Coherent States and Global Differential Geometry

  • Stefan Berceanu


The relationship between coherent states and geodesics is emphasized. It is found that CL 0 = Σ0, where CL 0 is the cut locus of 0 and Σ0 is the locus of coherent vectors othogonal to 0 >. The result is proved for manifolds on which the exponential from the Lie algebra to the Lie group equals the geodesic exponential. The conjugate loci on hermitian symmetric spaces are analyzed also in the context of the coherent state approach. The results are illustrated on the complex Grassmann manifold.


Line Bundle Coherent State Symmetric Space Geometric Quantization Grassmann Manifold 
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  1. 1.
    J. R. Klauder, B. S. Skagerstam (Eds.), “Coherent States”, Word Scientific, Singapore (1985)MATHGoogle Scholar
  2. 2.
    S. Berceanu, From quantum mechanics to classical mechanics and back, via coherent states, in “Quantization and Infinite-Dimensional Systems (Proc. Bialowieza 1994)”, p. 155, J-P. Antoine et al. (eds.), Plenum Press, New York (1994).CrossRefGoogle Scholar
  3. 3.
    A. M. Perelomov, Coherent states for arbitrary Lie groups, Commun. Math. Phys. 26: 222 (1972)MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    J. R. Rawnsley, Coherent states and Kähler manifolds, Quart. J. Math. Oxford 28: 403 (1977)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    B. Kostant, Quantization and unitary representations, in: “Lecture Notes in Mathematics”, Vol 170, p. 87; C. T. Taam (ed.), Springer-Verlag, Berlin (1970)Google Scholar
  6. 6.
    V. Ceausescu, A. Gheorghe, Classical limit and quantization of Hamiltonian systems, in: “Symmetries and Semiclassical Features of Nuclear Dynamics”, Lecture Notes in Physics, Vol 279, p. 69; Springer-Verlag, Berlin (1987)CrossRefGoogle Scholar
  7. 7.
    S. Berceanu, L. Boutet de Monvel, Linear dynamical systems, coherent state manifolds, flows and matrix Riccati equation, J. Math. Phys. 34: 2353 (1993)MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    S. Kobayashi, K. Nomizu, “Foundations of Differential Geometry”, Vol II, Interscience, New York (1969)MATHGoogle Scholar
  9. 9.
    S. Helgason, “Differential Geometry, Lie groups and Symmetric Spaces”, Academic Press, New York (1978)MATHGoogle Scholar
  10. 10.
    Y. C. Wong, Differential Geometry of Grassmann manifolds, Proc. Nat. Acad. Sci. U.S.A. 57: 589 (1967)MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    Y. C. Wong, Conjugate loci in Grassmann manifold, Bull Am. Math. Soc. 74: 240 (1968)MATHCrossRefGoogle Scholar
  12. 12.
    T. Sakai, On cut loci on compact symmetric spaces, Hokkaido Math. J. 6: 136 (1977)MathSciNetMATHGoogle Scholar
  13. 13.
    S. Kobayashi, On conjugate and cut loci, in “Global Differential Geometry”, M.A.A. Studies in Mathematics, 27, S. S. Chern (ed.), p. 140 (1989)Google Scholar
  14. 14.
    S. Berceanu, The coherent states: old geometrical methods in new quantum clothes, preprint Bucharest, Institute of Atomic Physics, FT-398-1994Google Scholar
  15. 15.
    V. Guillemin, S. Sternberg, “Symplectic Technics in Physics”, Cambridge University, Cambridge (1984)Google Scholar
  16. 16.
    A. Odzijewicz, Coherent states and geometric quantization, Commun. Math. Phys. 150: 385 (1992)MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    F. Hirzebruch, “Topological Methods in Algebraic Geometry”, Springer-Verlag, Berlin (1966).MATHCrossRefGoogle Scholar
  18. 18.
    H. Woodhouse, “Geometric Quantization”, Oxford University, Oxford (1980)MATHGoogle Scholar
  19. 19.
    A. W. Knapp, “Representation Theory of Semisimple Lie Groups”, Princeton University Press, Princeton (1986)Google Scholar
  20. 20.
    B. Shiffman, A. J. Sommese, “Vanishing Theorems on Complex Manifolds”, Progress in Mathematics, Vol. 56, Birkhäuser, Boston (1985)Google Scholar
  21. 21.
    P. Griffith, J. Harris, “Principles of Algebraic Geometry”, Wiley, New-York (1978)Google Scholar
  22. 22.
    S. Berceanu, A. Gheorghe, On the construction of perfect Morse functions on compact manifolds of coherent states, J. Math. Phys. 28: 2899 (1987)MathSciNetADSMATHCrossRefGoogle Scholar
  23. 23.
    M. Cahen, S. Gutt, J. Rawnsley, Quantization on Kähler manifolds, 11, Trans. Math. Soc. 337: 73 (1993)MathSciNetMATHGoogle Scholar
  24. 24.
    R. Crittenden, Minimum and conjugate points in symmetric spaces, Canad. J. Math. 14: 320 (1962)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    A. Thimm, Integrable geodesic flows on homogeneous spaces, Ergod. Theory Dyn. Syst. 1: 495 (1981)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    O. Kowalski, “Generalized Symmetric Spaces”, Lecture Notes in Mathematics 805, Springer-Verlag, Berlin (1980)Google Scholar
  27. 27.
    R. Montgomery, Isoholonomic problems and some applications, Comm. Math. Phys. 128: 565 (1990)MathSciNetADSMATHCrossRefGoogle Scholar
  28. 28.
    H. H. Wu, “The Equidistribution Theory of Holomorphic Curves”, Annals of Maths. Studies 164, Princeton Univ. Press, Princeton (1970)Google Scholar
  29. 29.
    A. Cayley, A sixth memoir upon quantics, Phil. Trans. Royal. Soc. London. 149: 61 (1859).CrossRefGoogle Scholar
  30. 30.
    S. Berceanu, On the geometry of complex Grassmann manifold, its noncompact dual and coherent states (in preparation)Google Scholar
  31. 31.
    S. S. Chern, “Complex Manifolds without Potential Theory”, Van Nostrand, Princeton (1967)MATHGoogle Scholar
  32. 32.
    L. C. Pontrjagin, Charakteristiceskie tzikly differentziruemyh mnogobrazia, Mat. sb. 21: 233 (1947)Google Scholar
  33. 33.
    F. P. Gantmacher, “Teoria Matritz”, Nauka, Moskwa (1966)Google Scholar
  34. 34.
    Y. C. Wong, A class of Schubert varieties, J. Diff. Geom. 4: 37 (1970)MATHGoogle Scholar
  35. 35.
    D. Husemoller, “Fibre Bundles”, Mc Graw-Hill, New York (1966)MATHCrossRefGoogle Scholar
  36. 36.
    C. Jordan, Sur la géométrie à n dimensions, Bull. Soc. Math. France t. III: 103 (1875)Google Scholar
  37. 37.
    B. Rosenfel’d, Vnnutrenyaya geometriya mnojestva m-mernyh ploskastei n-mernova ellipticeskovo prostranstva, Izv. Akad. Nauk. SSSR, ser. mat. 5: 353 (1941)MathSciNetGoogle Scholar
  38. 38.
    B. Rosenfel’d, “Mnogomernye Prostranstva”, Nauka, Moskwa (1966); “Neevklidovy Prostranstva”, Nauka, Moskwa (1969).Google Scholar

Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • Stefan Berceanu
    • 1
    • 2
  1. 1.Equipe de Physique Mathématique et Géométrie Institut de MathématiqueCNRS - Université Paris 7-Denis DiderotParis Cedex 05France
  2. 2.Department of Theoretical PhysicsInstitute of Atomic Physics Institute of Physics and Nuclear EngineeringBucharest-MagureleRomania

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