Subfactors, bimodules, and quantum mechanics

该项目涉及子因子和平面代数理论的继续工作，以及冯·诺伊曼代数上双模的Connes张量积与非常强交织量子系统之间关系的新研究。平面运算符可以用来公理化一大类超有限子因子，我们打算利用这个新的观点来更好地理解现有的例子和发现新的例子，以及探索超越子因子所要求的正条件的平面代数。我们说两个量子系统紧密地交织在一起，如果存在一个“共同可观测”的代数，这意味着一个系统的某些部分自动产生对另一个系统的测量。然后我们期望关节系统的希尔伯特空间是单个希尔伯特空间的康内斯张量积，它占据了von Neumann代数的可观测值。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。这个项目是对量子力学的数学结构的持续研究——在很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·诺伊曼代数是具有某些物理上相关闭包性质的算子的集合。“因子”是冯·诺伊曼代数，没有与代数中所有其他算子交换的算子。在一个时空区域中所有可观测的代数是一个因素。当考虑时空的因果几何时，子因子以有趣的方式出现。该项目侧重于子因子和结合两个量子系统的相关方法，称为Connes张量积，它能够识别一个系统上的冯诺依曼可观测代数与另一个系统上的这种代数。这些想法在量子计算中有潜在的应用，特别是通过迈克尔·弗里德曼的方法。

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.