Investigations in Metric Quantum Geometry

在过去的几年里，PI在Hilbert空间上的算子代数的设置下，发展了一种量子度量空间的理论。我的理论包括与经典的度量空间之间的Gromov-Hausdorff距离的类比。我给出了这些思想的几个应用，特别是在矩阵代数收敛到紧李群的余共轭轨道方面。我建议继续加强这一理论，并将其应用于量子化物理和数学中的许多情况所建议的几个方向，在这些情况下，一个人拥有一系列似乎正在收敛到另一个空间的量子空间，无论是量子空间还是经典空间。作为一个主要的新方向，我将尝试发展一种类似于向量丛、连接、杨-米尔斯作用量等超结构的量子版本的理论。我还将尝试将我的理论扩展到局部紧空间的量子模拟之外，以尝试处理量子场论模型的近似，特别是可积系统的近似，例如通过量子晶格模型。我们国家的技术和经济成就建立在我们周围世界的数学模型的基础上，科学家们开发这些模型是为了了解如何利用从实验科学家的实验室涌出的大量数据。但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，理解任何给定的近似值的有效性是至关重要的。关于个人计算，这一问题得到了广泛的研究。但对于复杂模型作为一个整体如何才能很好地被简单模型作为一个整体进行近似的研究较少。在量子物理模型的情况下，人们对这种“全局”近似的了解相对较少。量子物理模型是管理化学反应和生化反应、半导体的功能以及许多其他关键技术的物理学部分。在经典领域中，有一种重要的全局逼近形式，称为Gromov-Hausdorff距离。我已经开发了它的量子模拟，并成功地将其应用于几个例子。我建议加强这一理论，并将其应用于更广泛的例子类别，以便更好地理解如何有效地近似当前感兴趣的量子现象的各种模型。

Over the past several years the PI has developed a theory of quantum metric spaces, within the setting of algebras of operators on Hilbert space. My theory includes an analog of the classical Gromov-Hausdorff distance between metric spaces. I gave several applications of these ideas, notably to the convergence of matrix algebras to coadjoint orbits of compact Lie groups. I propose to continue to strengthen this theory, and to apply it in several directions suggested by the many situations in the physics and mathematics of quantization where one has a sequence of quantum spaces which appear to be converging to another space, either quantum or classical. As a major new direction I will try to develop an analogous theory for the quantum versions of the superstructure of vector bundles, connections, Yang-Mills actions, etc. I will also try to extend my theory beyond the quantum analog of locally compact spaces, so as to attempt to deal with the approximations of quantum field-theory models, especially those of integrable systems, say by quantum lattice models. Our nation's technological and economic success has at its foundation the mathematical models of the world around us which scientists develop in order to understand how to use the flood of data which flows from the laboratories of the experimental scientists. But human beings and computers can only deal 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. But 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, which is the part of physics which governs chemical and biochemical reactions, the functioning of semi-conductors, and many other key technologies. In the classical realm there is an important form of global approximation called Gromov-Hausdorff distance. I have developed a quantum analog of it, and successfully applied it to a few examples. I propose to strengthen this theory, and to apply it to a broader class of examples, so as to better understand how to effectively approximate various models of quantum phenomena of current interest.