Journal of Mathematical Sciences

, Volume 137, Issue 5, pp 5117–5136 | Cite as

Hamiltonian systems on complex Grassmann manifolds. Holonomy and Schrodinger equation

  • Z. Giunashvili


Differential geometric structures such as principal bundles for canonical vector bundles on complex Grassmann manifolds, canonical connection forms on these bundles, canonical symplectic forms on complex Grassmann manifolds, and the corresponding dynamical systems are investigated. Grassmann manifolds are considered as orbits of the co-adjoint action and symplectic forms are described as the restrictions of the canonical Poisson structure to Lie coalgebras. Holonomies of connections on principal bundles over Grassmannians and their relation with Berry phases is considered and investigated for integral curves of Hamiltonian dynamical systems.


Manifold Vector Bundle Poisson Structure Principal Bundle Adjoint Action 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. A. Agrachev and Yu. L. Sachkov, Lectures on Geometric Control Theory, SISSA Ref. 38/2001/M (2001).Google Scholar
  2. 2.
    W. Ambrose and I. M. Singer, “A theorem on holonomy,” Trans. Amer. Math. Soc., 75, 428–443 (1953).MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    K. Fujii, Note on coherent states and adiabatic connections and curvatures, Preprint q-ph/9910069 (1999).Google Scholar
  4. 4.
    K. Fujii, Mathematical foundations of holonomic quantum computer, Preprint q-ph/0004102 (2000).Google Scholar
  5. 5.
    K. Fujii, Mathematical foundations of holonomic quantum computer, II, Preprint q-ph/0101102 (2001).Google Scholar
  6. 6.
    K. Fujii, Introduction to Grassmann manifolds and quantum computations, Preprint q-ph/0103011 (2001).Google Scholar
  7. 7.
    C. Lorby, “Dynamical polysystems and control theory. Geometric methods in system theory,” in: Proc. NATO Adv. Stud. Inst. (1973), pp. 1–42.Google Scholar
  8. 8.
    J. Milnor and I. D. Stashe., Characteristic Classes, Princeton Univ. Press (1974).Google Scholar
  9. 9.
    K. Nomizu, Lie Groups and Differential Geometry, Math. Soc. Japan (1956).Google Scholar
  10. 10.
    J. Pachos, P. Zanardi, and M. Rasetti, Non-Abelian Berry connections for quantum computation, Preprint q-ph/9907103 (1999).Google Scholar
  11. 11.
    J. Pachos and P. Zanardi, Quantum holonomies for quantum computing, Preprint q-ph/0007110 (2000).Google Scholar
  12. 12.
    P. Zanardi and M. Rasetti, Holonomic quantum computation, Preprint q-ph/9904011 (1999).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Z. Giunashvili
    • 1
  1. 1.Department of Theoretical PhysicsInstitute of Mathematics, Georgian Academy of SciencesTbilisiGeorgia

Personalised recommendations