Foundations of Quantum Mechanics and Statistical Mechanics

量子力学和统计力学基础

• 批准号: 0504504
• 负责人: Sheldon Goldstein
• 金额: $ 12.6万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-09-01 至 2009-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0504504&HistoricalAwards=false
• 关键词: Foundations; Quantum; Mechanics; Statistical; Foundations; Quantum; Mechanics; Statistical

This project investigates mathematical problems arising from the foundations of quantum mechanics and from the foundations of statistical mechanics. The research on the foundations of quantum mechanics is concerned primarily with Bohmian mechanics -- a deterministic theory of particle motion naturally emerging as a completion of the Schrodinger dynamics for the wave function -- and related models. Among the problems proposed for exploration are (1) the extent to which the Bohmian dynamics can be utilized to facilitate the solution to the time-dependent Schrodinger equation, and (2) the development of mathematical structures providing a natural completion of quantum field theory. The research on the foundations of statistical mechanics is concerned primarily with the second law of thermodynamics and the status of the Boltzmann entropy as a nonequilibrium entropy, even for systems far from equilibrium and far from the low-density regime governed by the Boltzmann equation. These two lines of research will be merged through careful analysis of the validity of Boltzmann's ideas in quantum mechanics.Our world is a quantum world: the laws of nature describing the deepest levels of reality are quantum mechanical laws. While the validity of the predictions of quantum mechanics should not be seriously questioned, its standard formulation is conceptually inadequate. This project concerns the analysis of mathematical structures relevant to a nonstandard formulation of quantum mechanics called Bohmian mechanics, a formulation that is at once simpler than the standard one and conceptually much more coherent. Quantum mechanics has been the basis of remarkable technological innovations in recent years. Examples include quantum computers, which in principle would allow computations of a complexity previously regarded as impossible, and quantum cryptography, which affords a level of security that would otherwise be inconceivable. The research with which this project is concerned will lead to a deeper understanding of quantum theory and will facilitate the technological breakthroughs that quantum theory continues to make possible. The research is anticipated to generate novel mathematical structures, and it may lead to the resolution of longstanding puzzles and paradoxes that plague the foundations of quantum theory.

该项目研究了量子力学基础和统计力学基础引起的数学问题。 量子力学基础的研究主要与Bohmian力学有关 - 粒子运动的确定性理论自然出现，作为波动功能的Schrodinger动力学的完成 - 以及相关的模型。 在探索提出的问题中，（1）可以利用Bohmian动力学来促进解决时间依赖性的Schrodinger方程的解决方案，以及（2）数学结构的发展提供了量子场理论的自然完成。 关于统计力学基础的研究主要与热力学的第二定律和玻尔兹曼熵作为非平衡熵的状态有关，即使对于远离均衡的系统，远离鲍尔茨曼方程控制的低密度政权也很远。 这两条研究将通过仔细分析Boltzmann在量子力学中的思想的有效性进行合并。我们的世界是一个量子世界：描述现实最深水平的自然定律是量子力学定律。 尽管不应严重质疑量子力学预测的有效性，但其标准配方在概念上是不足的。 该项目涉及与称为Bohmian Mechanics的量子力学的非标准公式相关的数学结构的分析，该公式既比标准阶层都更简单，并且从概念上讲更一致。 近年来，量子力学一直是近年来技术创新的基础。 示例包括量子计算机，原则上将允许计算先前被视为不可能的复杂性和量子密码学，而量子密码学则提供了一定程度的安全性，否则这些安全性是不可想象的。 涉及该项目的研究将导致对量子理论的更深入的了解，并促进量子理论继续使其成为可能的技术突破。 预计这项研究将产生新颖的数学结构，并可能导致解决困扰量子理论基础的长期难题和悖论的解决方案。