Polymer quantum mechanics as a deformation quantization

We analyze the polymer representation of quantum mechanics within the deformation quantization formalism. In particular, we construct the Wigner function and the star-product for the polymer representation as a distributional limit of the Schrodinger representation for the Weyl algebra in a Gaussian weighted measure, and we observe that the quasi-probability distribution limit of this Schrodinger representation agrees with the Wigner function for loop quanttun cosmology. Further, the introduced polymer star-product fulfills Bohr's correspondence principle even though not all the operators are well defined in the polymer representation. Finally, within our framework, we also derive a generalized uncertainty principle which resembles the one appearing in different scenarios, including theories with a minimal length.