Subfactors, bimodules, and quantum mechanics

子因子、双模和量子力学

• 批准号: 0401734
• 负责人: Vaughan Jones
• 金额: --
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0401734&HistoricalAwards=false
• 关键词: Subfactors; bimodules; quantum; mechanics

The project involves continued work on subfactor and planar algebra theory and a new investigation of the relation between the Connes tensor product of bimodules over von Neumann algebras and very strongly intertwined quantum systems. The planar operad can be used to axiomatise a large class of hyperfinite subfactors and we intend to exploit this new point of view to better understand existing examples and discover new ones, as well as exploring planar algebras beyond the positivity condition required for subfactors. We say that two quantum systems are very strongly intertwined if there is an algebra of "common observables" which means that certain of one system automatically yield measurement of the other system. We would then expect the Hilbert space for the joint system to be the Connes tensor product of the individual Hilbert space, taken over the von Neumann algebra of common observables. We shall look for such systems and see if this kind of intertwining has observable consequences.The project is a continuing investigation of the mathematical structure of quantum mechanics-the study of the universe on a very small scale. The states of a system ("wave functions") are defined by a Hilbert space and operators on that Hilbert Space represent measurements. A von Neumann algebra is a collection of operators with certain physically relevant closure properties. "Factors" are von Neumann algebras with no operators commuting with all others in the algebra. The algebra of all observables localized in a region of space-time is a factor. Subfactors occur in interesting ways when considering the causal geometry of space-time. The project focuses on subfactors and a related way of combining two quantum systems called the Connes tensor product which is capable of identifying a von Neumann algebra of observables on one system with such an algebra on the other. There are potential applications of these ideas to quantum computing, especially through the approach of Michael Freedman.

该项目涉及在子因素和平面代数理论上继续进行工作，以及对von Neumann代数上双模模的Connes张量产品之间的关系的新研究，以及非常紧密的量子系统。平面作业可用于公理化一类大型的高铁子因子，我们打算利用这种新的观点，以更好地理解现有示例并发现新的示例，并探索超越子比例所需的积极条件的平面代数。我们说，如果存在“常见可观察物”的代数，这两个量子系统非常紧密地交织在一起，这意味着一个系统中的某些系统会自动对另一系统进行测量。然后，我们希望Hilbert空间使联合系统成为单个Hilbert Space的Connes张量产品，该产品占据了普通可观察物的von Neumann代数。我们将寻找这样的系统，看看这种交织是否具有可观察的后果。该项目是对量子力学的数学结构的持续研究 - 宇宙的研究在很小的范围内。系统（“波函数”）的状态是由希尔伯特空间定义的，而希尔伯特空间上的操作员则代表测量。冯·诺伊曼（Von Neumann）代数是具有某些物理相关闭合特性的运算符集合。 “因素”是von Neumann代数，没有操作员与代数中的所有其他代数通勤。所有可观察到位于时空区域中的观测值的代数是一个因素。当考虑时空的因果几何形状时，子因子以有趣的方式出现。该项目的重点是相关方法和一种相关的方法，将两个称为Connes Tensor产品的量子系统组合在一起，该系统能够在一个系统上识别一个可观察到的von Neumann代数，而另一种代数。这些想法有可能应用于量子计算，尤其是通过迈克尔·弗里德曼（Michael Freedman）的方法。