The Noncommutative Harmonic Oscillator based in Simplectic   Representation of Galilei Group
  In this work we study symplectic unitary representations for the Galilei group. As a consequence the Schr\"odinger equation is derived in phase space. The formalism is based on the non-commutative structure of the star-product, and using the group theory approach as a guide a physical consistent theory in phase space is constructed. The state is described by a quasi-probability amplitude that is in association with the Wigner function. The 3D harmonic oscillator and the noncommutative oscillator are studied in phase space as an application, and the Wigner function associated to both cases are determined. 
R. G. G. Amorim, S. C. Ulhoa, A. E. Santana
math-ph math.MP