Topics in Quantum Geometry

摘要：我提出发展和应用量子度量空间理论。我的出发点是阿兰·康内斯关于量子黎曼流形的思想。我的重点将是扩展到量子空间，在算子代数的情况下，度量空间之间的格罗莫夫-豪斯多夫距离的概念。我希望将这一理论应用到量子化的物理和数学中已经看到的许多情况中，其中一个量子空间序列似乎收敛到另一个空间，无论是量子空间还是经典空间。在我所知道的大多数情况下，目前的收敛概念要么只是启发式的，要么相当薄弱。我相信量子Gromov-Hausdorff收敛性通常可以用来给出一个更强的收敛形式。例子范围从有限维代数的近似，如经常出现在Berezin-Toeplitz量子化中，到量子场理论模型的近似，特别是那些可积系统的近似，比如通过晶格模型，一直到弦理论的假设“m理论”及其（假设的）近似“矩阵理论”。我们国家在技术和经济上的成功是建立在我们周围世界的数学模型的基础上的，科学家们为了理解如何使用从实验科学家的实验室流出的大量数据而开发了这些模型。但是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。但是，对于如何将复杂的模型作为一个整体很好地近似于简单的模型作为一个整体的研究却很少。相对而言，我们对量子物理模型中的这种“全局”近似所知甚少。量子物理是物理学中控制化学和生化反应、半导体功能和许多其他关键技术的部分。我建议将一种称为Gromov-Hausdorff距离的全局近似形式扩展到量子领域，近年来，它在经典领域中被证明是非常强大的，可以处理某些广泛的数学情况。我希望应用这种全局近似方法来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。

AbstractRieffelI propose to develop and apply a theory of quantum metric spaces. My point of departure will be the ideas of Alain Connes concerning quantum Riemannian manifolds. My emphasis will be on extending to quantum spaces, in the operator algebras setting, the notion of the Gromov-Hausdorff distance between metric spaces. I expect to apply this theory to the many situations already in view 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. In most such situations of which I am aware, the present notions of convergence are either just heuristic, or quite weak. I believe that a quantum Gromov-Hausdorff convergence can often be applied to give a stronger form of convergence. Examples range from approximation by finite dimensional algebras such as frequently occurs in Berezin-Toeplitz quantization, to approximation of quantum field-theory models, especially those of integrable systems, say by lattice models, all the way to the hypothesized "M-theory" of string theory and its (hypothesized) approximation by "Matrix theory".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. I propose to extend to the quantum realm one form of global approximation, called Gromov-Hausdorff distance, which has in recent years proved to be quite powerful in the classical realm for dealing with certain broad classes of mathematical situations. I expect to apply this global approximation method to better understand how to effectively approximate various models of quantum phenomena of current interest.