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On certain identities involving basic spinors and curvature spinors

Published online by Cambridge University Press:  09 April 2009

H. A. Buchdahl
H. A. Buchdahl
Affiliation:
Physics Department, University of Tasmania, Hobart, Tasmania
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The spinor analysis of Infeld and van der Waerden [1] is particularly well suited to the transcription of given flat space wave equations into forms which constitute possible generalizations appropriate to Riemann spaces [2]. The basic elements of this calculus are (i) the skew-symmetric spinor Γμν, (ii) the hermitian tensor-spinor σκ(img)ν(generalized Pauli matrices), and (iii) the curvature spinor Ρμνκι. When one deals with wave equations in Riemann spaces V4 one is apt to be confronted with expressions of somewhat bewildering appearance in so far as they may involve products of a large number of σ-symbols many of the indices of which may be paired in all sorts of ways either with each other or with the indices of the components of the curvature spinors. Such expressions are generally capable of great simplification, but how the latter may be achieved is often far from obvious. It is the purpose of this paper to present a number of useful relations between basic tensors and spinors, commonly known relations being taken for granted [3], [4], [5]. That some of these new relations appear as more or less trivial consequences of elementary identities is largely the result of a diligent search for their simplest derivation, once they had been obtained in more roundabout ways.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 2 , Issue 3 , February 1962 , pp. 369 - 379
Copyright
Copyright © Australian Mathematical Society 1962

References

[1]Infeld, L. and van der Waerden, B.L., Die Wellengleichung des Elektrons in der allgemeinen Relativitätstheorie, Sitz. Preu. Akad. Wiss., 9 (1930), 380–401.Google Scholar
[2]E.g. Buchdahl, H. A., On the compatibility of relativistic wave equations for particles of higher spin in the presence of a gravitational field, Nuovo Cimento, 10 (1958), 96–103.CrossRefGoogle Scholar
[3]Harish-Chandra, , A note on the σ-symbols, Proc. Ind. Acad. Sci., 23 (1946), 152–163.CrossRefGoogle Scholar
[4]Corson, E.M., Tensors, spinors, and relativistic wave equations. Blackie, London (1953), Chap. II, §§ 6—13.Google Scholar
[5]Bade, W. L., and Jehle, H., An introduction to spinors, Revs. Mod. Phys., 25 (1953), 714–729.CrossRefGoogle Scholar
[6]Eisenhart, L. P., Riemannian geometry, Princeton University Press (1949), Chap. 2, p. 90.Google Scholar
[7]Buchdahl, H. A., On extended conformal transformations of spinors and spinor equations, Nuovo Cimento, 11 (1959), 496–506.CrossRefGoogle Scholar