthogonal Cayley-Klein algebras, J. Math. Phys. 33 (1992), 1363-1373.
  9. Gromov N.A., Moskaliuk S.S., Special orthogonal groups in Cayley-Klein spaces, Hadronic J. 18 (1995), 451-483.
  10. Gromov N.A., Moskaliuk S.S., Classification of transitions between groups in Cayley-Klein spaces and kinematic groups, Hadronic J. 19 (1996), 407-435.
  11. Fjelstad P., Gal S.G., Two-dimensional geometries, topologies, trigonometries and physics generated by complex-type numbers, Adv. Appl. Clifford Algebr. 11 (2001), 81-107.
  12. Harkin A.A., Harkin J.B., Geometry of generalized complex numbers, Math. Mag. 77 (2004), 118-129.
  13. Herranz F.J., Ortega R., Santander M., Homogeneous phase spaces: the Cayley-Klein framework, Mem. Real Acad. Cienc. Exact. Fís. Natur. Madrid 32 (1998), 59-84, physics/9702030.
  14. Herranz F.J., Ortega R., Santander M., Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry, J. Phys. A: Math. Gen. 33 (2000), 4525-4551, math-ph/9910041.
  15. Herranz F.J., Santander M., Conformal symmetries of spacetimes, J. Phys. A: Math. Gen. 35 (2002), 6601-6618, math-ph/0110019.
  16. Herranz F.J., Santander M., Conformal compactification of spacetimes, J. Phys. A: Math. Gen. 35 (2002), 6619-6629, math-ph/0110019.
  17. Inonu E., Wigner E.P., On the contraction of groups and their representations, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 510-524.
  18. Katz V., A history of mathematics: an introduction, 2nd ed., Addison Wesley Longman, Inc., New York, 1998.
  19. Klein F., Über die sogenannte nicht-Euklidische geometrie, Gesammelte Math. Abh. I (1921), 254-305, 311-343, 344-350, 353-383.
  20. McRae A.S., The Gauss-Bonnet theorem for Cayley-Klein geometries of dimension two, New York J. Math. 12 (2006), 143-155.
  21. Penrose R., The road to reality, Alfred A. Knopf, New York, 2005.
  22. Pimenov R.I., Unified axiomatics of spaces with the maximum group of motions, Litovsk. Mat. Sb. 5 (1965), 457-486.
  23. Fernández Sanjuan M.A., Group contraction and the nine Cayley-Klein geometries, Internat. J. Theoret. Phys. 23 (1984), 1-14.
  24. Santander M., The Hyperbolic-AntiDeSitter-DeSitter triality, Pub. de la RSME 5 (2005), 247-260.
  25. Sattinger D.H., Weaver O.L., Lie groups and algebras with applications to physics, geometry, and mechanics, Springer-Verlag, New York, 1986.
  26. Sommerville D.M.Y., Classification of geometries with projective metrics, Proc. Edinb. Math. Soc. 28 (1910-1911), 25-41.
  27. Walker S., The non-Euclidean style of Minkowskian relativity, in The Symbolic Universe, Editor J. Gray, Oxford University Press, Oxford, 1999, 91-127.
  28. Yaglom I.M., A simple non-Euclidean geometry and its physical basis: an elementary account of Galilean geometry and the Galilean principle of relativity, Heidelberg Science Library, translated from the Russian by A. Shenitzer, with the editorial assistance of B. Gordon, Springer-Verlag, New York - Heidelberg, 1979.

Previous article   Next article   Contents of Volume 3 (2007)