A Possible Geometric Understanding of Gauge Field Theory: An Exploration Based on the Physical Space Curvature Hypothesis

This paper is an extension of a previous work, which proposed a core hypothesis: 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)) / ℏ². The present paper attempts to extend this geometric framework to gauge field theory. Based on the earlier curvature hypothesis, this paper starts from the scalar field Lagrangian and defines a relativistic curvature expression K_rel = (E² − m² c⁴) / (ℏ² c²), exploring the correspondence between the mass parameter and the spatial curvature. For fermions (quarks), a curvature-mass substitution can be established in heavy quark systems via the Foldy-Wouthuysen transformation; for scalar fields, the corresponding substitution can be derived from the Klein-Gordon equation. If the curvature hypothesis holds, in regions of constant negative curvature the Green's function of the Jacobi field equation is of the Yukawa type, a mathematical structure that may be related to the existence of a mass gap. For pure gauge fields, based on the fact that gluons and quarks coexist in the same microscopic space, this paper explores the possibility that gluon behavior is constrained by the geometry of curved space. The mathematical correspondence between the geometric quantities in the curvature framework and the known characteristic scales in standard physics is demonstrated. A heuristic calculation for glueballs provides preliminary indirect support for the extended hypothesis. Section 7 discusses a possible geometric understanding of the force range differences among the four fundamental interactions. This work is based on the curvature hypothesis and the extended hypothesis. The pure gauge field part constitutes a heuristic discussion.