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We give a definition for the Wigner function for quantum mechanics on theBohr compactification of the real line and prove a number of simpleconsequences of this definition. We then discuss how this formalism can beapplied to loop quantum cosmology. As an example, we use the Wigner function togive a new quantization of an important building block of the Hamiltonianconstraint.

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