Bargmann invariants, null phase curves, and a theory of the geometric phase

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 67, Issue 4, 2003, Pages 421141-4211416

  Mukunda, N ^a   Arvind, ^b,f   Ercolessi, E ^c   Marmo, G ^d   Morandi, G ^c   Simon, R ^e  

^a INDIAN INSTITUTE OF SCIENCE   (India)

^b CARNEGIE MELLON UNIVERSITY   (United States)

^c UNIVERSITY OF BOLOGNA   (Italy)

^d UNIVERSITY OF NAPLES FEDERICO II   (Italy)

^e INSTITUTE OF MATHEMATICAL SCIENCES   (India)

^f GURU NANAK DEV UNIVERSITY   (India)

Author keywords

[No Author keywords available]

Indexed keywords

GEOMETRICAL OPTICS; MATHEMATICAL TRANSFORMATIONS; ORDINARY DIFFERENTIAL EQUATIONS; PROBLEM SOLVING; VECTORS;

NULL PHASE CURVES;

QUANTUM THEORY;

EID: 17144426669     PISSN: 10502947     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (36)

1. M.V. Berry, Proc. R. Soc. London, Ser A 392, 45 (1984); A very useful reprint collection covering the period upto about 1989 is Geometric Phases in Physics, edited by J. Shapere and F. Wilczek, (World Scientific, Singapore, 1989).
2. M.V. Berry, Proc. R. Soc. London, Ser A 392, 45 (1984); A very useful reprint collection covering the period upto about 1989 is Geometric Phases in Physics, edited by J. Shapere and F. Wilczek, (World Scientific, Singapore, 1989).
3. B. Simon, Phys. Rev. Lett. 51, 2167 (1983).
4. Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987); J. Samuel and R. Bhandari, ibid. 60, 2339 (1988).
5. Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987); J. Samuel and R. Bhandari, ibid. 60, 2339 (1988).
6. N. Mukunda and R. Simon, Ann. Phys. (N.Y.) 228, 205 (1993); 228, 269 (1993).
7. N. Mukunda and R. Simon, Ann. Phys. (N.Y.) 228, 205 (1993); 228, 269 (1993).
8. V. Bargmann, J. Math. Phys. 5, 862 (1964).
9. S. Pancharatnam, Proc.-Indian Acad. Sci., Sect. A 44A, 247 (1956).
10. S. Ramaseshan and R. Nityananda, Curr. Sci. 55, 1225 (1986); N. Mukunda and R. Simon, Ref. [4] above.
11. S. Ramaseshan and R. Nityananda, Curr. Sci. 55, 1225 (1986); N. Mukunda and R. Simon, Ref. [4] above.
12. D. Page, Phys. Rev. A 36, 3479 (1987); S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Interscience, New York, 1969), Vol. II, Chap. 9.
13. D. Page, Phys. Rev. A 36, 3479 (1987); S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Interscience, New York, 1969), Vol. II, Chap. 9.
14. A concise account may be found, for example, V.I. Arnold, Mathematical Methods of Classical Mechanics (SpringerVerlag, Berlin, 1978), Appendix III.
15. J. Samuel and R. Bhandari, Ref. [3] above.
16. N. Mukunda and R. Simon in Ref. [4] above; R. Simon and N. Mukunda, Phys. Rev. Lett. 70, 880 (1993). For the use of BI's in the theory of off-diagonal geometric phases introduced by F. Pistolesi and N. Manini, ibid. 85, 1585 (2000); 85, 3067 (2000), see N. Mukunda, Arvind, S. Chaturvedi, and R. Simon, Phys. Rev. A 65, 012102 (2001).
17. N. Mukunda and R. Simon in Ref. [4] above; R. Simon and N. Mukunda, Phys. Rev. Lett. 70, 880 (1993). For the use of BI's in the theory of off-diagonal geometric phases introduced by F. Pistolesi and N. Manini, ibid. 85, 1585 (2000); 85, 3067 (2000), see N. Mukunda, Arvind, S. Chaturvedi, and R. Simon, Phys. Rev. A 65, 012102 (2001).
18. N. Mukunda and R. Simon in Ref. [4] above; R. Simon and N. Mukunda, Phys. Rev. Lett. 70, 880 (1993). For the use of BI's in the theory of off-diagonal geometric phases introduced by F. Pistolesi and N. Manini, ibid. 85, 1585 (2000); 85, 3067 (2000), see N. Mukunda, Arvind, S. Chaturvedi, and R. Simon, Phys. Rev. A 65, 012102 (2001).
19. N. Mukunda and R. Simon in Ref. [4] above; R. Simon and N. Mukunda, Phys. Rev. Lett. 70, 880 (1993). For the use of BI's in the theory of off-diagonal geometric phases introduced by F. Pistolesi and N. Manini, ibid. 85, 1585 (2000); 85, 3067 (2000), see N. Mukunda, Arvind, S. Chaturvedi, and R. Simon, Phys. Rev. A 65, 012102 (2001).
20. N. Mukunda and R. Simon in Ref. [4] above; R. Simon and N. Mukunda, Phys. Rev. Lett. 70, 880 (1993). For the use of BI's in the theory of off-diagonal geometric phases introduced by F. Pistolesi and N. Manini, ibid. 85, 1585 (2000); 85, 3067 (2000), see N. Mukunda, Arvind, S. Chaturvedi, and R. Simon, Phys. Rev. A 65, 012102 (2001).
21. E.M. Rabei, Arvind, N. Mukunda, and R. Simon, Phys. Rev. A 60, 3397 (1999).
22. These are described also in, for example, Refs. [2-4] above.
23. For discussions from the viewpoint of physical applications see, for instance, A.P. Balachandran, G. Marmo, B.S. Skagerstam, and A. Stem, Applications to Particle Dynamics (Springer-Verlag, Berlin, 1983); Classical Topology and Quantum States (World Scientific, Singapore, 1991).
24. For discussions from the viewpoint of physical applications see, for instance, A.P. Balachandran, G. Marmo, B.S. Skagerstam, and A. Stem, Applications to Particle Dynamics (Springer-Verlag, Berlin, 1983); Classical Topology and Quantum States (World Scientific, Singapore, 1991).
25. E. Ercolessi, G. Marmo, G. Morandi, and N. Mukunda, Int. J. Mod. Phys. A 16, 5007 (2001).
26. For a recent work on GP's in real Hubert spaces see, for instance, J. Samuel and A. Dhar, Phys. Rev. Lett. 87, 260401 (2001).
27. 1 using the unique geodesic running between them.
28. This may be regarded as the essential content of the discovery of S. Pancharatnam in Ref. [6] above.
29. For discussions of symplectic manifolds and their submanifolds see, for instance, R. Abraham and J.E. Marsden, Foundations of Mechanics (Benjamin-Cummings, Reading, MA, 1978); P. Liberman and C.M. Marle, Symplectic Geometry and Analytical Mechanics (Reidel, Dordrecht, 1987); G. Marmo, E.J. Saletan, A. Simoni, and B. Vitale, Dynamical Systems - A Differential Geometric Approach to Symmetry and Reduction (Wiley, New York, 1985).
30. For discussions of symplectic manifolds and their submanifolds see, for instance, R. Abraham and J.E. Marsden, Foundations of Mechanics (Benjamin-Cummings, Reading, MA, 1978); P. Liberman and C.M. Marle, Symplectic Geometry and Analytical Mechanics (Reidel, Dordrecht, 1987); G. Marmo, E.J. Saletan, A. Simoni, and B. Vitale, Dynamical Systems - A Differential Geometric Approach to Symmetry and Reduction (Wiley, New York, 1985).
31. For discussions of symplectic manifolds and their submanifolds see, for instance, R. Abraham and J.E. Marsden, Foundations of Mechanics (Benjamin-Cummings, Reading, MA, 1978); P. Liberman and C.M. Marle, Symplectic Geometry and Analytical Mechanics (Reidel, Dordrecht, 1987); G. Marmo, E.J. Saletan, A. Simoni, and B. Vitale, Dynamical Systems - A Differential Geometric Approach to Symmetry and Reduction (Wiley, New York, 1985).
32. R. Simon, E.C.G. Sudarshan, and N. Mukunda, Phys. Rev. A 37, 3028 (1988); Arvind, B. Dutta, N. Mukunda, and R. Simon, Pramana, J. Phys. 45, 471 (1995).
33. R. Simon, E.C.G. Sudarshan, and N. Mukunda, Phys. Rev. A 37, 3028 (1988); Arvind, B. Dutta, N. Mukunda, and R. Simon, Pramana, J. Phys. 45, 471 (1995).
34. For an exposition of Kaehler structures see, for instance, A. Weil, Introduction a l'Etude des Varietes Kaehleriennes (Hermann, Paris, 1971); S.S. Chern, Complex Manifolds Without Potential Theory (Springer-Verlag, New York, 1979).
35. For an exposition of Kaehler structures see, for instance, A. Weil, Introduction a l'Etude des Varietes Kaehleriennes (Hermann, Paris, 1971); S.S. Chern, Complex Manifolds Without Potential Theory (Springer-Verlag, New York, 1979).
36. For a detailed discussion see, for instance, N. Mukunda and R. Simon, Ref. [4] above.