On Dirac theory in the space with deformed Heisenberg algebra. Exact solutions

The Dirac equation has been studied in which the Dirac matrices \hat{\boldmath\alpha}, \hat\beta have space factors, respectively $f$ and $f_1$, dependent on the particle's space coordinates. The $f$ function deforms Heisenberg algebra for the coordinates and momenta operators, the function $f_1$ being treated as a dependence of the particle mass on its position. The properties of these functions in the transition to the Schr\"odinger equation are discussed. The exact solution of the Dirac equation for the particle motion in the Coulomnb field with a linear dependence of the $f$ function on the distance $r$ to the force centre and the inverse dependence on $r$ for the $f_1$ function has been found.Comment: 13 page

Theory Of Radiation And Absorption Of Deformed Field Quanta

We calculate the intensity of spontaneous radiation of a system of non-linear quantum field, where the non-linearity is due to deformations of the Poison brackets of the generalized coordinates and momenta.Comment: This paper is dedicated to Professor Igor Stasyuk on the occasion of his 70th birthday. 9 pages, 1 figur

Kepler problem in Dirac theory for a particle with position-dependent mass

Exact solution of Dirac equation for a particle whose potential energy and mass are inversely proportional to the distance from the force centre has been found. The bound states exist provided the length scale $a$ which appears in the expression for the mass is smaller than the classical electron radius $e^2/mc^2$. Furthermore, bound states also exist for negative values of $a$ even in the absence of the Coulomb interaction. Quasirelativistic expansion of the energy has been carried out, and a modified expression for the fine structure of energy levels has been obtained. The problem of kinetic energy operator in the Schr\"odinger equation is discussed for the case of position-dependent mass. In particular, we have found that for highly excited states the mutual ordering of the inverse mass and momentum operator in the non-relativistic theory is not important.Comment: 9 page

Comment on ``Kepler problem in Dirac theory for a particle with position-dependent mass''

Based on easy-to-follow considerations it is not difficult to be vehemently opposed not only the solutions found in that paper but also the conclusions manifested there.Comment: 4 page

Casimir effect in deformed field

The Casimir energy is calculated in one-, two-, and three-dimensional spaces for the field with generalized coordinates and momenta satisfying the deformed Poisson brackets leading to the minimal length.Comment: 12 pages, 1 figur