Weak equivalence principle in quantum space

Owing to the development of String Theory and Quantum Gravity, studies of quantized spaces described by deformed commutation relations for operators of coordinates and operators of momenta have received much attention. In this paper, the implementation of the weak equivalence principle is examined in the quantized spaces described by different types of deformed algebras, among them the noncommutative algebra of canonical type, Lie type, and the nonlinear deformed algebra with an arbitrary function of deformation depending on momenta. It is shown that the deformation of commutation relations leads to the mass-dependence of motion of a particle (a composite system) in a gravitational field, and, hence, to violation of the weak equivalence principle. We conclude that this principle is recovered in quantized spaces if one considers the parameters of the deformed algebras to be different for different particles (bodies) and to be determined by their masses.