Rotationally invariant noncommutative phase space of canonical type with recovered weak equivalence principle

Abstract

We study influence of noncommutativity of coordinates and noncommutativity of momenta on the motion of a particle (macroscopic body) in uniform and non-uniform gravitational fields in noncommutative phase space of canonical type with preserved rotational symmetry. It is shown that because of noncommutativity the motion of a particle in gravitational filed is determined by its mass. The trajectory of motion of a particle in uniform gravitational field corresponds to the trajectory of harmonic oscillator with frequency determined by the value of parameter of momentum noncommutativity and mass of the particle. The equations of motion of a macroscopic body in gravitational filed depend on its mass and composition. From this follows violation of the weak equivalence principle caused by noncommutativity. We conclude that the weak equivalence principle is recovered in rotationally invariant noncommutative phase space if we consider the tensors of noncommutativity to be dependent on mass. So, finally we construct noncommutative algebra which is rotationally invariant, equivalent to noncommutative algebra of canonical type, and does not lead to violation of the weak equivalence principle.

Weak equivalence principle in quantum space

“…A 3D algebra which is rotationally invariant and equivalent to the noncommutative algebra of canonical type was proposed in (Gnatenko K. P. and Tkachuk V. M., 2017a). It is important to mention that to recover the weak equivalence principle in the context of this algebra the idea to relate the parameters of noncommutativity with mass has to be considered [for details see (Gnatenko, 2018)].…”