Dimensional Reduction in $(1+2)$-dim Cylindrical Universes and Klein–Gordon Waves

This work develops the recent proposition to use dimensional reduction from the four-dimensional space-time ($D=1+3$) to the one with a smaller number of dimensions $D=(1+d)$; $d<3$ at small distances for construction of renormalizable quantum field theory.
To this goal, we study the Klein–Gordon equation on a few toy examples (educational toys) of space-time with variable dimension. The examples considered contain combination of parts of two manifolds with regular geometry (two-dimensional cylinder surfaces with non-equal radii) connected with some transition region.Here, the new trick of transforming the Klein–Gordon problem on variable geometry to the Schrödinger-type equation with potential generated by this variation is of help.
The nonstatic models of space geometry with cylinder symmetry are under consideration. Besides, to make these models closer to physical reality, we define the set of ‘admissible’ shape functions $\rho(t,z)$ by solving the Einstein equations in the $(1+2)$-dim space-time. Few explicit solutions of the Klein–Gordon equation in this set are given.
Our observations are:
1. The signal, related to degree of freedom specific for the higher-dimensional part does not penetrate into the smaller-dimensional part due to an effective inertial force inevitably arising at the junction region; in our models this is the centrifugal force.
2. The specific spectrum of scalar excitations resembles the spectrum of real particles. It reflects the junction geometry and can be treated as its ‘fingerprint’.
3. The interesting qualitative feature of these solutions relates to the DR points, their classification and time behavior. In particular, these new entities could provide us with novel insight into the nature of P-violation, T-violation, and Big Bang.