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.

该项目涉及继续工作的子因子和平面代数理论和一个新的调查之间的关系康纳斯张量积双模冯诺依曼代数和非常强烈的纠缠量子系统。平面运算可用于公理化一大类超有限子因子，我们打算利用这一新的观点，以更好地了解现有的例子，发现新的，以及探索平面代数以外的积极性条件所需的子因子。我们说，两个量子系统是非常强烈地交织在一起，如果有一个代数的“共同可观”，这意味着某些一个系统自动产生测量的其他系统。这样，我们就可以期望联合系统的希尔伯特空间是单个希尔伯特空间的康纳斯张量积，它覆盖了公共观测量的冯·诺依曼代数。我们将寻找这样的系统，看看这种交织是否有可观察到的后果。该项目是对量子力学数学结构的持续研究--在非常小的尺度上研究宇宙。系统的状态（“波函数”）由希尔伯特空间定义，希尔伯特空间上的算子表示测量。冯·诺依曼代数是具有某些物理相关闭包性质的算子的集合。“因子”是冯诺依曼代数，没有与代数中所有其他算子交换的算子。在一个时空区域中的所有可观测量的代数是一个因素。在考虑时空的因果几何时，子因子以有趣的方式出现。该项目的重点是子因子和一种相关的方法，结合两个量子系统称为康纳斯张量积，这是能够确定一个系统上的冯诺依曼代数的观测与这样的代数上的其他。这些想法在量子计算中有潜在的应用，特别是通过Michael Freedman的方法。