Covariant Approach of the Dynamics of Particles in External Gauge Fields, Killing Tensors and Quantum Gravitational Anomalies

TL;DR

This paper reviews covariant methods for analyzing particle dynamics in external gauge fields, emphasizing Killing tensors, and explores quantum anomalies related to classical conserved quantities, with applications to Kerr spacetime.

Contribution

It provides a covariant Hamiltonian framework for conserved quantities, details conditions for hidden symmetries in electromagnetic fields, and connects classical symmetries to quantum anomalies.

Findings

Explicit examples of Runge-Lenz type conserved quantities.

Conditions for electromagnetic fields to preserve hidden symmetries.

Construction of quantum symmetry operators and analysis of anomalies.

Abstract

We give an overview of the first integrals of motion of particles in the presence of external gauge fields in a covariant Hamiltonian approach. The special role of St\"ackel-Killing and Killing-Yano tensors is pointed out. Some nontrivial examples involving Runge-Lenz type conserved quantities are explicitly worked out. A condition of the electromagnetic field to maintain the hidden symmetry of the system is stated. A concrete realization of this condition is given by the Killing-Maxwell system and exemplified with the Kerr metric. Quantum symmetry operators for the Klein-Gordon and Dirac equations are constructed from Killing tensors. The transfer of the classical conserved quantities to the quantum mechanical level is analyzed in connection with quantum anomalies.