Deformed Heisenberg algebras of different types with preserved weak equivalence principle

Abstract

In the paper a review of results for recovering of the weak equivalence principle in a space with deformed commutation relations for operators of coordinates and momenta is presented. Different types of deformed algebras leading to a space quantization are considered among them noncommutative algebra of canonical type, algebra of Lie type, nonlinear deformed algebra with arbitrary function of deformation depending on momenta. A motion of a particle and a composite system in gravitational field is examined and the implementation of the weak equivalence principle is studied. The principle is preserved in quantized space if we consider parameters of deformed algebras to be dependent on mass. It is also shown that dependencies of parameters of deformed algebras on mass lead to preserving of the properties of the kinetic energy in quantized spaces and solving of the problem of significant effect of space quantization on the motion of macroscopic bodies (the problem is known as the soccer-ball problem).