Quantum mechanics of the closed collapsing Universe

TL;DR

This paper compares two quantum approaches to a closed universe model, showing that the universe's scale factor oscillates with damping and stabilizes over time.

Contribution

It introduces and contrasts Schrödinger and Klein-Gordon quantization methods for a closed universe, highlighting the advantages of the Klein-Gordon approach.

Findings

The universe's scale factor oscillates with damping.

The scale factor approaches a constant value at the end of evolution.

The Klein-Gordon approach enables constructing quasi-Heisenberg operators.

Abstract

Two approaches to quantization of Freedman's closed Universe are compared. In the first approach, the Shrodinger's norm of the wave function of Universe is used, and in the second approach, the Klein-Gordon's norm is used. The second one allows building the quasi-Heisenberg operators as functions of time and finding their average values. It is shown that the average value of the Universe scale factor oscillates with damping and approaches to some constant value at the end of the Universe evolution.