Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1992-05-20
J.Geom.Phys.11:559,1993
Physics
High Energy Physics
High Energy Physics - Theory
19 pages. Lectures at 28th Winter School of Theoretical Physics, Karpacz (Poland, 1992) by J.W. van Holten
Scientific paper
10.1016/0393-0440(93)90079-T
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.
Rietdijk R. H.
van Holten Jan-Willem
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