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

DMS-9504556戈尔茨坦研究人员打算继续他对玻色力学的分析，它定义了一种确定性的动力系统，涉及具有显著性质的常微分方程和偏微分方程组的新组合。建议的研究包括(1)分析玻色力学相对论扩展的可能性，(2)玻色力学与量子理论的主要竞争替代公式的比较，以及(3)从玻色力学的角度分析统计力学的基础。这项拟议的研究的重要性在于：1)量子理论的基础，作为所有物理理论中最基本的，在其成立约65年后，仍然深陷混乱和不连贯的泥潭。2)玻色力学是薛定谔方程的自然嵌入，薛定谔方程是几乎所有量子理论解释的数学核心，它是一个清晰、精确的物理理论。只要坚持这个方程与粒子的运动直接相关，它就出现了。换句话说，粗略地说，玻色力学起源于薛定谔方程，当时(也许是天真的)我们坚持最简单的本体论-由粒子的位置描述的粒子--并为这个本体论寻求自然的运动规律。3)所有量子理论的奥秘都在玻色力学中找到了令人信服的解释-实际上，在以显而易见的方式进化的显而易见的本体论中！4)当一个人试图将量子理论应用于新的方向--例如，理解宏观干扰效应的含义--以及新的领域，如量子宇宙学时，对玻色力学的欣赏可以带来巨大的灵活性和清晰度。5)玻色力学价值背后的基本原理是，它确实有助于以足够清晰的方式阐述物理理论，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.