Author(s): VG Kupriyanov
Noncommutative quantum mechanics can be considered as a firs t step in the construction of quantum field theory on noncommutative spac es of generic form, when the commutator between coordinates is a function of the se coordinates. In this paper we discuss the mathematical framework of such a th eory. The non- commutativity is treated as an external antisymmetric field satisfying the Jacoby identity. First, we propose a symplectic realization of a gi ven Poisson manifold and construct the Darboux coordinates on the obtained symplect ic manifold. Then we define the star product on a Poisson manifold and obtain the ex pression for the trace functional. The above ingredients are used to formulate a no nrelativistic quantum mechanics on noncommutative spaces of general form. All con sidered constructions are obtained as a formal series in the parameter of noncommut ativity. In particular, the complete algebra of commutation relations between coor dinates and conjugated momenta is a deformation of the standard Heisenberg algebra . As examples we con- sider a free particle and an isotropic harmonic oscillator o n the rotational invariant noncommutative space.