CONNECTIONS OF BERRY AND HANNAY TYPE FOR MOVING LAGRANGIAN SUBMANIFOLDS
CONNECTIONS OF BERRY AND HANNAY TYPE FOR MOVING LAGRANGIAN SUBMANIFOLDS
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DOI：
10.1016/0001-8708(90)90086-3
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发表时间：
1990-08-01
影响因子：
1.7
通讯作者：
WEINSTEIN, A
中科院分区：
文献类型：
--
作者：
WEINSTEIN, A
Berry相是射影化量子希尔伯特空间上正则圆丛上的自然联络的完整性。给定一个辛流形P，这个相的一个经典极限被构造为P中具有光滑密度的拉格朗日子流形空间上的有限余维等势叶（由作用积分的常数定义）上的Berry连通的完整性.如果密度由与辛结构相容的Kähler度量确定，则在拉格朗日子流形L上的Berry联络的曲率由一个涉及L曲率的简单公式给出。特别地，Berry联络定义了单连通Kähler流形中极小拉格朗日子流形的等压线环的同伦不变量。作者为辛同胚的环构造的一个类似的不变量也是经典Berry相的一个特例。贝里相位的一个经典类似物是由汉内发现的移动族的完全可积系统。根据Berry和Hannay的观点，我们将Hannay角解释为一束环面上的Hannay连接的完整性，该环面在拉格朗日环面层的空间上的等幅叶理上，该拉格朗日环面层由具有平坦仿射结构的拉格朗日环面和该结构到第一无穷小邻域的扩展组成。最后，我们证明了Hannay角是经典Berry相作用量的导数。
Berry's phase is the holonomy of the natural connection on the canonical circle bundle over projectivized quantum Hilbert space. Given a symplectic manifoldP, a classical limit of this phase is constructed as the holonomy of aBerry connectionover a finite-codimensionalisodrasticfoliation (defined by constancy of action integrals) on the space of lagrangian submanifolds inPequipped with smooth densities of total measure 1. If the densities are determined by a Kähler metric compatible with the symplectic structure, the curvature of the Berry connection at the lagrangian submanifoldLis given by a simple formula involving curvature ofL. In particular, the Berry connection defines a homotopy invariant for isodrastic loops of minimal lagrangian submanifolds in a simply connected Kähler manifold. A similar invariant constructed by the author for loops of symplectomorphisms is also a special case of the classical Berry phase. A classical analogue of Berry's phase was discovered by Hannay for moving families of completely integrable systems. Following Berry and Hannay, we interpret Hannay's angles as the holonomy of aHannay connectionon a bundle of tori over the isodrastic foliation on the space oflagrangian toral layersconsisting of lagrangian tori with a flat affine structure and an extension of this structure to the first infinitesimal neighborhood. Finally, we show that the Hannay angles are the derivatives with respect to action variables of the classical Berry phase.