Foundations of Quantum Mechanics and Statistical Mechanics

这个项目研究从量子力学基础和统计力学基础中产生的数学问题。 对量子力学基础的研究主要涉及Bohmian力学-粒子运动的确定性理论自然出现作为波函数的薛定谔动力学的完成-以及相关模型。 其中提出的探索问题是：（1）在何种程度上可以利用玻姆动力学来促进解决与时间有关的薛定谔方程，（2）数学结构的发展提供了一个自然完成量子场论。 对统计力学基础的研究主要涉及热力学第二定律和玻尔兹曼熵作为非平衡熵的地位，即使对于远离平衡和远离由玻尔兹曼方程控制的低密度区域的系统也是如此。 这两条研究路线将通过仔细分析玻尔兹曼在量子力学中的思想的有效性来融合。我们的世界是一个量子世界：描述现实最深层次的自然定律是量子力学定律。 虽然量子力学预言的有效性不应受到严重质疑，但它的标准公式在概念上是不够的。 该项目涉及与量子力学的非标准公式（称为Bohmian力学）相关的数学结构分析，该公式比标准公式更简单，概念上更连贯。 近年来，量子力学一直是引人注目的技术创新的基础。 例子包括量子计算机，它在原则上将允许计算的复杂性以前被认为是不可能的，和量子密码学，它提供了一个安全的水平，否则将是不可想象的。 该项目所涉及的研究将导致对量子理论的更深入理解，并将促进量子理论继续实现的技术突破。 这项研究预计将产生新的数学结构，并可能导致解决困扰量子理论基础的长期难题和悖论。

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.