Symmetry in noncommutative quantum mechanics

We reconsider the generalization of standard quantum mechanics in which the position operators do not commute. We argue that the standard formalism found in the literature leads to theories that do not share the symmetries present in the corresponding commutative system. We propose a general prescription to specify a Hamiltonian in the noncommutative theory that preserves the existing symmetries. We show that it is always possible to choose this Hamiltonian in such a way that the energy spectrum of the standard and non-commuting theories are identical, so that experimental differences between the predictions of both theories are to be found only at the level of the detailed structure of the energy eigenstates.