Position in Minimal Length Quantum Mechanics

Several approaches to quantum gravity imply the presence of a minimal measurable length at high energies. This is in tension with the Heisenberg Uncertainty Principle. Such a contrast is then considered in phenomenological approaches to quantum gravity by introducing a minimal length in quantum mechanics via the Generalized Uncertainty Principle. Several features of the standard theory are affected by such a modification. For example, position eigenstates are no longer included in models of quantum mechanics with a minimal length. Furthermore, while the momentum-space description can still be realized in a relatively straightforward way, the (quasi-)position representation acquires numerous issues. Here, we will review such issues, clarifying aspects regarding models with a minimal length. Finally, we will consider the effects of such models on simple quantum mechanical systems.

Primordial big bang nucleosynthesis and generalized uncertainty principle

The Generalized Uncertainty Principle (GUP) naturally emerges in several quantum gravity models, predicting the existence of a minimal length at Planck scale. Here, we consider the quadratic GUP as a semiclassical approach to thermodynamic gravity and constrain the deformation parameter by using observational bounds from Big Bang Nucleosynthesis and primordial abundances of the light elements $${}^4 He, D, {}^7 Li$$ 4 H e , D , 7 L i . We show that our result fits with most of existing bounds on $$\beta $$ β derived from other cosmological studies.

Quantum gravity corrections to the mean field theory of nucleons

In this paper, we analyze the correction to the mean field theory potential for a system of nucleons. It will be argued that these corrections can be obtained by deforming the Schrödinger’s equation describing a system of nucleons by a minimal length in the background geometry of space-time. This is because such a minimal length occurs due to quantum gravitational effects, and modifies the low energy quantum mechanical systems. In fact, as the mean field potential for the nucleons is represented by the Woods–Saxon potential, we will explicitly analyze such corrections to this potential. We will obtain the corrections to the energy eigenvalues of the deformed Schrödinger’s equation for the Woods–Saxon potential. We will also construct the wave function for the deformed Schrödinger’s equation.

A hypomorphic allele of telomerase uncovers the minimal functional length of telomeres in Arabidopsis

Despite the essential requirement of telomeric DNA for genome stability, the length of telomere tracts between species substantially differs, raising the question of the minimal length of telomeric DNA necessary for proper function. Here we address this question using a hypomorphic allele of the telomerase catalytic subunit, TERT. We show that although this construct partially restored telomerase activity to a tert mutant, telomeres continued to shorten over several generations, ultimately stabilizing at a bimodal size distribution. Telomeres on two chromosome arms were maintained at a length of 1kb, while the remaining telomeres were maintained at 400 bp. The longest telomeres identified in this background were also significantly longer in wild type populations, suggesting cis-acting elements on these arms either promote telomerase processivity or recruitment. Genetically disrupting telomerase processivity in this background resulted in immediate lethality. Thus, telomeres of 400 bp are both necessary and sufficient for Arabidopsis viability. As this length is the estimated minimal length for t-loop formation, our data suggest that telomeres long enough to form a t-loop constitute the minimal functional length.

Quantum gravitational decoherence from fluctuating minimal length and deformation parameter at the Planck scale

Schemes of gravitationally induced decoherence are being actively investigated as possible mechanisms for the quantum-to-classical transition. Here, we introduce a decoherence process due to quantum gravity effects. We assume a foamy quantum spacetime with a fluctuating minimal length coinciding on average with the Planck scale. Considering deformed canonical commutation relations with a fluctuating deformation parameter, we derive a Lindblad master equation that yields localization in energy space and decoherence times consistent with the currently available observational evidence. Compared to other schemes of gravitational decoherence, we find that the decoherence rate predicted by our model is extremal, being minimal in the deep quantum regime below the Planck scale and maximal in the mesoscopic regime beyond it. We discuss possible experimental tests of our model based on cavity optomechanics setups with ultracold massive molecular oscillators and we provide preliminary estimates on the values of the physical parameters needed for actual laboratory implementations.

Calculation of the minimal length of the high-speed line

This paper explores the minimum length of continuous sections of the high-speed line. The minimum length is examined in terms of the maximum speed of high-speed vehicles. Both traditional trainsets consisting of traction units and coaches, and train units were selected for examination. The graphs present the difference in the ability of the different vehicles to reach and use the maximumspeed.

Aharonov–Bohm-like scattering in the generalized uncertainty principle-corrected quantum mechanics

We discuss classical electrodynamics and the Aharonov–Bohm effect in the presence of the minimal length. In the former, we derive the classical equation of motion and the corresponding Lagrangian. In the latter, we adopt the generalized uncertainty principle (GUP) and compute the scattering cross-section up to the first-order of the GUP parameter [Formula: see text]. Even though the minimal length exists, the cross-section is invariant under the simultaneous change [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] are azimuthal angle and magnetic flux parameter. However, unlike the usual Aharonv–Bohm scattering, the cross-section exhibits discontinuous behavior at every integer [Formula: see text]. The symmetries, which the cross-section has in the absence of GUP, are shown to be explicitly broken at the level of [Formula: see text].