Explorations in Metric Quantum Geometry

在过去的几年里，首席研究员一直在发展量子度量空间的理论，也就是说，在普通空间上配备了距离（即度量）概念的模拟的量子空间。他在希尔伯特空间的算子代数集合中做了这个，这是物理学家用来模拟量子系统的集合。他的理论的核心部分是度量空间之间经典格罗莫夫-豪斯多夫距离的量子模拟。他给出了这些思想的几个应用，特别是矩阵代数收敛到紧李群的伴轨道，这是物理学家感兴趣的主要例子之一。最近，他在扩展量子Gromov-Hausdorff距离的概念方面取得了初步成功，从而可以处理量子度量空间上的向量束的类似物。（矢量束为物理学家广泛使用的规范理论提供了基础。）他最近还成功地利用狄拉克算子带来的技术优势，用狄拉克算子来表示量子距离。在这个项目中，他将继续加强这一理论，并将其应用于量子模型的许多物理和数学情况中提出的几个方向，其中一个量子空间序列似乎收敛于另一个空间，无论是量子空间还是经典空间。特别是，他将尝试为超越向量束的超结构的量子版本开发量子Gromov-Hausdorff收敛的类似物，例如连接、Dirac算子和Yang-Mills作用。他也希望开始处理动力学问题。我们国家在科技和经济上的成功是以我们周围世界的数学模型为基础的。科学家开发这样的模型是为了了解如何使用从实验科学家的实验室流出的大量数据。然而，人类和计算机一次只能处理有限的数字集合。因此，在应用这些数学模型时，几乎总是需要用有限的数字集合来近似我们世界的无限可变性。因此，理解任何给定近似的有效性是至关重要的。就个别计算而言，这个问题已得到广泛的研究。另一方面，关于如何将复杂模型作为一个整体很好地近似于简单模型作为一个整体的研究较少。在量子物理模型的情况下，对这种“全局”近似的了解相对较少。由于量子物理学是控制化学和生化反应、半导体功能和许多其他关键技术的物理学的一部分，因此理解复杂的量子模型如何能够被更简单的模型所近似是非常重要的。在经典领域，被称为“Gromov-Hausdorff距离”的全局近似概念被广泛使用。首席研究员已经开发了一个格罗莫夫-豪斯多夫距离的量子模拟，并成功地将其应用于几个重要的例子。在这个项目中，他试图加强这一理论，并将加强的理论应用于更广泛的例子。这将使我们更好地理解如何以有效的方式近似当前重要的量子现象的各种模型。

During the past few years the principal investigator have been developing a theory of quantum metric spaces, that is, of quantum spaces equipped with the analog of the notion of a distance (i.e., metric) on an ordinary space. He does this within the setting of algebras of operators on Hilbert space, which is the setting used by physicists to model quantum systems. A central part of his theory consists of a quantum analog of the classical Gromov-Hausdorff distance between metric spaces. He has given several applications of these ideas, notably to the convergence of matrix algebras to coadjoint orbits of compact Lie groups, which is one of the main examples of interest to physicists. Recently he has had initial success in extending his notion of quantum Gromov-Hausdorff distance so as to treat the analogs of vector bundles over quantum metric spaces. (Vectors bundles provide the foundation for the gauge theories widely used by physicists.) He has also had recent success going in the direction of expressing the quantum distances in terms of Dirac operators, with the technical advantages that Dirac operators bring. In this project he will continue to strengthen this theory and to apply it in several directions suggested by the many situations in the physics and mathematics of quantum models where one has a sequence of quantum spaces that appears to converge to another space, either quantum or classical. In particular, he will try to develop analogs of quantum Gromov-Hausdorff convergence for quantum versions of superstructures beyond vector bundles, such as for connections, Dirac operators, and Yang-Mills actions. He hopes, as well, to begin treating dynamical issues.Our nation's technological and economic success has at its foundation the mathematical models of the world around us. Scientists develop such models in order to understand how to use the flood of data that flows from the laboratories of the experimental scientists. However, human beings and computers can deal only with finite collections of numbers at a time. Thus, in applying these mathematical models, it is almost always necessary to approximate the infinite variability of our world by finite collections of numbers. It is then crucial to understand how valid any given approximation is. With respect to individual calculations, this matter has received extensive study. On the other hand, less study has been made of how complex models as a whole can be approximated well by simpler models as a whole. Relatively little is known about such "global" approximations in the case of the models of quantum physics. Since quantum physics is the part of physics that governs chemical and biochemical reactions, the functioning of semiconductors, and many other key technologies, it is of great importance to understand how well complex quantum models can be approximated by simpler models. In the classical realm, the global notion of approximation known as "Gromov-Hausdorff distance" is widely used. The principal investigator has developed a quantum analog of Gromov-Hausdorff distance and successfully applied it to a few significant examples. In this project he seeks to strengthen this theory and to apply the strengthened theory to a broader class of examples. This should lead to a better understanding of how to approximate in an effective manner various models of quantum phenomena of current importance.