Uncertainty Relations and Quantum Corrections in Noncommutative Quantum Mechanics on a Curved Space

Starting with the first-order singular Lagrangian describing the dynamical system with 2nd-class constraints, the noncommutative quantum mechanics on a curved space is investigated by the constraint star-product quantization formalism of the projection operator method. Imposing the additional constraints to eliminate the reduntant degrees of freedom, it is shown that the resultant noncommutative quantum system on the curved space is represented with two kinds of the constrained quantum systems, which are equivalent with each other. Then, it is shown that the resultant Hamiltonians contain the quantum corrections caused by the uncertainty relations among the constraint-operators in addition to those due to the projections of operators, which are missed in the usual approaches with the Dirac-bracket quantization formalism.