Visnyk of the Lviv University. Series Physics 55 (2018) ñ. 13-23
DOI: https://doi.org/10.30970/vph.55.2018.13

Influence of momentum noncommutativity on the motion of free particles system in rotationally-invariant noncommutative phase space

Kh. P. Gnatenko

Rotationally-invariant space with noncommutativity of coordinates and noncommutativity of momenta is considered. The noncommutative algebra is constructed involving additional coordinates and additional momenta which correspond to harmonic oscillators. The lengthes of the oscillators are considered to be equal to the Planck length. The frequencies of the oscillators are supposed to be very large. The algebra is invariant under rotations and is equivalent to noncommutative algebra of canonical type. In the rotationally-invariant noncommutative phase space we study the motion of a system of free particles. We find that the trajectory of free particle in rotationally-invar³ant noncommutative phase space depends on its mass. This dependence is caused by momentum noncommutativity. Up to the second order in the parameters of noncommutativity the trajectory of free particle corresponds to the trajectory of harmonic oscillator with corresponding mass and frequency determined by the mass of the particle and the parameter of momentum noncommutativity. Up to the second order in the parameters of noncommutativity the system of N particles in rotationally-invariant noncommutative phase space is described by Hamiltonian corresponding to the Hamiltonian of N harmonic oscillators. We find that because of momentum noncommutativity the system of free particles flies away even if the initial velocities of the particles are the same. We show that idea to relate tensor of momentum noncommutativity with mass opens possibility to solve the problem of dependence of free particle motion on its mass. We find that in the case when the tensor of momentum noncommutativity corresponding to a particle is proportional to its mass the trajectory of free particle in rotationally-invariant noncommutative phase space does not depend on the mass therefore a system of free particles does not fly away. It is important to mention that proportionality of the tensor of noncommutativity to mass is also important for preserving of the weak equivalence principle, for recovering of relations of noncommutative algebra for coordinates and momenta of the center-of-mass in rotationally-invariant noncommutative phase space.

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