Towards a definition of quantum integrability

International Journal of Geometric Methods in Modern Physics

Volumn 6, Issue 1, 2009, Pages 129-172

  Clemente Gallardo, Jesú ^a   Marmo, Giuseppe ^b  

^a UNIVERSITY OF ZARAGOZA   (Spain)

^b UNIVERSITY OF NAPLES FEDERICO II   (Italy)

Author keywords

Geometric quantum mechanics; Integrable system; Quantum integrability

Indexed keywords

EID: 65449136680     PISSN: 02198878     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0219887809003448     Document Type: Review

References (56)

1. R. Abraham and J. E. Marsden, Foundation of Mechanics (Benjamin, 1978).
2. D. Alekseevsky, J. Grabowski, G. Marmo and P. W. Michor, Completely integrable systems: A generalization, Mod. Phys. Lett. A 12 (1997) 1637-1648.
3. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, 1989).
4. A. D. Avanzo and G. Marmo, Reduction and unfolding: The Kepler problem, Int. J. Geom. Meth. Phys. 2 (2005) 83-109.
5. A. D. Avanzo, G. Marmo and A. Valentino, Reduction and unfolding for quantum systems: The Hydrogen atom, Int. J. Geom. Meth. Phys. 2 (2005) 1043-1062.
6. G. Bimonte, G. Marmo and A. Stern, Bicovariant calculus in quantum theory and a generalization of the Gauss law, Phys. Lett. B 478 (2000) 358.
7. J. F. Cariñena, J. Clemente-Gallardo and G. Marmo, Geometrization of quantum mechanics, Theor. Math. Phys. 152(1) (2007) 894-903.
8. J. F. Cariñena, J. Clemente-Gallardo and G. Marmo, Proc. XV International Workshop on Geometry and Physics, Vol. 11, Publicaciones de la RSME, chapter Introduction to Quantum Mechanics and the Quantum-Classical transition, 3-45. RSME, (2007).
9. J. F. Cariñena, J. Grabowski and G. Marmo, Lie-Scheffers Systems: A Geometric Approach (Bibliopolis, Napoli, 2000).
10. H. Cartan, Differential Forms (trans. from french) (Houghton Mifflin Co, Boston, 1970).
11. G. Cassinelli, E. de Vito and A. Levrero, Integrability of the quantum adiabatic evolution and geometric phases, J. Math. Phys. 38 (12) (1997) 6101-6118.
12. R. Cirelli, A. Maniá and L. Pizzocchero, A functional representation for noncommutative ℂ*-algebras, Rev. Math. Phys. 6(5) (1994) 675-697.
13. J. Clemente-Gallardo and G. Marmo, Differential Geometric Methods in Mechanics and Field Theory, The space of density states in geometrical quantum mechanics, pp. 35-56. (Gent Academia Press, 2007).
14. J. Clemente-Gallardo and G. Marmo, Basics of quantum mechanics, geometrization and some applications to quantum information, Int. J. Geom. Meth. Mod. Phys. 5(6) (2008).
15. A. Connes, Noncommutative Geomtry (Academic Press, 1994).
16. E. B. Davies, Hilbert space representations of Lie algebras, Comm. Math. Phys. 23 (1971) 159-168.
17. J. Dixmier, Les ℂ*-Algèbres et Leurs réprésentations, Gauthiers-Villars (Paris, 1964).
18. B. Dubrovin, G. Marmo and A. Simoni, Alternative hamiltonian description for quantum systems, Modern Physics Letters A 15(5) (1990) 1229-1234.
19. G. G. Emch, Foundations of 20th Century Physics (North Holland, 1984).
20. M. J. Engeldfeld, Group Theory and the Coulomb Problem (Wiley-Interscience, 1972).
21. E. Ercolessi, A. Ibort, G. Marmo and G. Morandi, Alternative linear structures for classical and quantum systems, Int. J. Mod. Phys. A 22(18) (2007) 3039-3064.
22. L. D. Faddeev, Nonlinear Equations and Spectral Theory, What is complete integrability in quantum mechanics, AMS Trans. Serv 2, 220, Providence (2007).
23. L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer, Berlin, 2007).
24. S. De Filippo, G. Vilasi and G. Marmo, A new characterization of completely integrable systems, Nuovo Cimento B 83(2) (1984) 97-112.
25. I. M. Gelfand and I. Ya. Dorfman, Hamiltonian operators and the classical yang-baxter equation (russian), Function. Anal. Appl. 16(4) (1982) 241-248.
26. J. Grabowski, M. Kus and G. Marmo, Geometry of quantum systems: Density states and entanglement, J. Phys. A 38 (2005) 10217-10244.
27. J. Grabowski, M. Kus and G. Marmo, Symmetries, group actions, and entanglement, Open Syst. Inf. Dyn. 13 (2006) 343-362.
28. J. Hietarinta, Classical versus quantum integrability, J. Math. Phys. 25 (1989) 1833.
29. A. A. Kirillov, Merits and demerits of the orbit method, Bull. AMS, 36 (1999) 433-488.
30. I. Kolar, P. W. Michor and J. Slovak, Natural Operations in Differential Geometry (Springer-Verlag, Berlin, 1993).
31. G. Landi, An Introduction to Noncommutative Spaces and their Geometries, Lecture Notes in Physics, Vol. 51 (Springer-Verlag, Berlin, 1997).
32. N. P. Landsman, Mathematical Topics Between Classical and Quantum Mechanics (Springer-Verlag, 1998).
33. S. Lang, Introduction to Differentiable Manifolds (Spinger-Verlag, New York, 2002).
34. P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968) 467-490.
35. S. Lie and G. Scheffers, Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen (Teubner, Leipzig, 1893). [Edited and revised by G. Scheffers].
36. F. Magri, A simple model of the integrable hamiltonian equation, J. Math. Phys. 19(5) (1978) 1156-1162.
37. V. I. Manko, G. Marmo, E. C. G. Sudarshan and F. Zaccaria, F oscillators and nonlinear coherent states, Physica Scripta 55(1997) 528-541.
38. V. I. Manko, G. Marmo, F. Zaccaria and E. C. G. Sudarshan, Wigner's problem and alternative commutation relations for quantum mechanics, Int. J. Modern Physics B 11(10) (1997) 1281-1296.
39. V. I. Man'ko, G. Marmo, S. Solimeno and F. Zaccaria, Physical non linear aspects of classical and quantum q-oscillators, Int. J. Mod. Phys. A 8(20) (1993) 3577-3597.
40. G. Marmo, G. Morandi, A. Simoni and F. Ventriglia, Alternative structures and bi-hamiltonian systems, J. Phys. A: Math. Gen. 35 (2002) 8393-8406.
41. G. Marmo, E. J. Saletan, A. Simoni and B. Vitale, Dynamical Systems, a Differential Geometric Approach to Symmetry and Reduction (John Wiley, Chicester, 1985).
42. G. Marmo, G. Scolarici, A. Simoni and F. Ventriglia, Alternative structures and bi-Hamiltonian systems on a Hilbert space, J. Phys. A: Math and Gen. 38(17) (2005) 3813-3821.
43. G. Marmo, G. Scolarici, A. Simoni and F. Ventriglia, The quantum-classical transition: The fate of the complex structure, Int. J. Geom. Meth. Mod. Phys. 2 (2005) 127-145.
44. G. Marmo and G. Vilasi, When do recursion operators generate new conservation laws? Phys. Lett. B 277 (1992) 137-140.
45. G. Marmo, P. Vitale and A. Zampini, Noncommutative differential calculus for Moyal subalgebras, J. Geom. Phys. 55 (2006) 611-622.
46. P. W. Michor and T. S. Ratiu, On the geometry of the Virasoro-Bott group, J. Lie Theory 8(2) (1998) 293-309.
47. G. Morandi, C. Ferrario, G. Lo Vecchio, G. Marmo and C. Rubano, The inverse problem in the calculus of variations and the geometry of the tangent bundle, Phys. Rep. 188(3-4) (1990) 147-284.
48. E. Nelson, Analytic vectors, Ann. Math. 70 (1959) 572-615.
49. M. B. Scheftel, P. Tempesta and P. Winternitz, Superintegrable systems in quantum mechanics and classical lie theory, J. Math. Phys. 42(2) (2001) 659-673.
50. I. E. Segal, Quantized differential forms, Topology 8 (1967) 147.
51. I. E. Segal, Quantization of the de Rham complex, Proc. Symp. Pure Math. (16) (1970) 205.
52. R. F. Streater, The representation of the oscillator group, Comm. Math. Phys. 4 (1967) 217-236
53. V. S. Varadarajan, An introduction to Harmonic Analysis on Semisimple Lie Groups, Cambridge studies in Advanced Mathematics (Cambrigde University Press, 1989).
54. G. Vilasi, Hamiltonian Dynamics (World Scientific, River Edge, NY, 2001).
55. J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton Univ. Press, 1996).
56. S. Weigert, The problem of quantum integrability, Physica D 56 (1) (1992) 107-119.