Noncommutative Quantum Mechanics and Quantum Cosmology

TL;DR

This paper develops a phase-space noncommutative quantum mechanics framework, applies it to quantum cosmology, and finds that momentum noncommutativity influences the universe's initial state by damping the wave function.

Contribution

It introduces a noncommutative extension of quantum mechanics to quantum cosmology, highlighting the role of momentum noncommutativity in the Wheeler-DeWitt equation.

Findings

Noncommutativity affects the Wheeler-DeWitt equation explicitly.

Momentum noncommutativity causes damping in the wave function.

Potential relevance for initial state selection in cosmology.

Abstract

We present a phase-space noncommutative version of quantum mechanics and apply this extension to Quantum Cosmology. We motivate this type of noncommutative algebra through the gravitational quantum well (GQW) where the noncommutativity between momenta is shown to be relevant. We also discuss some qualitative features of the GQW such as the Berry phase. In the context of quantum cosmology we consider a Kantowski-Sachs cosmological model and obtain the Wheeler-DeWitt (WDW) equation for the noncommutative system through the ADM formalism and a suitable Seiberg-Witten (SW) map. The WDW equation is explicitly dependent on the noncommutative parameters, $\theta$ and $\eta$. We obtain numerical solutions of the noncommutative WDW equation for different values of the noncommutative parameters. We conclude that the noncommutativity in the momenta sector leads to a damped wave function implying that this type of noncommmutativity can be relevant for a selection of possible initial states for the universe.