Curvature of Physical Space and Geometric Projection: A Possible Geometric Understanding of Quantum Mechanics and Relativity

Abstract

Starting from a formal comparison between the Schr\"{o}dinger 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 Schr\"{o}dinger 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.