Mathematical Sciences: Exact Results in Statistical Mechanics and the Foundations of Quantum Mechanics

研究人员打算继续他对波西米亚力学的分析，波西米亚力学定义了一个确定性动力系统，它涉及具有显著性质的常微分方程和偏微分方程的新组合。提出的研究包括(1)分析波西米亚力学的相对论扩展的可能性，(2)比较波西米亚力学与量子理论的主要竞争替代公式，以及(3)从波西米亚的角度分析统计力学的基础。提出的研究的重要性在于以下观察：1)量子理论的基础，所有物理理论中最基本的，在其诞生65年后仍然陷入混乱和不连贯的泥潭。2)波西米亚力学是将薛定谔方程自然嵌入到一个清晰、精确的物理理论中，而薛定谔方程是几乎所有量子理论解释的数学核心。如果一个人仅仅坚持认为这个方程与粒子的运动直接相关，它就会出现。换句话说，当我们坚持最简单的本体论——用它们的位置来描述的粒子——并为这个本体论寻找一种自然的运动规律时，粗略地说，波西米亚力学是从薛定谔方程中产生的。3)所有量子理论的奥秘都在波西米亚力学中找到了令人信服的解释——实际上，在以明显方式进化的明显本体论中！4)当一个人试图将量子理论应用于新的方向（例如，理解宏观干涉效应的含义）和新的领域（例如量子宇宙学）时，对波西米亚力学的欣赏可以带来巨大的灵活性和清晰度。5)波西米亚力学价值背后的基本原则是，它确实有助于以足够清晰的方式表述物理理论，使它们能够真正被理解。

DMS-9504556 Goldstein The investigator intends to continue his analysis of Bohmian mechanics, which defines a deterministic dynamical system involving a novel combination of ordinary and partial differential equations having remarkable properties. The proposed research involves (1) an analysis of the possibility of relativistic extensions of Bohmian mechanics, (2) comparisons of Bohmian mechanics with the major competing alternative formulations of quantum theory, and (3) an analysis of the foundations of statistical mechanics from a Bohmian perspective. The importance of the proposed research lies in the following observations: 1) The foundations of quantum theory, the most fundamental of all physical theories, continue to be mired in confusion and incoherence some sixty-five years after its inception. 2) Bohmian mechanics is the natural embedding of Schroedinger's equation, which is the mathematical core of almost all interpretations of quantum theory, into a clear, precise physical theory. It emerges if one merely insists that this equation be directly relevant to the motion of particles. In other words, Bohmian mechanics arises, roughly speaking, from Schroedinger's equation when (perhaps naively) we insist upon the simplest ontology---particles described by their positions---and seek a natural law of motion for this ontology. 3) All the mysteries of quantum theory find a compelling explanation in Bohmian mechanics---in effect, in the obvious ontology evolving in the obvious way! 4) An appreciation of Bohmian mechanics can be the source of enormous flexibility and clarity when one attempts to apply quantum theory in new directions---for example, to understand the implications of macroscopic interference effects---and to new domains, such as quantum cosmology. 5) The underlying principle behind the value of Bohmian mechanics is that it really does help to formulate physical theories with sufficient clarity that they can genuinely be understood.