Symmetries and Motions in Manifolds

TL;DR

This paper reviews the mathematical framework linking symmetries, Lie algebras, and Killing vectors in manifolds, extending to spinning spaces with Grassmann variables, and applies it to model electron motion in Schwarzschild spacetime.

Contribution

It generalizes symmetry concepts to include velocity-dependent transformations and spinning spaces, providing a formalism for pseudo-classical fermion dynamics in curved spacetime.

Findings

Derived the algebra of Killing tensors via Poisson brackets.

Extended formalism to include Grassmann variables for spinning particles.

Solved for electron motion in Schwarzschild spacetime using the developed approach.

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.