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Geometric structure of a classifying space in the sector theory

扇形理论中分类空间的几何结构

基本信息

批准号：
15540117
负责人：
OJIMA Izumi
金额：
$ 1.98万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540117/
关键词：
Micro-Macro duality Reconstruction of Micro system from Macro data Kac-Takesaki operator selection criterion adjunction crossed product Takesaki duality symmetry breaking Micro-Macri duality セクター内構造極大可換部分環双対作用 adjunctions and dua

项目摘要

At the starting point, the aim of this project on the mathematical structures of infinite quantum systems was to construct a mathematical framework for classifying, describing and interpreting the quantum states according to the symmetries and their breakings or to the thermodynamic properties in terms of geometric structures of classifying spaces of sectors ; this goal has been achieved by my research in these three years. The attained results lead to a new perspective concerning not only quantum states but the algebra and the group action on it to characterize a specific physical system as a whole. The classification scheme of states is divided into two levels, the first one consisting of sectors specified by order parameters in the centre algebra and the second one concerning the internal structure of a sector. Generalizing the Doplicher-Haag-Roberts sector theory for unbroken symmetries, I proposed in [1-4] a unified scheme for generalized sectors applicable also to broken symmetries and thermal situations. To jump into the inside of a sector, we need to replace the centre by a maximal abelian subalgebra and to introduce a Kac-Takesaki operator controlling the group duality, by which all the necessary ingredients are implemented [6] to construct a measurement' process, such as the notion of instrument. Once this is done, it becomes possible to recover the infinite-dimensional non-commutative algebra of microscopic quantum system from the mathematical structure of measured data through the Takesaki duality for crossed products. It is evident here that the dynamics with an outer action is crucial for a local subalgebra of quantum fields of type III to be restored, which forces us to face mathematically the problem of determining a dynamics in an operational way (I.Ojima & M.Takeori, in preparation).

在起点上，这个关于无限量子系统数学结构的项目的目的是构建一个数学框架，用于根据对称性和它们的破缺或根据扇形分类空间的几何结构的热力学性质对量子态进行分类、描述和解释；通过这三年的研究，我已经达到了这个目标。所获得的结果带来了一个新的视角，不仅涉及量子态，而且涉及代数和对它的群作用，以表征一个整体的特定物理系统。将状态分类方案分为两个层次，第一级由中心代数中由序参数指定的扇区组成，第二级涉及扇区的内部结构。我在[1-4]中推广了完整对称的doplicher - hag - roberts扇形理论，提出了一个适用于破碎对称和热情况的广义扇形统一方案。为了进入扇区的内部，我们需要用一个极大的阿贝尔子代数取代中心，并引入一个控制群对偶性的Kac-Takesaki算子，通过它实现所有必要的成分来构建一个测量过程，例如仪器的概念。一旦做到这一点，就有可能通过交叉积的Takesaki对偶从测量数据的数学结构中恢复微观量子系统的无限维非交换代数。很明显，具有外部作用的动力学对于III型量子场的局部子代数的恢复至关重要，这迫使我们面对以操作方式确定动力学的数学问题（I.Ojima & m.a okori，在准备中）。