Position-dependent mass in strong quantum gravitational background fields

Abstract

More recently, we have proposed a set of noncommutative space that describes the quantum gravity at the Planck scale [J. Phys. A: Math. Theor. 53, 115303 (2020)]. The interesting significant result, we found is that, the generalized uncertainty principle induces a maximal measurable length of quantum gravity. This measurement revealed strong quantum gravitational effects at this scale and predicted a detection of gravity particles with low energies. In the present paper, to make evidence this prediction, we study in this space, the dynamics of a particle with position-dependent mass (PDM) trapped in an infinite square well. We show that by increasing the quantum gravitational effect, the PDM of the particle increases and induces deformations of the quantum energy levels. These deformations are more pronounced as one increases the quantum levels allowing, the particle to jump from one state to another with low energies and with high probability densities.

Path integral in position-deformed Heisenberg algebra with strong quantum gravitational measurement

“…To circumvent this requirement, one of us recently has proposed a position-deformed Heisenberg algebra [14] in two dimensions (2D) that introduces the simultaneous existence of minimal and maximal length uncertainties. The emergence of this maximal length demonstrated strong quantum gravitational effects in this space and predicted the detection of low-energy gravity particles [1]. In continuation of this work, we construct the position space representation describing this maximal length, as well as the corresponding Fourier transform and its inverse representations.…”

“…They satisfy the following relation [1] [ X, P ] = i (I − τ X + τ 2 X2 ), (10) where τ ∈ (0, 1) is the generalized uncertainty principle parameter related to quantum gravitational effects in this space which describes the Planck scale [2,3]. Obviously by taking τ → 0, we recover the algebra (1). The action of these operators on the following unit basis vectors {|x } and {|p } reads as follows…”

“…Figure (1) illustrates the deformed action (67) of the free particle versus the position x (with x = 0) and the time ∆t for fixed values of parameter τ . Figure 1a shows that, for any ∆t > 0 the values of the deformed action S f p over the position decrease from the non deformed action S 0 f p as one increases the quantum gravitational parameter τ .…”

“…We compute the propagators and actions of a free particle and a simple harmonic oscillator as applications. Since the quantum gravity is strongly measured in this space [1], we show that the classical trajectories of particles described by their actions are deformed, allowing particles to take the shortest path between two points in the minimum time. This result strengthens the claim that the recently proposed position-deformed algebra [1] induces strong quantum gravitational fields with features close to those of classical ones of general relativity.…”

Photon-added BarutGirardello like coherent states of time-dependent Landau problem

“…In the following discussion we will consider various properties of these states including the non-orthogonality, the usual conditions of continuity in the label, normalizability, the resolution of identity by finding the weight function ω [39] .…”