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首页> 外文期刊>JP journal of geometry and topology >TOWARD QUANTIZING QUANTUM MECHANICAL SYSTEMS USING HODGE-DE RHAM THEORY
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TOWARD QUANTIZING QUANTUM MECHANICAL SYSTEMS USING HODGE-DE RHAM THEORY

机译：利用HODGE-DE RHAM理论对量子力学系统进行量化

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摘要

Empirical evidence of quantization is found in experiments demonstrating the Aharonov-Bohm and integer and fractional Quantum Hall effects. In the associated ongoing open areas of research there have been numerous attempts to explain the observed nature of such quantization. Of particular note, and one motivation for the topological concepts of space-time addressed here, is the occurrence of certain sequences of plateaus in fractional Quantum Hall results, represented by positive integer multiples of quantum units where nature selects certain integers as multipliers of fundamental quantum measures of electrical charge and magnetic flux. The micro-origins of such selections are unknown. Our recently deceased colleague, Evert Jan Post, espoused a universal view of integer and fractional QH impedance characterized by the ratio of period integrals for flux and charge, leading to a ratio of the corresponding quantum integers, often referred to as filling factors. Our main purpose in the present article is to build upon previous topological results toward the ultimate goal of accommodating singularities in a space-time Riemannian manifold, representing the experimentally observed specific sequences of integers and fractions as an extension of the familiar manoeuvres such as the residue theorem of Cauchy in complex analysis, or, in a more general topological setting of exterior calculus, Hodge-de Rham cohomology, and the Mittag-Leffler theorems. It is our ultimate intention to shed light on the nature of particles and space by examining such singular features through extensions of classical singularity theory to the space-time pseudo-Riemannian manifold.
机译：在证明Aharonov-Bohm以及整数和分数量子霍尔效应的实验中发现了量化的经验证据。在相关的正在进行的开放研究领域中，已经进行了许多尝试来解释这种量化的观察性质。需要特别注意的是，这里提出的时空拓扑概念的动机是，分数量子霍尔结果中某些平稳平台的出现，由量子单位的正整数倍表示，其中自然选择某些整数作为基本量子的乘数电荷和磁通量的度量。这种选择的微起源是未知的。我们最近去世的同事Evert Jan Post拥护一个整数和分数QH阻抗的通用视图，其特征在于通量和电荷的周期积分之比，从而得出相应量子整数之比，通常称为填充因子。本文的主要目的是在先前的拓扑结果的基础上，朝最终的目标实现时空黎曼流形中的奇异性，该形式表示实验观察到的整数和分数的特定序列，是诸如残基之类的常见操作的扩展复杂分析中的柯西定理，或者在更复杂的外部演算拓扑结构中，采用Hodge-de Rham同调论和Mittag-Leffler定理。我们的最终目的是通过将经典奇点理论扩展到时空伪黎曼流形来研究这种奇点特征，从而阐明粒子和空间的本质。

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