Mathematical Studies of Quantum Dynamics of Matter and Radiation

物质和辐射的量子动力学的数学研究

• 批准号: RGPIN-2017-06588
• 负责人: Sigal, IsraelMichael
• 金额: $ 2.7万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=665176
• 关键词: Mathematical; Studies; Quantum; Dynamics; Matter

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.

这个提议的主题是非相对论性粒子系统的动力学，与量子化场相互作用。具体来说，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;（ii）固体中的电子与固体晶体原子和量子化晶格振荡（声子）相互作用的动力学。这类系统的量子动力学由薛定谔方程描述，由狄拉克、费米和泡利发现，下面用SE表示。我们在这里解决的关键问题是描述SE解的长期行为。对于n个粒子（n可以是1024），如果忽略量子场，SE是3 n个变量的线性演化方程，如果不忽略量子场，SE是∞个变量的线性演化方程。令人惊讶的是，我们可以用这个方程理解多少，即使n=2，没有量子场，它也不能用计算机求解，除非有一个隐藏的降维。为了解决具体的问题，我们必须使用一个更简单、近似的动态。后者给出了一个粗略的描述，归零的一个理想的性质类别;它被称为有效的动态。因此，我们要解决的第二个问题是导出有效方程并描述它们的解的长时间行为。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结构和性质的基础。毫不奇怪，他们获得了不成比例的诺贝尔物理学奖和化学奖。然而，处理这些问题的数学工具还只是处于初步的建设阶段，特别是与过去几个世纪中为处理经典物理学问题而发展起来的数学技术相比。虽然该项目的主要目标是理论，但其主题与当前纳米器件，所有光学器件和量子计算背后的理论密切相关。在微观和介观尺度上，不仅电磁场，而且热也是量子化的，热的“粒子”正是我们建议中讨论的声子。引用著名的哈佛物理学家M。D. Lukin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，这是一项极具挑战性的任务。对量子动力学更好的数学理解可以帮助我们完成这项任务。