Abstract
We consider a special extension of Clifford algebras and show that these generalized Clifford algebras are naturally equipped with a metric defined by a fundamental form of degreen which isSL(n, φ) ⊗SL(n,φ) invariant. Using the embedding of the quaternions in the generalized Clifford algebras, in the Hermitian limit, we obtain an algebraic description of the inclusion of the Minkowski space into the hyperspin manifold.
Similar content being viewed by others
References
Brauer, R., and Weyl, H. (1935).American Journal of Mathematics,57, 425.
Cartan, E. (1898).Annales de la Faculté des Sciences de Toulouse, Mémoire 2.
Cartan, E. (1909). InEncyclopédie des Sciences Pures et Appliquées, Vol. I, Gauthier-Villars, Paris.
Cartan, E. (1938).The Theory of Spinors, Hermann, Paris [Reprinted, Hermann, Paris, 1960].
Duff, M. J., Nilsson, B. E. W., and Pope, C. N. (1986).Physics Reports,130, 1.
Edérlyi, A. (1955).Higher Transcendental Functions, Vol. III, McGraw-Hill, New York, pp. 212–217.
Finkelstein, D. (1986).Physical Review Letters,56, 1532.
Finkelstein, D., Finkelstein, S. R., and Holm, C. (1986).International Journal of Theoretical Physics,25, 441.
Fleury, N., and Rausch de Traubenberg, M.Journal of Mathematical Physics,33, 3356.
Fleury, N., Rausch de Traubenberg, M. and Yamaleev, R. M. (1991). Extended complex numbers and connected trigonometry,Journal of Mathematics Analysis and Applications (to appear).
Jordan, P. (1932).Zeitschrift für Physik,80, 285.
Kwasniewski, A. K. (1985).Journal of Mathematical Physics,26, 2234.
Kwasniewski, A. K. (1992).Advances in Applied Clifford Algebras,2, 107.
Long, F. W. (1976).Journal of the London Mathematical Society,13, 438.
Morinaga, K., and Nono, T. (1952).Journal of Science of Hiroshima University Series A Mathematics Physics Chemistry,16, 13.
Morris, A. O. (1967).Quarterly Journal of Mathematics (Oxford),18, 7.
Morris, A. O. (1968).Quarterly Journal of Mathematics (Oxford),19, 289.
Penrose, R., and Rindler, W. (1984).Spinors and Space-Time, Cambridge University, Cambridge.
Popovici, I., and Ghéorghe, C. (1966a).Comptes Rendus de l'Académie des Sciences Paris Série A,262, 682.
Popovici, I. and Ghéorghe, C. (1966b).Revue Roumaine de Mathématiques Pures et Appliquées,11, 989.
Ramakrishnan, A., Chandrasekaran, P. S., Ranganathan, N. R., Santhanan, T. S., and Vesudevan, R. (1969).Journal of Mathematical Analysis and Applications,27, 164.
Schwinger, J. (1960).Proceedings of the National Academy of Science U.S.A.,46, 570.
Sylvester, J. J. (1884).American Journal of Mathematics,6, 270.
Thomas, E. (1974).Glasgow Mathematical Journal,15, 74.
Wene, G. P. (1984).Journal of Mathematical Physics,25, 414.
Weyl, H. (1932).The Theory of Groups and Quantum Mechanics, E. P. Dutton, pp. 272–280 [Reprinted, Dover, New York, 1950].
Witten, E. (1981).Nuclear Physics B,186, 412.
Yamazaki, K. (1964).Journal of the Faculty of Science University of Tokyo Section 1,10, 147.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fleury, N., de Traubenberg, M.R. & Yamaleev, R.M. Generalized Clifford algebras and hyperspin manifolds. Int J Theor Phys 32, 503–516 (1993). https://doi.org/10.1007/BF00673754
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00673754