Symmetries and Motions in Manifolds

Abstract

In these lectures the relations between symmetries, Lie algebras, Killing vectors and Noether's theorem are reviewed. A generalisation of the basic ideas to include velocity-dependend co-ordinate transformations naturally leads to the concept of Killing tensors. Via their Poisson brackets these tensors generate an {\em a priori} infinite-dimensional Lie algebra. The nature of such infinite algebras is clarified using the example of flat space-time. Next the formalism is extended to spinning space, which in addition to the standard real co-ordinates is parametrized also by Grassmann-valued vector variables. The equations for extremal trajectories (`geodesics') of these spaces describe the pseudo-classical mechanics of a Dirac fermion. We apply the formalism to solve for the motion of a pseudo-classical electron in Schwarzschild space-time.

19 pages. Lectures at 28th Winter School of Theoretical Physics, Karpacz (Poland, 1992) by J.W. van Holten

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• Stability of circular orbits of spinning particles in Schwarzschild-like space-times
• An index theorem for non-standard Dirac operators
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• Uses of Killing and Killing-Yano Tensors
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