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

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 with a few toy examples (“educational toys”) of a space–time with a variable spatial 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 for transforming the study of solutions of the Klein–Gordon problem on a space with variable geometry into solution of a one-dimensional stationary Schrödinger-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 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 the CP symmetry violation.