Geometric significance of the spinor Lie derivative. I
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Abstract
In a previous article, the writer explored the geometric foundation of the generally covariant spinor calculus. This geometric reasoning can be extended quite naturally to include the Lie covariant differentiation of spinors. The formulas for the Lie covariant derivatives of spinors, adjoint spinors, and operators in spin space are deduced, and it is observed that the Lie covariant derivative of an operator in spin space must vanish when taken with respect to a Killing vector. The commutator of two Lie covariant derivatives is calculated; it is noted that the result is consistent with the geometric interpretation of the Jacobi identity for vectors. Lie current conservation is seen to spring from the result that the operator of spinor affine covariant differentiation commutes with the operator of spinor Lie covariant differentiation with respect to a Killing vector. It is shown that differentiations of the spinor field defined geometrically are Lorentz-covariant.
Keywords
Covariant Derivative Geometric Interpretation Previous Article Killing Vector Jacobi IdentityPreview
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References
- 1.V. Jhangiani,Found. Phys. 7, 111 (1977).Google Scholar
- 2.M. Ryan and L. C. Shepley,Homogeneous Relativistic Consmologies, (Princeton Univ. Press, Princeton, 1975).Google Scholar
- 3.S. W. Hawking and R. Ellis,The Large-Scale Structure of Space-Time (Cambridge Univ. Press, London, 1973).Google Scholar
- 4.C. W. Misner, K. S. Thorne, and J. A. Wheeler,Gravitation (Freeman, San Francisco, 1973).Google Scholar
- 5.M. M. Hatalkar,Phys. Rev. 94, 1472 (1954).Google Scholar
- 6.W. Unruh,Phys. Rev. D 10, 3194 (1974).Google Scholar
- 7.E. Cartan,The Theory of Spinors (MIT Press, Hermann, Paris, 1966).Google Scholar
- 8.B. K. Berger, Homothetic and Conformal Motions in Spacelike Slices of Solutions of Einstein's Equations, Yale Univ., preprint.Google Scholar
- 9.K. Yano,Theory of Lie Derivatives (North-Holland, Amsterdam, 1955).Google Scholar
- 10.S. S. Schweber, Relativistic Quantum Mechanics, inThe Mathematics of Physics and Chemistry, Vol. 2, H. Margenau and M. Murphy, eds. (Van Nostrand, New York, 1964).Google Scholar
- 11.W. L. Bade and H. Jehle,Rev. Mod. Phys. 25, 714 (1953).Google Scholar
- 12.Sci. Am. 234(1), 61–62 (January 1976).Google Scholar
- 13.V. Jhangiani, PhD thesis, University of California at Santa Barbara (June 1978).Google Scholar