Quantum Effects of an Extra Compact Dimension on the Wave Function of the Universe

TL;DR

This paper extends quantum cosmology to five dimensions, revealing that a compact extra dimension leads to quantized momentum and results in a two-dimensional Schrödinger equation for the universe's wave function.

Contribution

It introduces a Hamiltonian formulation for 5D quantum cosmology and demonstrates how compactness of the extra dimension quantizes momentum and simplifies the wave function analysis.

Findings

Quantized spectrum for the fifth dimension's momentum

Wave function satisfies a 2D Schrödinger equation

Compact extra dimension influences quantum properties of the universe

Abstract

We extend the direct quantum approach of the standard FRW cosmology from 4D to 5D and obtain a Hamiltonian formulation for a wave-like 5D FRW cosmology. Using a late-time approximation we isolate out a y-part from the full wave function of the 5D Universe. Then we find that the compactness of the fifth dimension y yields a quantized spectrum for the momentum $P_{5}$ along the fifth dimension, and we show that the whole space-part of the wave function of the 5D Universe satisfies a two-dimensional Schr\"odinger equation.