Abstract
An algebraic description of basic discrete symmetries (space reversal P, time reversal T, and their combination PT) is studied. Discrete subgroups of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. In accordance with a division ring structure, a complete classification of automorphism groups is established for the Clifford algebras over the field of real numbers. The correspondence between eight double coverings (Dąbrowski groups) of the orthogonal group and eight types of the real Clifford algebras is defined with the use of isomorphisms between the automorphism groups and finite groups. Over the field of complex numbers there is a correspondence between two nonisomorphic double coverings of the complex orthogonal group and two types of complex Clifford algebras. It is shown that these correspondences associate with a well-known Atiyah–Bott–Shapiro periodicity. Generalized Brauer–Wall groups are introduced on the extended sets of the Clifford algebras. The structure of the inequality between the two Clifford–Lipschitz groups with mutually opposite signatures is elucidated. The physically important case of the two different double coverings of the Lorentz groups is considered in details.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Abłamowicz, R. (1996). Clifford algebra computations with Maple. In Proc. Clifford (Geometric) Algebras, Banff, Alberta Canada, 1995. W. E. Baylis, ed., Birkhäuser, Boston, pp. 463–501.
Abłamowicz, R. (1998). Spinor representations of Clifford algebras: A symbolic approach, CPC thematic issue ‘Computer Algebra in Physics Research.’ Physics Communications 115, 510–535.
Abłamowicz, R. (2000). CLIFFORD—Maple V package for Clifford algebra computations, Ver. 4, http://math.tntech.edu/rafal/cliff4/.
Atiyah, M. F., Bott, R., and Shapiro, A. (1964). Clifford modules. Topology 3(Suppl. 1), 3–38.
Berestetskii, V. B., Lifshitz, E. M., and Pitaevskii, L. P. (1982). Quantum Electrodynamics Course of Theoretical Physics, Vol. 4, 2nd edn., Pergamon Press, Oxford.
Blau, M. and Dabrowski, L. (1989). Pin structures on manifolds quotiented by discrete groups. Journal of Geometry and Physics 6, 143–157.
Budinich, P. and Trautman, A. (1987). An introduction to the spinorial chessboard. Journal of Geometry and Physics 4, 363–390.
Budinich, P. and Trautman, A. (1988). The Spinorial Chessboard, Springer, Berlin.
Cahen, M., Gutt, S., and Trautman, A. (1995). Pin structures and the modified Dirac operator. Journal of Geometry and Physics 17, 283–297.
Cahen M., Gutt, S., and Trautman, A. (1998). Pin structures and the Dirac operator on real projective spaces and quadrics. In Clifford Algebras and Their Applications in Mathematical Physics, K. Habetha, V. Dietrich, and G. Tank, eds., Kluwer Academic Publishers, 391–399.
Carlip, S. and De Witt-Morette, C. (1988). Where the sign of the metric makes a difference. Physical Review Letters 60, 1599–1601.
Chamblin, A. (1994). On the obstructions to non-Cliffordian Pin structures. Communications in Mathematical Physics 164, 67–87.
Chevalley, C. (1954). The Algebraic Theory of Spinors, Columbia University Press, New York.
Chevalley, C. (1955). The construction and study of certain important algebras. Publications of Mathematical Society of Japan, No. 1, Herald Printing, Tokyo.
Choquet-Bruhat, Y., De Witt-Morette, C., and Dillard-Bleick, M. (1982). Analysis, Manifolds and Physics, North-Holland, Amsterdam.
Dabrowski, L. (1988). Group Actions on Spinors, Bibliopolis, Naples.
De Witt-Morette, C. and De Witt, B. (1990). The Pin groups in physics. Physical Review D 41, 1901–1907.
De Witt-Morette, C. (1989). Topological obstructions to Pin structures. In Geom. and Algebraic Aspects of Nonlinear Field Theory: Proc. Meet. Amalfi, May, 23–28 (1988), 113–118.
De Witt-Morette, C. and Gwo, Shang-Jr. (1990). One Spin Group; Two Pin Groups, Gauss Symposium, Gaussanium Inst.
De Witt-Morette, C., Gwo, Shang-Jr., and Kramer, E. (1997). Spin or Pin?, preprint; http://www. rel.ph.utexas.edu/Members/cdewitt/SpinOrPin1.ps.
Friedrich, Th. and Trautman, A. (1999). Spin Spaces, Lipschitz Groups, and Spinor Bundles, preprint SFB 288, N 362, TU-Berlin.
Hurwitz, A. (1923). Uber die Komposition der quadratischen Formen. Math. Ann. 88, 1–25.
Karoubi, M. (1979). K-Theory. An Introduction, Springer-Verlag, Berlin.
Kirby, R. C. and Taylor, L. R. (1989). Pin structures on low-dimensional manifolds. In London Math. Soc. Lecture Notes, No. 151, Cambridge University Press, Cambridge.
Lipschitz, R. (1886). Untersuchungen über die Summen von Quadraten, Max Cohen und Sohn, Bonn.
Lounesto, P. (1981). Scalar products of spinors and an extension of Brauer—Wall groups. Found. Phys. 11, 721–740.
Lounesto, P. (1993). Clifford algebras and Hestenes spinors. Found. Phys. 23, 1203–1237.
Lounesto, P. (1997). Clifford Algebras and Spinors, Cambridge University Press, Cambridge.
Minkowski, H. (1909). Raum und Zeit, Phys. Zs. 10, 104.
Miralles, D., Parra, J. M., and Vaz, Jr., J. (in press). Signature change and Clifford algebras, preprint math-ph/0003041; Int. J. Theor. Phys.
Porteous, I. R. (1969). Topological Geometry, van Nostrand, London.
Radon, J. (1922). Lineare Scharen orthogonaler Matrizen. Abh. Math. Seminar Hamburg 1, 1–24.
Rashevskii, P. K. (1957). The theory of Spinors. (in Russian) Uspekhi Mat. Nauk 10, 3–110 (1955); (in English) Amer. Math. Soc. Transl. (Ser.2) 6, 1.
Salingaros, N. (1981). Realization, extension, and classification of certain physically important groups and algebras. J. Math. Phys. 22, 226–232.
Salingaros, N. (1982). On the classification of Clifford algebras and their relation to spinors in n dimensions. J. Math. Phys. 23(1).
Salingaros, N. (1984). The relationship between finite groups and Clifford algebras. J. Math. Phys. 25, 738–742.
Varlamov, V. V. (1999). Fundamental automorphisms of Clifford algebras and an extension of Dabrowski Pin groups. Hadronic Journal 22, 497–533.
Wall, C. T. C. (1964). Graded Brauer groups. J. Reine und Angew. Math. 213, 187–199.
Wigner, E. P. (1964). Unitary representations of the inhomogeneous Lorentz group including reflections. In Group Theoretical Concepts and Methods in Elementary Particle Physics, F. Gürsey, ed., Gordon and Breach, New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Varlamov, V.V. Discrete Symmetries and Clifford Algebras. International Journal of Theoretical Physics 40, 769–805 (2001). https://doi.org/10.1023/A:1004122826609
Issue Date:
DOI: https://doi.org/10.1023/A:1004122826609