The $2$ -- D free particle in the phase space quantum mechanics

The Wigner function of a free quantum particle propagating on a plane is derived. Two options are discussed. First the particle with fixed components of the momentum is mentioned. Then the particle of given energy and angular momentum is considered. The analysis does not refer to the Hilbert space formulation of the problem at all. One starts from a choice of coordinates on the symplectic space $(\mathbb{R}^{4},\,\omega)$ suitable for representation of eigenstates of the discussed particle. Then the Moyal $\star_{(\text{M})}$ -- product on the phase space is derived with the use of the Fedosov algorithm adapted to curvilinear coordinates on a flat phase space. Next the eigenvalue equations for the Wigner eigenfunction are solved. Finally the physically acceptable solutions are identified.