Geometric phases for SU(3) representations and three level quantum systems

Annals of Physics

Volumn 253, Issue 1, 1997, Pages 55-82

  Khanna, G ^a   Mukhopadhyay, S ^b   Simon, R ^c,d   Mukunda, N ^e  

^a INDIAN INSTITUTE OF TECHNOLOGY KANPUR   (India)

^b TATA INSTITUTE OF FUNDAMENTAL RESEARCH   (India)

^c INSTITUTE OF MATHEMATICAL SCIENCES   (India)

^d S N BOSE NATIONAL CENTRE FOR BASIC SCIENCES   (India)

^e INDIAN INSTITUTE OF SCIENCE   (India)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030636914     PISSN: 00034916     EISSN: None     Source Type: Journal    
DOI: 10.1006/aphy.1997.5601     Document Type: Article

References (22)

1. 1. For a comprehensive overview and reprints of many original papers, see, for instance, A. Shapere and F. Wilczek, "Geometric Phases in Physics," World Scientific, Singapore, 1989.
2. 2. N. Mukunda and R. Simon, Ann. Phys. (NY) 228 (1993), 205.
3. 3. M. V. Berry, Proc. Roy. Soc. (London) A 392 (1984), 45; reprinted in Ref. (1).
4. 4. N. Mukunda and R. Simon, Ann. Phys. (NY) 228 (1993), 269.
5. 5. Such a listing has been provided in, for instance, L. Michel and L. A. Radicati, Ann. Inst. Henri Poincaré 18 (1973), 185; Ph. Combe, A. Sciarrino, and P. Sorba, Nucl. Phys. B 158 (1979), 452. However, for the convenience of the reader and in conformity with our notations, we provide a complete and independent discussion in this paper as well.
6. 5. Such a listing has been provided in, for instance, L. Michel and L. A. Radicati, Ann. Inst. Henri Poincaré 18 (1973), 185; Ph. Combe, A. Sciarrino, and P. Sorba, Nucl. Phys. B 158 (1979), 452. However, for the convenience of the reader and in conformity with our notations, we provide a complete and independent discussion in this paper as well.
7. 6. See, for instance, J. J. de Swart, Rev. Mod. Phys. 35 (1963), 916.
8. 7. M. Gell-Mann and Y. Neeman, "The Eightfold Way", Benjamin, New York 1964.
9. 8. See, for instance, Ref. (6) above.
10. 9. L. C. Biedenharn, Phys. Lett. 3 (1962), 69; N. Mukunda and L. K. Pandit, J. Math. Phys. 6 (1963), 746.
11. 9. L. C. Biedenharn, Phys. Lett. 3 (1962), 69; N. Mukunda and L. K. Pandit, J. Math. Phys. 6 (1963), 746.
12. 10. Attempts to parametrise elements of SU(3), in the spirit of the Euler angles parametrisation of SU(2) or SO(3), may be found in M. A. B. Beg and H. Ruegg, J. Math. Phys. 6 (1965), 677; T. J. Nelson, J. Math. Phys. 8 (1967), 857; D. A. Akyeampong and M. A. Rashid, J. Math. Phys. 13 (1972), 1218.
13. 10. Attempts to parametrise elements of SU(3), in the spirit of the Euler angles parametrisation of SU(2) or SO(3), may be found in M. A. B. Beg and H. Ruegg, J. Math. Phys. 6 (1965), 677; T. J. Nelson, J. Math. Phys. 8 (1967), 857; D. A. Akyeampong and M. A. Rashid, J. Math. Phys. 13 (1972), 1218.
14. 10. Attempts to parametrise elements of SU(3), in the spirit of the Euler angles parametrisation of SU(2) or SO(3), may be found in M. A. B. Beg and H. Ruegg, J. Math. Phys. 6 (1965), 677; T. J. Nelson, J. Math. Phys. 8 (1967), 857; D. A. Akyeampong and M. A. Rashid, J. Math. Phys. 13 (1972), 1218.
15. 11. For information on Maurer-Cartan one-forms see, for instance, B. Schutz, "Geometrical Methods of Mathematical Physics," Cambridge Univ. Press, Cambridge, 1980; Y. Choquet-Bruhat and C. Dewitt Morette, "Analysis, Manifolds and Physics," Part I, 2nd ed., North-Holland, Amsterdam 1982; C. J. Isham, "Modern Differential Geometry for Physicists," World Scientific, Singapore, 1989.
16. 11. For information on Maurer-Cartan one-forms see, for instance, B. Schutz, "Geometrical Methods of Mathematical Physics," Cambridge Univ. Press, Cambridge, 1980; Y. Choquet-Bruhat and C. Dewitt Morette, "Analysis, Manifolds and Physics," Part I, 2nd ed., North-Holland, Amsterdam 1982; C. J. Isham, "Modern Differential Geometry for Physicists," World Scientific, Singapore, 1989.
17. 11. For information on Maurer-Cartan one-forms see, for instance, B. Schutz, "Geometrical Methods of Mathematical Physics," Cambridge Univ. Press, Cambridge, 1980; Y. Choquet-Bruhat and C. Dewitt Morette, "Analysis, Manifolds and Physics," Part I, 2nd ed., North-Holland, Amsterdam 1982; C. J. Isham, "Modern Differential Geometry for Physicists," World Scientific, Singapore, 1989.
18. 12. Explicit expressions may be found in Ref. (4) above.
19. 13. See, for instance, E. C. G. Sudarshan and N. Mukunda, "Classical Dynamics: A Modern Perspective," Chap. 9, Wiley, New York, 1974. A similar procedure in the case of the Lorentz group has been described in M. V. Atre and N. Mukunda, J. Math. Phys. 27 (1986), 2908.
20. 13. See, for instance, E. C. G. Sudarshan and N. Mukunda, "Classical Dynamics: A Modern Perspective," Chap. 9, Wiley, New York, 1974. A similar procedure in the case of the Lorentz group has been described in M. V. Atre and N. Mukunda, J. Math. Phys. 27 (1986), 2908.
21. 14. For excellent discussions of monopole solutions in classical non-Abelian gauge theories and their topological classification, see the following reviews: P. Goddard and D. I. Olive, Rep. Prog. Phys. 41 (1978), 1357; S. Coleman, "The Magnetic Monopole Fifty Years Later," lectures, 1981 "Ettore Majorana" International School of Subnuclear Physics.
22. 14. For excellent discussions of monopole solutions in classical non-Abelian gauge theories and their topological classification, see the following reviews: P. Goddard and D. I. Olive, Rep. Prog. Phys. 41 (1978), 1357; S. Coleman, "The Magnetic Monopole Fifty Years Later," lectures, 1981 "Ettore Majorana" International School of Subnuclear Physics.