Phase space quantization and Loop Quantum Cosmology: A Wigner function for the Bohr-compactified real line

TL;DR

This paper introduces a Wigner function for quantum mechanics on the Bohr-compactified real line and applies it to loop quantum cosmology, providing a new quantization method for the Hamiltonian constraint component.

Contribution

It defines a Wigner function suitable for the Bohr-compactified real line and demonstrates its application in loop quantum cosmology for the first time.

Findings

Defined a Wigner function for the Bohr-compactified real line

Derived simple consequences of the new Wigner function

Applied the formalism to a new quantization of the Hamiltonian constraint component

Abstract

We give a definition for the Wigner function for quantum mechanics on the Bohr compactification of the real line and prove a number of simple consequences of this definition. We then discuss how this formalism can be applied to loop quantum cosmology. As an example, we use the Wigner function to give a new quantization of an important building block of the Hamiltonian constraint.