The Classical and Commutative Limits of noncommutative Quantum   Mechanics: A Superstar \bigstar Wigner-Moyal Equation
  We are interested in the similarities and differences between the quantum-classical (Q-C) and the noncommutative-commutative (NC-Com) correspondences. As one useful platform to address this issue we derive the superstar Wigner-Moyal equation for noncommutative quantum mechanics (NCQM). A superstar $\bigstar$-product combines the usual phase space $\ast$ star and the noncommutative $\star$ star-product. Having dealt with subtleties of ordering present in this problem we show that the classical correspondent to the NC Hamiltonian has the same form as the original Hamiltonian, but with a non-commutativity parameter $\theta$-dependent, momentum-dependent shift in the coordinates. Using it to examine the classical and the commutative limits, we find that there exist qualitative differences between these two limits. Specifically, if $\theta \neq 0$ there is no classical limit. Classical limit exists only if $\theta \to 0$ at least as fast as $\hbar \to 0$, but this limit does not yield Newtonian mechanics, unless the limit of $\theta/\hbar$ vanishes as $\theta \to 0$. Another angle to address this issue is the existence of conserved currents and the Noether's theorem in the continuity equation, and the Ehrenfest theorem in the NCQM context. 
Ardeshir Eftekharzadeh, B. L. Hu