Geometrical derivation of the Klein-Gordon equation

Abstract

In order to integrate quantum mechanics into geometrical theories of physics, new quantities must be included in the description of the infinitesimal structure of space-time. Using gauge transformations, Weyl's coefficients of connection are generalized to include not just the metric tensor and the electromagnetic potential, but the wave function as well. As the field equations are developed, it is found that invariance of the scalar of curvature under the appropriate gauge and coordinate transformations implies the Klein-Gordon equation. Since the terms required for invariance correspond with known quantum effects, no invariant classical limit is possible. The selection of appropriate fixed gauges and the elimination of small terms does lead directly to the Hamilton-Jacobi equations. Trajectories are defined for all cases, and as an example, the Lorentz force law is derived from these trajectories. The probability density is related to the particle trajectories and a conformal parameter of an additional metric tensor. Because of the gauge transformation, gravitational, electromagnetic, and quantum effects must be described as aspects of the same geometrical structure.