44 research outputs found
Kinematic variables in noncommutative phase space and parameters of noncommutativity
We consider a space with noncommutativity of coordinates and noncommutativity
of momenta. It is shown that coordinates in noncommutative phase space depend
on mass therefore they can not be considered as kinematic variables. Also,
noncommutative momenta are not proportional to a mass as it has to be. We find
conditions on the parameters of noncommutativity on which these problems are
solved. It is important that on the same conditions the weak equivalence
principle is not violated, the properties of kinetic energy are recovered, and
the motion of the center-of-mass of composite system and relative motion are
independent in noncommutative phase space
Harmonic oscillator chain in noncommutative phase space with rotational symmetry
We consider a quantum space with rotationally invariant noncommutative
algebra of coordinates and momenta. The algebra contains tensors of
noncommutativity constructed involving additional coordinates and momenta. In
the rotationally invariant noncommutative phase space harmonic oscillator chain
is studied. We obtain that noncommutativity affects on the frequencies of the
system. In the case of a chain of particles with harmonic oscillator
interaction we conclude that because of momentum noncommutativity the spectrum
of the center-of-mass of the system is discrete and corresponds to the spectrum
of harmonic oscillator
Features of description of composite system's motion in twist-deformed space-time
Composite system made of particles is considered in twist-deformed
space-time. It is shown that in the space the motion of the center-of-mass of
the system depends on the relative motion. Influence of deformation on the
motion of the center-of-mass of composite system is less than on the motion of
individual particles and depends on the system's composition. We conclude that
if we consider commutation relations for coordinates of a particle to be
proportional inversely to its mass, the commutation relations for coordinates
of composite system do not depend on its composition and are proportional
inversely to system's total mass, besides the motion of the center-of-mass is
independent of the relative motion. In addition we find that inverse
proportionality of parameters of noncommutativity to mass is important for
considering coordinates in twist-deformed space as kinematic variables and for
preserving of the weak equivalence principle
System of interacting harmonic oscillators in rotationally invariant noncommutative phase space
Rotationally invariant space with noncommutativity of coordinates and
noncommutativity of momenta of canonical type is considered. A system of
interacting harmonic oscillators in uniform filed and a system of particles
with harmonic oscillator interaction are studied. We analyze effect of
noncommutativity on the energy levels of these systems. It is found that
influence of coordinates noncommutativity on the energy levels of the systems
increases with increasing of the number of particles. The spectrum of free
particles in uniform field in rotationally-invariant noncommutative phase space
is also analyzed. It is shown that the spectrum corresponds to the spectrum of
a system of harmonic oscillators with frequency determined by the parameter
of momentum noncommutativity
Composite system in noncommutative space and the equivalence principle
The motion of a composite system made of N particles is examined in a space
with a canonical noncommutative algebra of coordinates. It is found that the
coordinates of the center-of-mass position satisfy noncommutative algebra with
effective parameter. Therefore, the upper bound of the parameter of
noncommutativity is re-examined. We conclude that the weak equivalence
principle is violated in the case of a non-uniform gravitational field and
propose the condition for the recovery of this principle in noncommutative
space. Furthermore, the same condition is derived from the independence of
kinetic energy on the composition.Comment: 12 page
Parameters of noncommutativity in Lie-algebraic noncommutative space
We find condition on the parameters of noncommutativity on which a list of
important results can be obtained in a space with Lie-algebraic
noncommutativity. Namely, we show that the weak equivalence principle is
recovered in the space, the Poisson brackets for coordinates and momenta of the
center-of-mass of a composite system do not depend on its composition and
reproduce relations of noncommutative algebra for coordinates and momenta of
individual particles if parameters of noncommutativity corresponding to a
particle are proportional inversely to its mass. In addition in particular case
of Lie-algebraic noncommutativity (space coordinates commute to time) on this
condition the motion of the center-of-mass is independent of the relative
motion and problem of motion of the center-of-mass and problem corresponding to
the internal motion can be studied separately
Effect of coordinate noncommutativity on the mass of a particle in a uniform field and the equivalence principle
We consider the motion of a particle in a uniform field in noncommutative
space which is rotationally invariant. On the basis of exact calculations it is
shown that there is an effect of coordinate noncommutativity on the mass of a
particle. A particular case of motion of a particle in a uniform gravitational
field is considered and the equivalence principle is studied. We propose the
way to solve the problem of violation of the equivalence principle in the
rotationally invariant noncommutative space
Effect of noncommutativity on the spectrum of free particle and harmonic oscillator in rotationally invariant noncommutative phase space
We consider rotationally invariant noncommutative algebra with tensors of
noncommutativity constructed with the help of additional coordinates and
momenta. The algebra is equivalent to well known noncommutative algebra of
canonical type. In the noncommutative phase space with rotational symmetry
influence of noncommutativity on the spectrum of free particle and spectrum of
harmonic oscillator is studied up to the second order in the parameters of
noncommutativity. We find that because of momentum noncommutativity the
spectrum of free particle is discrete and corresponds to the spectrum of
harmonic oscillator in the ordinary space (space with commutative coordinates
and commutative momenta). We obtain the spectrum of the harmonic oscillator in
the rotationally invariant noncommutative phase space and conclude that
noncommutativity of coordinates affects on its mass. The frequency of the
oscillator is affected by the coordinate noncommutativity and the momentum
noncommutativity. On the basis of the results, the eigenvalues of squared
length operator are found and restrictions on the value of length in
noncommutative phase space with rotational symmetry are analyzed
Composite system in rotationally invariant noncommutative phase space
Composite system is studied in noncommutative phase space with preserved
rotational symmetry. We find conditions on the parameters of noncommutativity
on which commutation relations for coordinates and momenta of the
center-of-mass of composite system reproduce noncommutative algebra for
coordinates and momenta of individual particles. Also, on the conditions the
coordinates and the momenta of the center-of-mass satisfy noncommutative
algebra with effective parameters of noncommutativity which depend on the total
mass of the system and do not depend on its composition. Besides, it is shown
that on these conditions the coordinates in noncommutative space do not depend
on mass and can be considered as kinematic variables, the momenta are
proportional to mass as it has to be. A two-particle system with Coulomb
interaction is studied and the corrections to the energy levels of the system
are found in rotationally invariant noncommutative phase space. On the basis of
this result the effect of noncommutativity on the spectrum of exotic atoms is
analyzed
Hydrogen atom in rotationally invariant noncommutative space
We consider the noncommutative algebra which is rotationally invariant. The
hydrogen atom is studied in a rotationally invariant noncommutative space. We
find the corrections to the energy levels of the hydrogen atom up to the second
order in the parameter of noncommutativity. The upper bound of the parameter of
noncommutativity is estimated on the basis of the experimental results for
1s-2s transition frequency