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Paper   IPM / Physic / 6523
School of Physics
  Title:   Geometric Phases, Bundle Classification, and Group Representation
  Author(s):  A. Mostafazadeh
  Status:   Published
  Journal: J. Math. Phys.
  No.:  27
  Vol.:  37
  Year:  1996
  Pages:   1218-1233
  Supported by:  IPM
The line bundles that arise in the holonomy interpretations of the geometric phase display curious similarities to those encountered in the statement of the Borel-Weil-Bott theorem of the representation theory. The remarkable relationship between the mathematical structure of the geometric phase and the classification theorem for complex line bundles provides the necessary tools for establishing the relevance of the Borel-Weil-Bott theorem to Berry's adiabatic phase. This enables one to define a set of topological charges for arbitrary compact connected semi-simple dynamical Lie groups. These charges signify the topological content of the phase. They can be explicitly computed. In this paper, the problem of the determination of the parameter space of the Hamiltonian is also addressed. It is shown that, in general, the parameter space is either a flag manifold or one of its submanifolds. A simple topological argument is presented to indicate the relation between the Riemannian structure on the parameter space and Berry's connection. The results about the fiber bundles and gorup theory are used to introduce a procedure to reduce the problem of the nonadiabatic (geometric) phase to Berry's adiabatic phase for cranked Hamiltonians. Finally, the possible relevance of the topological charges of the geometric phase to those of the non-Abelian monopoles is pointed out.

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