Conformal transformations and Clifford algebras
HTML articles powered by AMS MathViewer
- by Pertti Lounesto and Esko Latvamaa PDF
- Proc. Amer. Math. Soc. 79 (1980), 533-538 Request permission
Abstract:
A spinor representation for the conformal group of the real orthogonal space ${R^{p,q}}$ is given. First, the real orthogonal space ${R^{p,q}}$ is compactified by adjoining a (closed) isotropic cone at infinity. Then the nonlinear conformal transformations are linearized by regarding the conformal group as a factor group of a larger orthogonal group. Finally, the spin covering group of this larger orthogonal group is realized in the Clifford algebra ${R_{1 + p,q}}$ containing the Clifford algebra ${R_{p,q}}$ on the orthogonal space ${R^{p,q}}$. Explicit formulas for orthogonal transformations, translations, dilatations and special conformal transformations are given in Clifford language.References
- M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 (1964), no. suppl, suppl. 1, 3–38. MR 167985, DOI 10.1016/0040-9383(64)90003-5
- Claude C. Chevalley, The algebraic theory of spinors, Columbia University Press, New York, 1954. MR 0060497 J. Haantjes, Conformal representations of an n-dimensional euclidean space with a non-definite fundamental form on itself, Nederl. Akad. Wetensch. Proc. 40 (1937), 700-705.
- Max Karoubi, $K$-theory, Grundlehren der Mathematischen Wissenschaften, Band 226, Springer-Verlag, Berlin-New York, 1978. An introduction. MR 0488029
- H. A. Kastrup, Zur Physikalischen Deutung und darstellungsthoretischen Analyse der konformen Transformationen von Raum und Zeit, Ann. Physik (7) 9 (1961/62), 388–428 (German). MR 144690, DOI 10.1002/andp.19624640706 —, Some experimental consequences of conformal invariance at extremely high energies, Phys. Letters 3 (1962), 78-80.
- Ian R. Porteous, Topological geometry, Van Nostrand Reinhold Co., London-New York-Melbourne, 1969. MR 0254852
- Irving Ezra Segal, Mathematical cosmology and extragalactic astronomy, Pure and Applied Mathematics, Vol. 68, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0496337
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 79 (1980), 533-538
- MSC: Primary 15A66; Secondary 81C40
- DOI: https://doi.org/10.1090/S0002-9939-1980-0572296-4
- MathSciNet review: 572296