Solutions of the Klein-Gordon equation on manifolds with variable geometry including dimensional reduction

TL;DR

This paper investigates the solutions of the Klein-Gordon equation on manifolds with variable geometry, demonstrating how dimensional reduction affects scalar field behavior and spectrum, with implications for particle physics and symmetry violations.

Contribution

It introduces a new technique transforming Klein-Gordon solutions on variable geometries into a Schrödinger-type problem, and explores the effects of dimensional reduction on scalar spectra and symmetry.

Findings

Higher-dimensional signals do not penetrate the smaller-dimensional region due to inertial forces.

Scalar excitation spectra reflect the geometry of the transition region.

Asymmetry in models may relate to CP symmetry violation.

Abstract

We develop the recent proposal to use dimensional reduction from the four-dimensional space-time D=(1+3) to the variant with a smaller number of space dimensions D=(1+d), d < 3, at sufficiently small distances to construct a renormalizable quantum field theory. We study the Klein-Gordon equation on a few toy examples ("educational toys") of a space-time with variable special geometry, including a transition to a dimensional reduction. The examples considered contain a combination of two regions with a simple geometry (two-dimensional cylindrical surfaces with different radii) connected by a transition region. The new technique of transforming the study of solutions of the Klein-Gordon problem on a space with variable geometry into solution of a one-dimensional stationary Schr\"odinger-type equation with potential generated by this variation is useful. We draw the following conclusions: (1) The signal related to the degree of freedom specific to the higher-dimensional part does not penetrate into the smaller-dimensional part because of an inertial force inevitably arising in the transition region (this is the centrifugal force in our models). (2) The specific spectrum of scalar excitations resembles the spectrum of the real particles; it reflects the geometry of the transition region and represents its "fingerprints". (3) The parity violation due to the asymmetric character of the construction of our models could be related to violation of the CP symmetry.