Geometric structure of a classifying space in the sector theory

扇形理论中分类空间的几何结构

• 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

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，该项目对无限量子系统的数学结构的目的是构建一个数学框架，用于根据对称性及其断裂或对热力学特性进行分类，描述和解释量子状态，以分类扇区的几何结构来分类；在这三年中，我的研究已经实现了这个目标。所达到的结果导致了关于量子状态的新观点，还导致了代数及其对整个特定物理系统的群体作用。状态的分类方案分为两个级别，第一个级别由中心代数中的订单参数指定的扇区，第二个级别，第二个级别涉及一个部门的内部结构。我在[1-4]中提出了适用于破裂的对称性和热处的通用行业的统一方案。要跳入一个部门的内部，我们需要用最大的Abelian sublgebra替换中心，并引入控制组偶性的Kac-Takesaki操作员，通过该操作员[6]通过该二重性来构建测量过程，例如仪器的概念。完成此操作后，就可以通过跨产品的Takeaki二元性回收微观量子系统的无限二二维非交通性代数。在这里很明显，外部动作的动力学对于要恢复的III型量子场的局部亚距离至关重要，这迫使我们在数学上面对以操作方式确定动力学的问题（i.ojima＆M.takeori，准备制备）。

项目成果

K.Nakamura, K.Watanabe, I.Ojima, A.Tonomura, et al.(共編著): "A Garden of Quanta --Essays in Honor of Hiroshi Ezawa--"World Scientific Publishing Company. 502 (2003)

K.Nakamura、K.Watanabe、I.Ojima、A.Tonomura 等人（共同编辑）：“量子花园——纪念江泽宏的论文——”世界科学出版公司 502 (2003)。

Generalized Sectors and Adjunctions to Control Micro-Macro Transitions

控制微观-宏观转变的广义部门和附属机构

• 发表时间: 2006
• 期刊: Quantum Information and Computing, QP-PQ : Quantum Probability and White Noise Analysis Vol.19(to appear)
• 作者: Izumi Ojima

Temperature as order parameter of broken scale invariance

温度作为破坏尺度不变性的有序参数

• 发表时间: 2004
• 期刊: Publ.RIMS (Kyoto University) 40, No.3
• 作者: Izumi Ojima (with C.J.Fewster;M.Porrmann);Izumi Ojima
• 通讯作者: Izumi Ojima

Izumi Ojima: "Generalized sectors and adjunctions to control micro-macro transitions"Proc.of Intern.Conference on Quantum Information,2003 --mathematical, physical engineering and industrial aspects--. (to appear). (2004)

Izumi Ojima：“控制微观宏观转变的广义部门和辅助”Proc.of Intern.Conference on Quantum Information，2003 - 数学、物理工程和工业方面 -。

A Garden of Quanta--Essays in Honor of Hiroshi Ezawa--

量子花园--江泽宏纪念文--

• 发表时间: 2003
• 作者: K.Nakamura;K.Watanabe;I.Ojima;A.Tonomura et al.(共編著)
• 通讯作者: A.Tonomura et al.(共編著)