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

DMS-9504556 Goldstein 研究人员打算继续他的分析Bohmian力学，它定义了一个确定性的动力系统，涉及一个新的组合的普通和偏微分方程具有显着的性质。拟议的研究包括：（1）分析Bohmian力学的相对论扩展的可能性;（2）比较Bohmian力学与量子理论的主要竞争替代公式;（3）从Bohmian的角度分析统计力学的基础。 所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理论，在其成立约65年后继续陷入混乱和不连贯。2)玻姆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，成为一个清晰，精确的物理理论。只要坚持这个方程与质点运动直接相关，它就会出现。换句话说，粗略地说，当我们（也许是天真地）坚持最简单的本体论--用位置描述粒子--并为这个本体论寻求运动的自然规律时，玻姆力学就从薛定谔方程中产生了。3)量子理论的所有奥秘都在玻姆力学中找到了令人信服的解释-实际上，在以明显的方式进化的明显的本体论中！4)当人们试图将量子理论应用于新的方向时，对玻姆力学的理解可以是巨大的灵活性和清晰度的来源-例如，理解宏观干涉效应的含义-以及新的领域，如量子宇宙学。5)玻姆力学的价值背后的基本原则是，它确实有助于以足够清晰的方式制定物理理论，以便真正理解它们。

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.