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

Ivan Franko National University of Lviv, Department for Theoretical Physics - 12 Drahomanov St., Lviv, 79005, Ukraine and Laboratory for Statistical Physics of Complex Systems, Institute for Condensed Matter Physics, NAS of Ukraine 79011, Lviv, Ukraine

Received: 21 July 2018
Accepted: 3 September 2018

Abstract

We study the influence of noncommutativity of coordinates and noncommutativity of momenta on the motion of a particle (macroscopic body) in uniform and nonuniform 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 a gravitational field is determined by its mass. The trajectory of motion of a particle in a uniform gravitational field corresponds to the trajectory of a harmonic oscillator with frequency determined by the value of the parameter of momentum noncommutativity and mass of the particle. The equations of motion of a macroscopic body in a gravitational field depend on its mass and composition. From this follows a 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.