Differential geometry on SU(3) with applications to three state systems

Journal of Mathematical Physics

Volumn 39, Issue 11, 1998, Pages 6125-6136

  Byrd, Mark ^a  

^a The University of Texas at Austin   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032356646     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.532618     Document Type: Article

References (25)

1. S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, Cambridge, 1996), Vol. II Sec. 18.7.
2. M. Gell-Mann and Y. Ne'eman, The Eight-Fold Way (Benjamin, New York, 1964).
3. J. P. Elliot, "Collective motion in the nuclear shell model I. Classification schemes for states of mixed configurations," Proc. R. Soc. London, Ser. A 245, 128-145 (1958); and "Collective motion in the nuclear shell, model II. The introduction of intrinsic wave-functions," ibid. 245, 562-581 (1958).
4. J. P. Elliot, "Collective motion in the nuclear shell model I. Classification schemes for states of mixed configurations," Proc. R. Soc. London, Ser. A 245, 128-145 (1958); and "Collective motion in the nuclear shell, model II. The introduction of intrinsic wave-functions," ibid. 245, 562-581 (1958).
5. Arvind, K. S. Mallesh, and N. Mukunda, "A generalized Pancharatnam geometric phase formula fo three-level systems," J. Phys. A 30, 2417-2431 (1997).
6. G. Khanna, S. Mukhopadhyay, R. Simon, and N. Mukunda, "Geometric phases for SU(3) representations and three level quantum systems," Ann. Phys. (N.Y.) 253, 55-82 (1997).
7. There have been other methods of decomposing the group into an Euler angle parametrization. See. Ref. 16 and D. A. Akyeampong and M. A. Rashid, "On the finite transformations of SU(3)," J. Math. Phys. 13, 1218-1224 (1972).
8. For an excellent review and a collection of reprints, see Geometric Phases in Physics, edited by A. Shapere and F. Wilczek (World Scientific, Singapore, 1989).
9. L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics Theory and Application (Cambridge University Press, Cambridge, 1977), p. 57.
10. R. Hermann, Lie Groups for Physicists (Benjamin, New York, 1966), pp. 30-36.
11. L. C. Biedenharn (private communication). See also L. C. Biedenharn and Y. Dothan, "Monopolar harmonics in SU(3) as eigenstates of the Skyrme-Witten model for baryons," in From SU(3) to Gravity (Cambridge University Press, Cambridge, 1986), pp. 19-34.
12. L. C. Biedenharn (private communication). See also L. C. Biedenharn and Y. Dothan, "Monopolar harmonics in SU(3) as eigenstates of the Skyrme-Witten model for baryons," in From SU(3) to Gravity (Cambridge University Press, Cambridge, 1986), pp. 19-34.
13. T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups (Springer-Verlag, New York, 1995) pp. 10, 214.
14. It is conventional to use D* for the wave functions in SU(2) (see, for instance, Ref. 8, p. 58). This convention is followed here. The necessary distinction between the two fundamental representations of SU(3) is made when needed in Ref. 17.
15. See Ref. 8, p. 61.
16. Y. Choquet-Bruhat and C. DeWitt-Morette, Analysis, Manifolds and Physics (North-Holland, Amsterdam, 1982), Part I, p. 164.
17. C. Isham, Modern Differential Geometry for Physicists (World Scientific, Singapore, 1989), p. 72.
18. T. J. Nelson, "A set of harmonic functions for the group SU(3) as specialized matrix elements of a general finite transformation," J. Math. Phys. 8, 857-863 (1967).
19. M. Byrd and E. C. G. Sudarshan, "SU(3) revisited," physics/9803029.
20. D. F. Holland, "Finite transformations of SU(3)," J. Math. Phys. 10, 531-535 (1969).
21. M. S. Marinov, "Invariant volumes of compact groups," J. Phys. A 13, 3357-3366 (1980).
22. M. Nakahara, Geometry, Topology and Physics (IOP Publishing, Philadelphia, 1990), pp. 29, 364.
23. J. Anandan, J. Christian, and K. Wanelik, "Resource Letter GPP-1: geometric phases in physics," quant-ph/9702011.
24. Reference 14, p. 138.
25. A private conversation with D. Cook.