Probability and quantum symmetries. II. The theorem of nœther in quantum mechanics

Journal of Mathematical Physics

Volumn 47, Issue 6, 2006, Pages

  Albeverio, S ^a   Rezende, J ^b   Zambrini, J C ^b  

^a Institut für Angewandte Mathematics   (Germany)

^b UNIVERSITY OF LISBON   (Portugal)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33745683242     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.2199087     Document Type: Article

References (71)

1. M. Thieullen and J.-C. Zambrini, Ann. I.H.P. Phys. Theor. 67(3), 297 (1997).
2. R. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
3. W. Thirring, A Course in Mathematical Physics Vol. 3: Quantum Mechanics of Atoms and Molecules (Springer-Verlag, New York, 1981).
4. M. Reed and B. Simon, Fourier Analysis, Self-Adjointness (Academic, New York, 1975).
5. T. Tsuchida, Nagoya Math. J. 136, 157 (1994).
6. D. Fujiwara and T. Tsuchida, J. Math. Soc. Jpn. 49, 299 (1997).
7. Y. Oshima, "Time dependent Dirichlet form and its applications to a transformation of space-time processes," Dirichlet Forms and Stochastic Processes (de Gruyter, New York, 1993), pp. 305-320.
8. V. Arnold, Méthodes Mathématiques de la Mécanique Classique (Mir, Moscow, 1976).
9. S. Albeverio and R. Høegh-Krohn, Mathematical Theory of Feynman Path Integrals, Lect. Notes Math. 523 (Springer, New York, 1976). New edition, with S. Mazzucchi, in preparation, Springer, Berlin (2006).
10. S. Albeverio and Z. Brzeźniak, J. Funct. Anal. 113, 177 (1993).
11. B. S. DeWitt, Rev. Mod. Phys. 29, 377 (1957).
12. S. Albeverio, A. M. Beutet de Monvel-Berthier, and Z. Brzeźniak, Potential Anal. 4, 469 (1995).
13. L. Shiff, Quantum Mechanics (McGraw-Hill, Kogakusha Tokyo, 1968).
14. V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Commande Optimale (Mir, Moscow, 1982).
15. W. Miller, Symmetry and Separation of Variables (Addison-Wesley, Reading, MA, 1977).
16. P. J. Olver, Applications of Lie Groups to Differential Equations (Springer-Verlag, New York, 1986).
17. V. Guillemin and S. Stemberg, Symplectic Techniques in Physics (Cambridge University Press, Cambridge, 1984).
18. E. Nelson, Ann. Math. 70, 572 (1959).
19. G. B. Folland, Harmonic Analysis in Phase Space (Princeton University Press, Princeton, NJ, 1989).
20. R. Berndt, An Introduction to Symplectic Geometry, Grad. Stud. Math. 26 (American Mathematical Society, Providence, RI, 2001).
21. S. Zelditch, Commun. Math. Phys. 90, 1 (1993).
22. A. Weinstein, Am. J. Math. 107, 1 (1985).
23. D. Fujiwara, Duke Math. J. 47, 559 (1980).
24. Y. Colin de Verdière, Chem. Phys. 105, 327 (1986).
25. H. Tamura, Nagoya Math. J. 105, 40 (1987).
26. A. Mohamed and J. Nourrigat, J. Math. Pures Appl. 70, 87 (1991).
27. H. Matsumoto, "The asymptotic distributions of the eigenvalues for the Schrödinger operators with magnetic fields," in Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotics, edited by K. D. Elworthy and N. Ikeda, Pitman Lecture Notes, 284 (Longman, Harlow, 1993), pp. 169-193.
28. R. Kuwabara, Rep. Math. Phys. 19, 27 (1984).
29. B. Doubrovine, S. Novikov, and A. Fomenko, Géométrie Contemporaine, 1st ed. (Mir, Moscow, 1982).
30. R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed. (Benjamin, Reading, MA, 1978).
31. T. Kolsrud, "Quantum constants of motion and the heat Lie algebra in a Riemanniann manifold," Trita-Mat, 1996, Stockholm (unpublished).
32. J.-C. Zambrini, "Probability and quantum symmetries in a Riemannian manifold," in Progress in Probability, edited by R. Dalang, M. Dozzi, and F. Russo (Birkhäuser, Boston, 1999), Vol. 45.
33. J. Von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, NJ, 1955).
34. N. Kumano-go, Bull. Sci. Math. 128, 197 (2004).
35. "The Mathematical Legacy of R. P. Feynman," in Stochastic, Analysis and Mathematical Physics, edited by R. Rebolledo, J. Rezende, and J.-C. Zambrini (World Scientific, Singapore, 2004).
36. R. H. Cameron, J. Math. Phys. 39, 126 (1960).
37. S. Albeverio, "Wiener and Feynman path integrals and their applications," Proceedings of Symposium in Applied Mathematics (American Mathematical Society, Providence, RI), Vol. 52 (1997).
38. A. D. Wentzell, A Course in the Theory of Stochastic Processes (McGraw-Hill, New York, 1981).
39. S. Albeverio, A. Hahn, and A. Sengupta, "Rigorous Feynman path integrals with applications to quantum theory, gauge fields and topological invariants," in Ref. 35.
40. S. Albeverio and S. Mazzucchi, J. Stat. Phys. 115, 191 (2004).
41. J. Rezende, Commun. Math. Phys. 101, 187 (1985).
42. G. W. Johnson and M. L. Lapidus, The Feynman Integral and Feynman's Operational Calculus (Oxford University Press, Oxford, 2000).
43. M. Kac, "On some connections between probability theory and differential and integral equations," Proceedings of 2nd Berkeley Symposium on Probability and Statistics, edited by J. Neyman (University of California Press, Berkeley, 1951).
44. K.-L. Chung and J.-C. Zambrini, Introduction to Quantum Time and Quantum Randomness, 2nd ed. (Word Scientific, Singapore, 2003).
45. S. Albeverio, K. Yasue, and J.-C. Zambrini, Ann. I.H.P. Phys. Theor. 49, 259 (1989).
46. A. B. Cruzeiro, W. Liming, and J.-C. Zambrini, "Bernstein processes associated with a Markov process," in Stochastic Analysis and Mathematical Physics, edited by R. Rebolledo, Santiago, (Birkhäuser, Boston, 2000).
47. E. Nelson, Dynamical Theories of Brownian Motion, Math. Notes (Princeton University Press, Princeton, NJ, 1967).
48. L. C. G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales, Vol. 2, Itô Calculus (Wiley, Chichester, 1987).
49. K. Itô and H. P. McKean, Jr., Diffusion Processes and Their Sample Paths (Springer, Berlin, 1965).
50. K. Itô, "Extension of Stochastic Integrals," in Proceedings of International Symposium in Stochastic Differential Equations - Kyoto 1976, edited by K. Itô, (Kinokuniya, Tokyo, 1976).
51. A. B. Cruzeiro and J.-C. Zambrini, J. Funct. Anal. 96, 62 (1991).
52. H. R. Lewis and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969).
53. J.-C. Zambrini, "A special time-dependent quantum invariant and a general theorem on quantum symmetries," in Stochastic Analysis and Mathematical Physics (Anestoc' 96), edited by R. Rebolledo (World Scientific, Singapore, 1996).
54. N. H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations (CRC Press, Boca Raton, FL), Vol. 2 (1994).
55. A. Galindo, P. Pascual, "Quantum mechanics I," Texts and Monographs in Physics (Springer-Verlag, New York, 1990).
56. 2 Euclidean (quantum) field theory, Princeton Series in Physics," (Princeton University Press, NJ, 1974).
57. N. Privault and J.-C. Zambrini, Ann. I.H.P. Probab. Stat. 40, 599 (2004).
58. P. Lescot and J.-C. Zambrini, "Probabilistic deformation of contact geometry, diffusion processes and their quadratures," Seminar on Stochastic Analysis, Random Fields and Applications V, Ascona (Birkhäuser, Boston), to appear.
59. M. Thieullen and J.-C. Zambrini, Probab. Theory Relat. Fields 107, 401 (1997).
60. K. Yajima, Commun. Math. Phys. 181, 605 (1996).
61. S. Albeverio and S. Mazzucchi, J. Funct. Anal. 221, 83 (2005).
62. G. Ben Arous and F. Castell, J. Funct. Anal. 137, 243 (1996).
63. M. De Faria, J. Potthoff, and L. Streit, J. Math. Phys. 32, 2123 (1991).
64. D. Elworthy and A. Truman, Ann. I.H.P. Phys. Theor. 41, 115 (1984).
65. M. Grothaus, D. Khandekar, J. L. da Silva, and L. Streit, J. Math. Phys. 38, 3278 (1997).
66. V. N. Kolokoltsov, Semiclassical Analysis for Diffusions and Stochastic Processes, Lect. Notes Math. 1724 (Springer, Berlin, 2000).
67. T. Hida, Brownian Motion (Springer-Verlag, New York, 1980).
68. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes (North-Holland, Amsterdam, 1981).
69. P. Malliavin, Stochastic Analysis, Grund, der Math. Wiss 313 (Springer-Verlag, Berlin, 1997).
70. S. Albeverio, "Theory of Dirichlet forms and applications," in S. Albeverio, W. Schachermayer, and M. Talagrand, Lectures on Probability Theory and Statistics (Ecole d'Eté de Probabilités de Saint-Flour XXX-2000), edited by P. Bernard, Lect. Notes Math. 1816 (Springer, New York, 2003), pp. 4-110.
71. T. Kolsrud, "Position dependent NLS hierarchies: Involutivity, commutation relations, renormalisation and classical invariants," to appear in Bull. Sci. Math.