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

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.

26 pages, 3 figures

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• Loop Quantum Cosmology
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• Quasi-probability distributions in Loop Quantum Cosmology
• Uncertainty Principle in Loop Quantum Cosmology by Moyal Formalism
• An open scattering model in polymerized quantum mechanics
• Born-Oppenheimer meets Wigner-Weyl in Quantum Gravity