Wigner function for polymer particle and Galileo relativity

We put forward a Wigner function of a free non relativistic particle in one spatial dimension within polymer quantization. To do so we use the algebra of the Euclidian group E(2) to which the “flux-holonomy” algebra of the polymer quantization can be reduced. The Wigner function of energy eigenstates fulfills an energy equation that is also introduced. Although not invariant itself the Wigner function combines with other quantity to form an invariant under deformed Galileo boosts with deformation parameter given by the length scale μ of the discrete particle position. Our analysis sheds light on the fate of spacetime symmetries within polymer quantization in line with recent results for relativistic particle. Open problems are commented upon in the discussion.