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

SIGMA 7 (2011), 037, 14 pages      arXiv:1102.0095
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

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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  1. Benenti S., Intrinsic characterization of the variable separation in the Hamilton-Jacobi equation, J. Math. Phys. 38 (1997), 6578-6602.
  2. Yano K., Some remarks on tensor fields and curvature, Ann. of Math. (2) 55 (1952), 328-346.
  3. Gibbons G.W., Rietdijk R.H., van Holten J.W., SUSY in the sky, Nuclear Phys. B 404 (1993), 42-64, hep-th/9303112.
  4. Carter B., McLenaghan R.G., Generalized total angular momentum operator for the Dirac equation in curved space-time, Phys. Rev. D 19 (1979), 1093-1097.
  5. Cariglia M., Quantum mechanics of Yano tensors: Dirac equation in curved spacetime, Classical Quantum Gravity 21 (2004), 1051-1077, hep-th/0305153.
  6. Frolov V.P., Hidden symmetries of higher-dimensional black hole spacetimes, Progr. Theor. Phys. Suppl. (2008), no. 172, 210-219, arXiv:0712.4157.
  7. Ianus S., Visinescu M., Vîlcu G.-E., Conformal Killing-Yano tensors on manifolds with mixed 3-structures, SIGMA 5 (2009), 022, 12 pages, arXiv:0902.3968.
  8. van Holten J.W., Covariant Hamiltonian dynamics, Phys. Rev. D 75 (2007), 025027, 9 pages, hep-th/0612216.
  9. Carter B., Separability of the Killing-Maxwell system underlying the generalized singular momentum constant in the Kerr-Newman black hole metrics, J. Math. Phys. 28 (1987), 1535-1538.
  10. Horváthy P.A., Ngome J.-P., Conserved quantities in anon-abelian monopole field, Phys. Rev. D 79 (2009), 127701, 4 pages, arXiv:0902.0273.
  11. Ngome J.-P., Curved manifolds with conserved Runge-Lenz vectors, J. Math. Phys. 50 (2009), 122901, 13 pages, arXiv:0908.1204.
  12. Visinescu M., Higher order first integrals of motion in a gauge covariant Hamiltonian framework, Modern Phys. Lett. A 25 (2010), 341-350, arXiv:0910.3474.
  13. Igata T., Koike T., Isihara H., Constants of motion for constrained hamiltonian systems, arXiv:1005.1815.
  14. Crampin M., Hidden symmetries and Killing tensors, Rep. Math. Phys. 20 (1984), 31-40.
  15. Vaman D., Visinescu M., Spinning particles in Taub-NUT space, Phys. Rev. D 57 (1998), 3790-3793, hep-th/9707175.
  16. Marquette I., Generalized Kaluza-Klein monopole, quadratic algebras and ladder operators, arXiv:1103.0374.
  17. Iwai T., Katayama N., On extended Taub-NUT metrics, J. Geom. Phys. 12 (1993), 55-75.
  18. Kashiwada T., On conformal Killing tensor, Natur. Sci. Rep. Ochanomizu Univ. 19 (1968), 67-74.
  19. Kress J., Generalized conformal Killing-Yano tensors: applications to electrodynamics, PhD Thesis, University of Newcastle, 1997.
  20. Floyd R., The dynamics of Kerr fields, PhD Thesis, London University, 1973.
  21. Penrose R., Naked singularities, Ann. New York Acad. Sci. 224 (1973), 125-134.
  22. Gibbons G.W., Ruback R.J., The hidden symmetries of multi-center metrics, Comm. Math. Phys. 115 (1988), 267-300.
  23. Visinescu M., Generalized Taub-NUT metrics and Killing-Yano tensors, J. Phys. A: Math. Gen. 33 (2000), 4383-4391, hep-th/9911126.
  24. Jezierski J., CYK tensors, Maxwell field and conserved quantities for the spin-2 field, Classical Quantum Gravity 19 (2002), 4405-4429, gr-qc/0211039.
  25. Frolov V.P., Hidden symmetries and black holes, J. Phys. Conf. Ser. 189 (2009), 012015, 13 pages, arXiv:0901.1472.
  26. Hughston L.P., Penrose R., Sommers P., Walker M., On a quadratic first integral for the charged particle orbits in the charged Kerr solution, Comm. Math. Phys. 27 (1972), 303-308.
  27. Tanimoto M., The role of Killing-Yano tensors in supersymmetric mechanics on a curved manifold, Nuclear Phys. B 442 (1995), 549-560, gr-qc/9501006.
  28. McLenaghan R.G., Spindel Ph., Quantum numbers for Dirac spinor fields on a curved space-time, Phys. Rev. D 20 (1979), 409-413.
  29. Carter B., Global structure of the Kerr family of gravitational fields, Phys. Rev. 174 (1968), 1559-1571.
  30. Jezierski J., Lukasik M., Conformal Yano-Killing tensor for the Kerr metric and conserved quantities, Classical Quantum Gravity 23 (2006), 2895-2918, gr-qc/0510090.
  31. Visinescu M., Hidden conformal symmetries and quantum gravitational anomalies, Europhys. Lett. EPL 90 (2010), 41002, 4 pages.
  32. Carter B., Killing tensors quantum numbers and conserved currents in curved spaces, Phys. Rev. D 16 (1977), 3395-3414.
  33. Benn I.M., Charlton P., Dirac symmetry operators from conformal Killing-Yano tensors, Classical Quantum Gravity 14 (1997), 1037-1042, gr-qc/9612011.
  34. Cariglia M., Krtous P., Kubiznák D., Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets, arXiv:1102.4501.
  35. Kamran N., McLenaghan R.G., Symmetry operators for the neutrino and Dirac fields on curved spacetime, Phys. Rev. D 30 (1984), 357-362.
  36. van Holten J.W., Waldron A., Peeters K., An index theorem for non-standard Dirac operators, Classical Quantum Gravity 16 (1999), 2537-2544, hep-th/9901163.
  37. Moroianu S., Visinescu M., Finiteness of the L2-index of the Dirac operator on generalized Euclidean Taub-NUT metrics, J. Phys. A: Math. Gen. 39 (2006), 6575-6581, math-ph/0511025.
  38. Moroianu A., Moroianu S., The Dirac operator on generalized Taub-NUT space, arXiv:1003.5364.
  39. Visinescu M., Integrable systems and higher rank Killing tensors, in preparation.
  40. Houri T., Kubiznák D., Warnick C.M., Yasui Y., Generalized hidden symmetries and the Kerr-Sen black hole, J. High Energy Phys. 2010 (2010), no. 7, 055, 33 pages, arXiv:1004.1032.

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