Quantum integrability of quadratic Killing tensors

Journal of Mathematical Physics

Volumn 46, Issue 5, 2005, Pages

  Duval, C ^a   Valent, G ^b  

^a CENTRE DE PHYSIQUE THÉORIQUE   (France)

^b LPTHE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 18844459421     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1899986     Document Type: Article

References (38)

1. Babelon, O., Bernard, D., and Talon, M., Introduction to Classical Integrable Systems (Cambridge University Press, New York, 2003).
2. Bellon, M. and Talon, M., hep-th0407005.
3. Benenti, S., " Inertia tensors and Stäckel systems in the Euclidean spaces., " Rend. Semin. Mat. Torino 50, 315-341 (1992).
4. Benenti, S., Chanu, C., and Rastelli, G., " Remarks on the connection between the additive separation of the Hamilton-Jacobi equation and the multiplicative separation of the Schrödinger equation. I. The completeness and Robertson conditions., " J. Math. Phys. 43, 5183-5222 (2002).
5. Benenti, S., Chanu, C., and Rastelli, G., " Remarks on the connection between the additive separation of the Hamilton-Jacobi equation and the multiplicative separation of the Schrödinger equation. II. First integrals and symmetry operators., " J. Math. Phys. 43, 5223-5253 (2002).
6. Bordemann, M., " Sur l'existence d'une prescription d'ordre naturelle projectivement invariante., " math.DG0208171.
7. Bouarroudj, S., " Projectively equivariant quantization map., " Lett. Math. Phys. 51, 265-274 (2000).
8. Cariglia, M., " Quantum mechanics of Yano tensors: Dirac equation in curved spacetime., " Class. Quantum Grav. 21, 1051-1078 (2004).
9. Carter, B., " Hamilton-Jacobi and Schrödinger separable solutions of Einstein's equations., " Commun. Math. Phys. 10, 280-310 (1968).
10. Carter, B., " Black hole equilibrium states., " in Black HolesLes Astres Occlus (Ecole d'́t́ de Physique Th́orique, Les Houches, 1972), pp. 57-214; Gordon and Breach, New York (1973).
11. Carter, B., " Killing tensor quantum numbers and conserved currents in curved space., " Phys. Rev. D 16, 3395-3414 (1977).
12. Chong, Z. W., Gibbons, G. W., Lü, H., and Pope, C. N., " Separability and Killing tensors in Kerr-Taub-NUT-de Sitter metrics in higher dimensions., " hep-th0405061.
13. Cordani, B., Feh́r, L., and Horváthy, P., " O(4, 2) dynamical symmetry of the Kaluza-Klein monopole., " Phys. Lett. B 201, 481-486 (1988).
14. Dullin, H. R., and Matveev, V. S., " A new integrable system on the sphere., " Math. Res. Lett. 11, 715-722 (2004).
15. Duval, C., Lecomte, P., and Ovsienko, V., " Conformally equivariant quantization: Existence and uniqueness., " J. Microwave Power Electromagn. Energy 49, 1999-2029 (1999).
16. Duval, C., and Ovsienko, V., " Conformally equivariant quantum Hamiltonians., " Selecta Math., New Ser. 7, 291-320 (2001).
17. Duval, C., and Ovsienko, V., " Projectively equivariant quantization and symbol calculus: Noncommutative hypergeometric functions., " Lett. Math. Phys. 57, 61-67 (2001).
18. Eisenhart, L. P., " Separable systems of Stäckel., " Ann. Math. 35, 284-305 (1934).
19. Gibbons, G. W., Hartnoll, S. A., and Yasui, Y., " Properties of some five-dimensional Einstein metrics., " Class. Quantum Grav. 21, 4697-4730 (2004).
20. Gibbons, G. W., Rietdijk, R. H., and van Holten, J. W., " SUSY in the sky., " Nucl. Phys. 404, 42-64 (1993).
21. Gibbons, G. W., and Ruback, P. J., " The hidden symmetries of Multi-Centre metrics., " Commun. Math. Phys. 115, 267-300 (1988).
22. Harnad, J., and Winternitz, P., " Classical and quantum integrable systems in gl̃ (2)+* and separation of variables., " Commun. Math. Phys. 172, 263-285 (1995).
23. Kunduri, H. K., and Lucietti, J., " Integrability and the Kerr-(A) dS black hole in five dimensions., " hep-th0502124.
24. Lecomte, P. B. A., and Ovsienko, V., " Projectively invariant symbol calculus., " Lett. Math. Phys. 49, 173-196 (1999).
25. Loubon Djounga, S. E., " Conformally invariant quantization at order three., " Lett. Math. Phys. 64, 203-212 (2003).
26. Moser, J., " Various aspects of the integrable Hamiltonian systems., " in Dynamical Systems (C.I.M.E. Summer School Bressanone; 1978), Progress in Mathematics Vol. 8 (Birkhäuser, Boston, 1981), pp. 233-289.
27. Moser, J., " Geometry of quadrics and spectral theory., " in Proceedings of the Chern Symposium, Berkeley 1979 (Springer, New York, 1980), pp. 147-188.
28. Perelomov, A. M., Integrable Systems of Classical Mechanics and Lie Algebras (Birkhäuser, Boston, 1990), Vol. 1, and references therein.
29. Plebanski, J. F., " A class of solutions of Einstein-Maxwell equations., " Ann. Phys. 90, 196-255 (1975).
30. Plebanski, J. F., and Demianski, M., " Rotating, charged, and uniformly accelerating mass in General Relativity., " Ann. Phys. 98, 98-127 (1976).
31. Robertson, H. P., " Bermerkung über separierbare systeme in der Wellenmechanik., " Math. Ann. 98, 749-752 (1927).
32. Sakaguchi, M., and Yasui, Y., " Notes on five-dimensional Kerr black holes., " hep-th/0502182.
33. Souriau, J.-M., Structure des Syst̀mes Dynamiques, Dunod (1970, ©1969);
34. Structure of Dynamical Systems. A Symplectic View of Physics, translated by, C. H. Cushman-de Vries, R. H. Cushman, and, G. M. Tuynman, translation editors (Birkhäuser, Boston, 1997).
35. Toth, J. A., " Various quantum mechanical aspects of quadratic forms., " J. Funct. Anal. 130, 1-42 (1995).
36. Valent, G., " Integrability versus separability for the multi-centre metrics., " Commun. Math. Phys. 244, 571-594 (2004).
37. Vasudevan, M., Stevens, K. A., and Page, D. N., " Separability of the Hamilton-Jacobi and Klein-Gordon equations in Kerr-de Sitter metrics., " Class. Quantum Grav. 22, 339-352 (2005).
38. Weigert, S., " The problem of quantum integrability., " Physica D 56, 107-119 (1992).