×

Remarks on pure spinors. (English) Zbl 0602.15021

The authors work on Clifford algebras and spinor structures in higher dimensions corresponding to recent researches on fundamental interactions and their unification in physics. In this note, they briefly (i) describe the relation between spinor connections on low-dimensional spheres and simple gauge configurations, (ii) outline an approach to the notion of pure spinors in spaces of nonneutral signature and, in particular, in conformal extensions of spacetime, (iii) indicate some topological nontrivial configurations associated with pure spinors, and (iv) show how the ’purity constraint’ may lead to ’mass term’ in the Weyl equation for pure spinors in a space of signature (3,4).
Reviewer: T.Nôno

MSC:

15A66 Clifford algebras, spinors
53C27 Spin and Spin\({}^c\) geometry
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] RodriguesO.,Journal de Math. (Liouville) 5, 380 (1840).
[2] HamiltonW., manuscript of 16 October 1843, printed inThe Mathematical Papers of Sir William Rowan Hamilton, vol. 3, University Press, Cambridge, 1967, pp. 103-105.
[3] CayleyA.,Phil. Mag. 7, 133 (1854).
[4] Lipschitz, R.,Untersuchungen ueber die Summen von Quadraten, Bonn, 1886; cf. alsoAnn. Math. 69, 247 (1959).
[5] CliffordW. K.,Amer. J. Math. 1, 350 (1878). · JFM 10.0297.02 · doi:10.2307/2369379
[6] CartanE.,Bull. Soc. Math. France 41, 53 (1913).
[7] CartanE.,La théorie des spineurs I and II, Actualités Sci. et Industr., No. 643 and 701, Hermann, Paris, 1938.
[8] ChevalleyC.,The Algebraic Theory of Spinors, Columbia University Press, New York, 1954.
[9] PenroseR. and RindlerW.,Spinors and Space-Time, I and II, Cambridge University Press, Cambridge, 1984 and 1986.
[10] BrauerR. and WeylH.,Amer. J. Math. 57, 425 (1935). · Zbl 0011.24401 · doi:10.2307/2371218
[11] NewmanE. T. and PenroseR.,J. Math. Phys. 3, 566 (1962). · Zbl 0108.40905 · doi:10.1063/1.1724257
[12] WittenE.,Commun. Math. Phys. 80, 381 (1981). · Zbl 1051.83532 · doi:10.1007/BF01208277
[13] Trautman, A., ?Differential Geometry and Spinors?, Lectures at SISSA, Trieste, May 1984 (unpublished).
[14] Dabrowski, L. and Trautman, A., ?Spinor Structures on Spheres and Projective Spaces?, SISSA preprint, Trieste, 1985. · Zbl 0599.53030
[15] DeAlfaroV., FubiniS., and FurlanG.,Phys. Lett. 65B, 163 (1976).
[16] ChernS. S. and SimonsJ.,Ann. Math. 99, 48 (1974). · Zbl 0283.53036 · doi:10.2307/1971013
[17] BelavinA. A. et al.,Phys. Lett. 59B, 85 (1975).
[18] Landi, G., ?The Natural Spinor Connection onS 8 is a Gauge Field?, SISSA preprint, Trieste, 1985.
[19] GrossmanB., KephartT. K., and StasheffJ. D.,Commun. Math. Phys. 96, 431 (1984). · Zbl 0575.58034 · doi:10.1007/BF01212529
[20] BourbakiN.,Algèbre, Ch. IX, §4 and 9, Hermann, Paris, 1959.
[21] ReggeT., in B. S.DeWitt and R.Stora (eds.),Relativité, Groupes et Topologie II, North-Holland, Amsterdam, 1984.
[22] PorteousI. R.,Topological Geometry, 2nd edn., Van Nostrand-Reinhold, London, 1979.
[23] Furlan, P. and Raçzka, R., ?A Pure Spinor Non-Linear Sigma-Type Model?,Phys. Lett. B (in print) and ?Non-Linear Spinor Representations?, SISSA preprint, Trieste, 1985.
[24] Benn, I. M. and Tucker, R. W., ?Pure Spinors and Real Clifford Algebras?, University of Lancaster preprint, 1984.
[25] Giler, S., Kosi?ski, P., and Rembieli?ski, J., ?On SO(p, q) Pure Spinors?, University of ?ód? preprint, 1985.
[26] TyrrellJ. A. and SempleJ. G.,Generalized Clifford Parallelism, Cambridge University Press, Cambridge, 1971.
[27] HughstonL. P. and HurdT. R.,Phys. Rep. 100, 273 (1983). · doi:10.1016/0370-1573(83)90003-0
[28] HughstonL. P., article in W.Rindler and A.Trautman (eds.),Gravitation and Geometry, a volume in honour of Ivor Robinson, Bibliopolis, Naples, 1986.
[29] ZundJ. D.,Ann. Math. Pura Appl. 82, 381 (1969);110, 29 (1976). · Zbl 0185.50001 · doi:10.1007/BF02410802
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.