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.