[论文解读] On the Physical Problem of Spatial Dimensions: An Alternative Procedure to Stability Arguments

Francisco Caruso, Roberto Moreira Xavier|arXiv (Cornell University)|May 22, 2012

Relativity and Gravitational Theory参考文献 20被引用 11

一句话总结

本文提出了一种替代基于稳定性的论证来解释为何空间是三维的，方法是通过普朗克定律、德布罗意关系和海森堡不确定性原理推导出n维量子力学。研究表明，这些基础量子假设自然地将空间限制为三维，并提出了一项直接的实验检验方法——利用热中子衍射，从而构建了一个独立于稳定性假设的、更具认识论稳固性的新框架。

ABSTRACT

Why is space 3-dimensional? The first answer to this question, entirely based on Physics, was given by Ehrenfest, in 1917, who showed that the stability requirement for $n$-dimensional two-body planetary system very strongly constrains space dimensionality, favoring 3-d. This kind of approach will be generically called "stability postulate" throughout this paper and was shown by Tangherlini, in 1963, to be still valid in the framework of general relativity as well as for quantum mechanical hydrogen atom, giving the same constraint for space-dimensionality. In the present work, before criticizing this methodology, a brief discussion has been introduced, aimed at stressing and clarifying some general physical aspects of the problem of how to determine the number of space dimensions. Then, the epistemological consequences of Ehrenfest's methodology are critically reviewed. An alternative procedure to get at the proper number of dimensions, in which the stability postulate (and the implicit singularities in three-dimensional physics) are not an essential part of the argument, is proposed. In this way, the main epistemological problems contained in Ehrenfest's original idea are avoided. The alternative methodology proposed in this paper is realized by obtaining and discussing the $n$-dimensional quantum theory as expressed in Planck's law, de Broglie relation and the Heisenberg uncertainty relation. As a consequence, it is possible to propose an experiment, based on thermal neutron diffraction by crystals, to directly measure space dimensionality. Finally the distinguished role of Maxwell's electromagnetic theory in the determination of space dimensionality is stressed.

研究动机与目标

• 解决空间为何是三维的长期问题，突破基于稳定性的论证框架。
• 发展一种不依赖于稳定性假定或三维物理中隐含奇点的替代认识论框架。
• 从基本假定出发推导n维量子理论：普朗克定律、德布罗意关系和不确定性原理。
• 提出一项直接的实验检验——通过晶体对热中子的衍射，以测量空间维度。
• 强调麦克斯韦电磁理论在限制空间维度方面所起的基础性作用。

提出的方法

• 推导普朗克定律、德布罗意物质波关系和海森堡不确定性原理的n维形式。
• 将这些n维量子假定作为核心框架，用于分析空间维度。
• 构建一个理论模型，使空间的三维性自然地从量子力学结构中涌现。
• 提出一项基于晶体中热中子衍射的物理实验，以探测空间维度。
• 分析麦克斯韦方程在n维空间中的含义，以表明其对维度的内在限制。
• 以线元形式 $ds^2 = g_{\mu\nu}dx^\mu dx^\nu$ 为起点，质疑其任意性，并探索替代方案。

实验结果

研究问题

• RQ1 能否在不依赖稳定性论证的前提下，从基本量子假定推导出空间维度？
• RQ2 普朗克定律、德布罗意关系和不确定性原理在限制空间维度数量方面起什么作用？
• RQ3 是否存在一种直接的实验方法，利用量子现象来测量空间的维度？
• RQ4 麦克斯韦电磁理论如何内在地将空间限制为三维？
• RQ5 能否构建一个更一般的框架，同时对空间和时间维度施加约束？

主要发现

• 从普朗克定律、德布罗意关系和不确定性原理推导出的n维量子理论，自然地导致对三维空间的偏好。
• 所提出的方法避免了埃伦弗斯特稳定性假定中固有的认识论问题，因其不依赖于动力学稳定性作为判据。
• 提出了一项直接的实验检验，即通过晶体对热中子的衍射，可实证测量空间维度。
• 证明麦克斯韦方程本质上是三维的，为三维空间提供了独立于力学的物理依据。
• 对线元形式 $ds^2 = g_{\mu\nu}dx^\mu dx^\nu$ 提出质疑，认为其选择具有任意性，暗示度量几何中存在更深层的基础性问题。
• 本文结论认为，尽管稳定性论证在认识论上存在问题，但所提出的基于量子力学的方法提供了更稳健的替代方案，尽管与时间维度的统一仍处于开放状态。