Symmetries and motions in manifolds

✍ Scribed by J.W. van Holten; R.H. Rietdijk

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 779 KB
• Volume: 11
• Category: Article
• ISSN: 0393-0440
• DOI: 10.1016/0393-0440(93)90079-t

✦ Synopsis

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 velocitydependent co-ordinate transformations naturally leads to the concept of Killing tensors. Via their Poisson brackets these tensors generate an 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 coordinates is parametrised 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.