Physics in Low-Dimensional Space and Noncommutative Geometry

最近，非对易空间中的场论引起了人们的广泛关注，这里的坐标是非对易的，[x，y]=-iθ。相应的场论是通过用所谓的星积代替两个场的普通乘积而得到的。最简单的非对易空间有望在2维空间中实现，尽管它在粒子理论中得到了广泛的研究，但唯一受非对易几何支配的现实物理系统是量子霍尔(QH)系统。QH系统是一个平面电子的世界，被限制在强磁场下的最低朗道能级，其中x和y坐标变得非对易。因此，当电子具有SU(N)对称性时，系统的代数结构成为W_∞代数的SU(N)扩张，我们称之为W_∞(N)代数。从这个角度出发，我们建立了QH系统的量子场论。由于非对易，我们…更多的研究表明，在相邻电子之间的交换作用的驱动下，量子相干性自发地发展起来。通过观察与量子相干性相关的拓扑孤子，这一理论结果在实验上是可验证的。有趣的是，在QH系统中，拓扑孤子是一种非对易孤子。到目前为止，拓扑孤子是经典的场结构。作为主要结果，我们通过使空穴状态W_∞(N)旋转来构造非对易孤子(天空)的量子力学状态。我们还推导出了天空离子的电子密度和拓扑密度之间的精确关系。此外，我们还计算了单层QH系统和双层QH系统中天空离子的激发能。我们还将我们的结果与现有的实验数据进行了比较。以QH系统为例，对非对易空间中的物理进行了分析，并证明了非对易几何中各种概念的正确性。较少

Recently much attentions have been paid to the field theory in the noncommutative space, where the coordinates are assumed to be noncommutative, [x,y]=-iθ. The corresponding field theory is obtained by replacing the ordinary product of two fields with the so-called star product. The simplest noncommutative space is expected to be realized in the 2-dimensional space.Though it is studied extensively in particle theories, the only realisitic physical system governed by the noncommutative geometry is the quantum Hall (QH) system. The QH system is a world of planar electrons confined to the lowest Landau level under a strong magnetic field, where the x and y coordinates become noncommutative. As a result, when the electron possesses the SU(N) symmetry, the algebraic structure of the system becomes the SU(N) extension of the W_∞ algebra, which we have named the W_∞(N) algebra. We have constructed the quantum field theory of the QH system from this point of view.Due to the noncommutativity we … More have shown that a quantum coherence develops spontaneously driven by the exchange interaction between neighboring electrons. This theoretical result is experimentally testable by observing topological solitons associated with the quantum coherence. It is intriguing that the topological soliton is a noncommutative soliton in QH systems.Topological solitons have been known so far to be classical field configurations. As a main result, we have constructed a quantum mechanical state of a noncommutative soliton (skyrmion) by making a W_∞(N) rotation of a hole state. We have also derived an exact relation between the electron density and the topological density of a skyrmion. Furthermore, we calculated the excitation energy of a skyrmion both in the monolayer QH system and the bilayer QH system. We have also compared our results with the available experimental data. In this way we have analyzed physics in the noncommutaive space, taking an instance of the QH system, and demonstrated the validity of various concepts on the noncommutative geometry. Less