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Physics in Low-Dimensional Space and Noncommutative Geometry

低维空间物理与非交换几何

基本信息

批准号：
14540237
负责人：
EZAWA Zyun F.
金额：
$ 2.18万
依托单位：
TOHOKU UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540237/
关键词：
Noncommutative Geometry Noncommutative Soliton Quantum Hall Effects W_∞ Algebra Quantum Coherence Composite Boson Topological Soliton Skyrmion W(∞)代数複合フェルミオン分数統計

项目摘要

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

近年来，非对易空间中的场论受到了广泛的关注，其中坐标被假定为非对易的，[x，y]=-iθ。相应的场论是用所谓的星星积代替两个场的普通积而得到的。最简单的非对易空间是在二维空间中实现的，虽然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实在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，在强磁场下被限制在最低的朗道能级，其中x和y坐标变得非对易。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，我们构造了QH系统的量子场论，由于非对易性， ...更多信息已经表明，量子相干性在相邻电子之间的交换相互作用的驱动下自发地发展。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检验的。有趣的是，拓扑孤子是QH系统中的非对易孤子，拓扑孤子是迄今为止已知的经典场组态。作为主要结果，我们通过对空穴态进行W_∞（N）旋转，构造了一个非对易孤子（Skyrmion）的量子力学态。我们还导出了Skyrmion的电子密度与拓扑密度之间的精确关系。此外，我们还计算了单层QH和双层QH体系中skyrmion的激发能。我们还比较了我们的结果与现有的实验数据。以QH系统为例，分析了非对易空间中的物理，论证了非对易几何中各种概念的正确性。少