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产生的数学问题。对量子力学基础的研究主要涉及玻色力学--一种粒子运动的确定性理论，作为波函数薛定谔动力学的补充而自然出现--以及相关的模型。提出要探索的问题包括(1)Bohmian动力学在多大程度上可以用来促进求解含时薛定谔方程，以及(2)提供量子场论自然完善的数学结构的发展。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玻尔兹曼方程所支配的低密度区的系统也是如此。这两条研究路线将通过仔细分析玻尔兹曼的量子力学思想的正确性而融合在一起。我们的世界是一个量子世界：描述现实最深层次的自然规律是量子力学定律。虽然量子力学预测的正确性不应受到严重质疑，但其标准表述在概念上是不充分的。这个项目涉及与被称为玻色力学的非标准量子力学公式相关的数学结构的分析，这种公式比标准公式更简单，在概念上也更连贯。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，它原则上可以计算以前被认为不可能实现的复杂性，以及量子密码术，它提供了否则无法想象的安全级别。这一项目所涉及的研究将导致对量子理论的更深层次的理解，并将促进量子理论继续实现的技术突破。这项研究有望产生新的数学结构，并可能导致长期困扰量子理论基础的谜题和悖论的解决。

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.