A geometrical setting for geometric phases on complex Grassmann manifolds

JOURNAL OF GEOMETRY AND PHYSICS

卷:57

Article

Abbati, M. C. ; Mania, A.

关键词: Garassmann manifolds;    geometric phases;    geodesics;    Stiefel bundles;    holonomy;   

DOI  :  10.1016/j.geomphys.2006.06.002

来源: Elsevier

【 摘 要 】

The geometry of Grassmann manifolds Gr(K)(H), of orthogonal projection manifolds P-K(H) and of Stiefel bundles St(K,H) is reviewed for infinite dimensional Hilbert spaces K and H. Given a loop of projections, we study Hamiltonians whose evolution generates a geometric phase, i.e. the holonomy of the loop. The simple case of geodesic loops is considered and the consistence of the geodesic holonomy group is discussed. This group agrees with the entire U(K) if H is finite dimensional or if dim(K) <= dim(K-perpendicular to). In the remaining case we show that the holonomy group is contained in the unitary Fredholm group U-infinity(K) and that the geodesic holonomy group is dense in U-infinity(K). (c) 2006 Elsevier B.V. All rights reserved.

【 授权许可】

Free