AI 中文总结
本文对比薛定谔方程与雅可比场方程，提出微观物理空间截面曲率表达式，尝试为量子力学与相对论提供统一几何理解，并拓展至统计力学。
AI 中文摘要
本文从薛定谔方程与雅可比场方程的形式对比出发，提出如下假设：在非相对论量子力学框架内，微观物理空间的截面曲率可表示为\
英文摘要
Starting from a formal comparison between the Schrödinger equation and the Jacobi field equation, this paper proposes the hypothesis that, within the framework of non-relativistic quantum mechanics, the sectional curvature of microscopic physical space can be expressed as \(K(x) = 2m(E-V(x))/\hbar^2\), and attempts to offer a possible unified geometric understanding of quantum mechanics and relativity. The paper is divided into three parts. The first part identifies the Schrödinger equation, in its mathematical form, as a Jacobi field equation. The wave function is made to correspond to a Jacobi field, giving it the geometric meaning of the deviation of geodesics in physical space. Phenomena such as quantum tunneling, energy quantization, and the path integral are then interpreted, from this perspective, as geometric manifestations of different spatial curvatures. The second part proposes a geometric projection scheme based on semi-geodesic coordinates. The transverse metric factor of a curved space, which satisfies a Jacobi field equation, is taken as a unified geometric framework. It is demonstrated that the metric correction terms in classical mechanics, special relativity, general relativity, and quantum mechanics can all be derived from the same equation, with the curvature \(K\) arising from different physical sources. The third part extends this framework to statistical mechanics.
Commentsv2: Corrected the statement on isometric embedding of constant-curvature spaces and added the Hilbert1901 reference. Added a clarification at wave function nodes in the probability interpretation section. Updated the DOI of the subsequent work to 10.5281/zenodo.22108495
