Quantum Field Theory on Noncommutative Space and Its Applications

Ezawa研究了量子霍尔（QH）系统作为量子场论在非交换空间中的应用。QH系统是平面电子的世界，其中x和y坐标不可交换。当电子具有SU(N)对称性时，代数结构成为W_∞代数的SU(N)扩展，他将其命名为W_∞(N)代数。由于非交换性，他证明了量子相干性是由电子之间的交换相互作用自发发展的。这一理论结果可以通过观察与量子相干性相关的拓扑孤子进行实验验证。令人感兴趣的是，在QH系统中拓扑孤子是一个非交换孤子。作为主要成果之一，他通过对空穴态进行W_∞(N)旋转，构建了非对易孤子（skyrmion）的量子力学态。此外，他还计算了单层QH系统和双层QH系统中粒子的激发能，并将理论结果与实验数据进行了比较。Watamura研究了超弦理论中出现的非（反）交换空间的规范理论中的非平凡组态。这项研究对了解d膜的性质具有重要意义。特别地，他利用投影模构造构造了模糊球上的非平凡组态——单极束。通过推广该方法，他澄清了可交换CPn上束的结构。一方面，在四维空间中，我们可以使用所谓的ADHM结构来解决这个问题。然后，他将ADHM方法推广到非反交换的超空间，并通过构造实例解来分析模空间。即使在交换情况下，超场方法在该问题上的应用也是一个开放问题。他成功地在超空间中构造了包括量规固定在内的完整的瞬变解。他证明了费米子和玻色子模空间的变形对应于变形的瞬子解推广到非反交换的情况。少

Z. F. Ezawa made an investigation on quantum Hall (QH) systems as an application of the quantum field theory on noncommutative space. The QH system is a world of planar electrons, where the x and y coordinates become noncommutative. When the electron possesses the SU(N)symmetry,the algebraic structure becomes the SU(N)extension of the W_∞ algebra,which he has named the W_∞(N)algebra. Due to the noncommutativity he has shown that quantum coherence develops spontaneously driven by the exchange interaction between 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. As one of the main results, he has constructed the quantum mechanical state of a noncommutative soliton (skyrmion) by making a W_∞(N)rotation of a hole state. Furthermore, calculating the excitation energy of a skyrmion both in the monolayer QH system and the … More bilayer QH system, he has compared successfully his theoretical results with the experimental data.S. Watamura has studied non-trivial configurations in the gauge theory on the non-(anti)commutative spaces which are emerging from the superstring theory. This research is important to understand the properties of a D-brane. Especially, he has constructed a non-trivial configuration, the monopole bundle on the Fuzzy sphere by using the projective module construction. Generalizing that method, he could clarify the structure of the bundles on concommutative CPn. On the one hand, in 4-dimensional space one can solve this problem by using the so-called ADHM construction. He then generalized the ADHM method to the non-anticommutative superspace and analyzed the moduli space by constructing the instanton solutions. Applying the superfield method on this problem was an open problem even in the commutative case. He has succeeded to construct the Instanton solutions completely in the superspace, including the gauge fixing. He has shown the deformation of the fermionic and bosonic moduli space corresponding to the deformed instanton solution generalized to the nonanticommutative case. Less