Mathematical Studies of Quantum Dynamics of Matter and Radiation

这个提议的主题是非相对论粒子系统与量子化的场相互作用的动力学。具体地说，它涉及(I)真空中的电子与原子核和量子化电磁场(光子)相互作用的动力学；以及(Ii)固体中的电子与固体晶体原子相互作用和量子化晶格振荡(声子)的动力学。 这类系统的量子动力学由狄拉克、费米和保利发现的薛定谔方程描述，下面记为SE。我们在这里解决的关键问题是描述SE解决方案的长期行为。 对于n个粒子(n可以大到1024)，如果忽略量子场，则SE是3n个变量的线性演化方程，如果不忽略，则SE是变量个数的线性演化方程。令人惊讶的是，使用这个方程我们可以理解得如此之多，因为即使对于n=2并且没有量子场，它也无法使用计算机来求解，除非存在隐藏的降维。 为了解决特定的问题，我们必须使用一个更简单、近似的动态。后者给出了一个更粗略的描述，将重点放在所需的一类属性上；它被称为有效动态。因此，我们解决的第二个问题是推导有效方程并描述其解的长期行为。 构建量子非相对论物质和辐射的严格理论，并为此开发相应的数学工具，对于理解我们周围的物理世界至关重要。它也是基础数学问题的丰富来源，这些问题可能会在分析中产生新的分支。 提案中描述的问题是物质的结构和性质的基础。毫不奇怪，他们在物理和化学领域获得了不成比例的诺贝尔奖。然而，处理它们的数学装置只处于建造的初始阶段，特别是与过去几个世纪为处理经典物理问题而发展的数学技术相比。 虽然这个项目的主旨是理论，但它的主题与当前在纳米设备、所有光学设备和量子计算方面的努力背后的理论密切相关。在微观和介观尺度上，不仅电磁场是量子化的，热也是量子化的，而热的粒子正是我们所讨论的声子。 引用哈佛大学著名物理学家M·D·卢金的话说，量子信息的实际实现需要精确操纵来自许多耦合量子力学系统的光和物质的量子态，这是一项极其具有挑战性的任务。对量子动力学的更好的数学理解可以帮助我们管理这项任务。

The subject of this proposal is the dynamics of systems of non-relativistic particles, interacting with quantized fields. Specifically, it deals with the dynamics of (i) electrons in vacuum, interacting with the nuclei and with the quantized electromagnetic field (photons); and (ii) electrons in solids interacting with the solid crystal atoms and with the quantized crystal lattice oscillations (phonons). The quantum dynamics of such systems are described by the Schrödinger-type equation, discovered by Dirac, Fermi and Pauli and denoted below as SE. The key problem we address here is to describe the long-time behaviour of the solutions of SE. For n particles (n could be as large as 1024), SE is a linear evolution equation in 3n variables if one neglects quantum fields, and in number of variables if one does not. It is surprising how much we can understand using this equation, given that even for n=2 and without quantum fields, it cannot be solved using computers, unless there is a hidden dimensional reduction. To address specific questions we have to use a simpler, approximate dynamic. The latter gives a rougher description, zeroing on a desired class of properties; it is called the effective dynamic. Hence, the second problem we address is to derive the effective equations and describe their long- time behaviour of the solutions. Constructing rigorous theory of quantum non-relativistic matter and radiation and developing the corresponding mathematical tools for this is critical for understanding the physical world around us. It is also a rich source of fundamental mathematical problems which could potentially give rise to new branches in Analysis. The issues described in the proposal underlie the structure and properties of matter. Not surprisingly, they have garnered a disproportionate number of Nobel prizes in Physics and Chemistry. Nevertheless the mathematical apparatus dealing with them is only in the initial stages of construction, especially compared with the mathematical techniques developed in the last few centuries to deal with problems of classical physics. Though the main thrust of this project is theoretical, its subject matter is closely related to the theory behind the current effort in nano devices, all optical devices and in quantum computing. On micro- and meso-scopic scales, not only the electromagnetic field, but also heat is quantized, and particles' of heat are exactly the phonons discussed in our proposal. To quote from a well-known Harvard physicist M. D. Lukin, “the practical implementation of quantum information requires precise manipulation of quantum states of light and matter coming from many coupled quantum-mechanical systems, which is an extremely challenging task.” A better mathematical understanding of quantum dynamics can help us to manage this task.