Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 037, 14 pages      arXiv:1102.0095      https://doi.org/10.3842/SIGMA.2011.037
Contribution to the Proceedings of the Conference “Symmetries and Integrability of Difference Equations (SIDE-9)”

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

Mihai Visinescu
Department of Theoretical Physics, National Institute for Physics and Nuclear Engineering, P.O. Box M.G.-6, Magurele, Bucharest, Romania

Received February 02, 2011, in final form March 28, 2011; Published online April 05, 2011

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 Stäckel-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.

Key words: hidden symmetries; Killing tensors; Killing-Maxwell system; quantum anomalies.

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