Effect of the generalized uncertainty principle on Galilean and Lorentz transformations

Generalized Uncertainty Principle (GUP) was obtained in string theory and quantum gravity and suggested the existence of a fundamental minimal length which, as was established, can be obtained within the deformed Heisenberg algebra. We use the deformed commutation relations or in classical case (studied in this paper) the deformed Poisson brackets, which are invariant with respect to the translation in configurational space. We have found transformations relating coordinates and times of moving and rest frames of reference in the space with GUP in the first order over parameter of deformation. For the non-relativistic case we find the deformed Galilean transformation which is similar to the Lorentz one written for Euclidean space with signature $(+,+,+,+)$. The role of the speed of light here plays some velocity $u$ related to the parameter of deformation, which as we estimate is many order of magnitude larger than the speed of light $u\simeq 1.2 \times 10^{22} c$. The coordinates of the rest and moving frames of reference for relativistic particle in the space with GUP satisfy the Lorentz transformation with some effective speed of light. We estimate that the relative deviation of this effective speed of light $\tilde c$ from $c$ is ${(\tilde c-c)/ c}\simeq 3.5\times 10^{-45}$. The influence of GUP on the motion of particle and the Lorentz transformation in the first order over parameter of deformation is hidden in $1/c^2$ relativistic effects.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1301.189

Supersymmetric Method for Constructing Quasi-Exactly Solvable Potentials

We propose a new method for constructing the quasi-exactly solvable (QES) potentials with two known eigenstates using supersymmetric quantum mechanics. General expression for QES potentials with explicitly known energy levels and wave functions of ground state and excited state are obtained. Examples of new QES potentials are considered.Comment: 11 pages, latex, to appear in Cond. Matt. Phys. (Lviv) (Proceedings of INTAS-Ukraine Workshop on Condensed Matter Physics, May, Lviv, 1998

Classical electrodynamics in a space with spin noncommutativity of coordinates

We propose a new relativistic Lorentz-invariant spin-noncommutative algebra. Using the Weyl ordering of noncommutative position operators, we build an analogue of the Moyal-Groenewald product for the proposed algebra. The Lagrange function of an electromagnetic field in the space with spin noncommutativity is constructed. In such a space electromagnetic field becomes non-abelian. A gauge transformation law of this field is also obtained. Exact nonlinear field equations of noncommutative electromagnetic field are derived from the least action principle. Within the perturbative approach we consider field of a point charge in a constant magnetic field and interaction of two plane waves. An exact solution of a plane wave propagation in a constant magnetic and electric fields is found.Comment: 15 page

Dirac oscillator with nonzero minimal uncertainty in position

In the context of some deformed canonical commutation relations leading to isotropic nonzero minimal uncertainties in the position coordinates, a Dirac equation is exactly solved for the first time, namely that corresponding to the Dirac oscillator. Supersymmetric quantum mechanical and shape-invariance methods are used to derive both the energy spectrum and wavefunctions in the momentum representation. As for the conventional Dirac oscillator, there are neither negative-energy states for $E=-1$, nor symmetry between the $l = j - {1/2}$ and $l = j + {1/2}$ cases, both features being connected with supersymmetry or, equivalently, the $\omega \to - \omega$ transformation. In contrast with the conventional case, however, the energy spectrum does not present any degeneracy pattern apart from that associated with the rotational symmetry. More unexpectedly, deformation leads to a difference in behaviour between the $l = j - {1/2}$ states corresponding to small, intermediate and very large $j$ values in the sense that only for the first ones supersymmetry remains unbroken, while for the second ones no bound state does exist.Comment: 28 pages, no figure, submitted to JP

More on a SUSYQM approach to the harmonic oscillator with nonzero minimal uncertainties in position and/or momentum

We continue our previous application of supersymmetric quantum mechanical methods to eigenvalue problems in the context of some deformed canonical commutation relations leading to nonzero minimal uncertainties in position and/or momentum. Here we determine for the first time the spectrum and the eigenvectors of a one-dimensional harmonic oscillator in the presence of a uniform electric field in terms of the deforming parameters $\alpha$, $\beta$. We establish that whenever there is a nonzero minimal uncertainty in momentum, i.e., for $\alpha \ne 0$, the correction to the harmonic oscillator eigenvalues due to the electric field is level dependent. In the opposite case, i.e., for $\alpha = 0$, we recover the conventional quantum mechanical picture of an overall energy-spectrum shift even when there is a nonzero minimum uncertainty in position, i.e., for $\beta \ne 0$. Then we consider the problem of a $D$-dimensional harmonic oscillator in the case of isotropic nonzero minimal uncertainties in the position coordinates, depending on two parameters $\beta$, $\beta'$. We extend our methods to deal with the corresponding radial equation in the momentum representation and rederive in a simple way both the spectrum and the momentum radial wave functions previously found by solving the differential equation. This opens the way to solving new $D$-dimensional problems.Comment: 26 pages, no figure, new section 2.4 + small changes, accepted in J. Phys. A, Special issue on Supersymmetric Quantum Mechanic

Lorentz-covariant deformed algebra with minimal length and application to the 1+1-dimensional Dirac oscillator

The $D$-dimensional $(\beta, \beta')$-two-parameter deformed algebra introduced by Kempf is generalized to a Lorentz-covariant algebra describing a ($D+1$)-dimensional quantized space-time. In the D=3 and $\beta=0$ case, the latter reproduces Snyder algebra. The deformed Poincar\'e transformations leaving the algebra invariant are identified. It is shown that there exists a nonzero minimal uncertainty in position (minimal length). The Dirac oscillator in a 1+1-dimensional space-time described by such an algebra is studied in the case where $\beta'=0$. Extending supersymmetric quantum mechanical and shape-invariance methods to energy-dependent Hamiltonians provides exact bound-state energies and wavefunctions. Physically acceptable states exist for $\beta < 1/(m^2 c^2)$. A new interesting outcome is that, in contrast with the conventional Dirac oscillator, the energy spectrum is bounded.Comment: 20 pages, no figure, some very small changes, published versio