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arXiv:1611.00001 · Physics Letters B

The most general form of deformation of the Heisenberg algebra from the generalized uncertainty principle

Syed Masood, Mir Faizal, M. Zaid Zaz, Ahmed Farag Ali, Jamil Raza, Mushtaq B. Shah

Published Oct 24, 2016• 49 citations
arXivHTMLPDF

Abstract

In this paper, we will propose the most general form of the deformation of Heisenberg algebra motivated by the generalized uncertainty principle. This deformation of the Heisenberg algebra will deform all quantum mechanical systems. The form of the generalized uncertainty principle used to motivate these results will be motivated by the space fractional quantum mechanics, and non-locality in quantum mechanical systems. We also analyse a specific limit of this generalized deformation for one dimensional system, and in that limit, a nonlocal deformation of the momentum operator generates a local deformation of all one dimensional quantum mechanical systems. We analyse the low energy effects of this deformation on a harmonic oscillator, Landau levels, Lamb shift, and potential barrier. We also demonstrate that this deformation leads to a discretization of space.

DOI
10.1016/j.physletb.2016.10.047

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In this paper, we will propose the most general form of the deformation of Heisenberg algebra motivated by the generalized uncertainty principle. This deformation of the Heisenberg algebra will deform all quantum mechanical systems. The form of the generalized uncertainty principle used to motivate these results will be motivated by the space fractional quantum mechanics, and non-locality in quantum mechanical systems. We also analyse a specific limit of this generalized deformation for one dimensional system, and in that limit, a nonlocal deformation of the momentum operator generates a local deformation of all one dimensional quantum mechanical systems. We analyse the low energy effects of this deformation on a harmonic oscillator, Landau levels, Lamb shift, and potential barrier. We also demonstrate that this deformation leads to a discretization of space.
This paper provides a concise overview of the comprehensive version of the Generalized Uncertainty Principle (GUP) derived from nonlocal quantum mechanics and extensively addressed by S. Masood, et al. We utilize the specific constraint of this GUP formulation to compute the energy correction and the modified wave function for the linear potential in one dimension. In addition, we examine the impact of deformation on a one-dimensional delta potential well and a one-dimensional delta potential barrier. Furthermore, we specifically focus on the impact of deformation on hydrogen atoms existing in one dimension, and we conduct a distinct investigation into the Stark effect on the atom.
This work establishes a link between deformed angular momentum operators and modifications of the quantum uncertainty principle. Starting from an algebra in which the standard angular momentum operators are deformed by two parameters, [Formula: see text] and [Formula: see text], we construct the corresponding momentum operators via [Formula: see text] where the radial operator [Formula: see text] is determined by the requirement that the standard momentum operator is recovered in the limit [Formula: see text], [Formula: see text]. The resulting commutation relations take the form [Formula: see text] where [Formula: see text] is a position-dependent deformation matrix. This leads to an uncertainty relation [Formula: see text] which is state-dependent, anisotropic and position-dependent. We use the term Generalized Uncertainty Principle (GUP) in its broad sense referring to any modification of the Heisenberg relation motivated by quantum-gravity ideas. However, the model does not yield a fixed minimal length of the kind familiar from conventional GUP models (e.g. [Formula: see text]). In the isotropic, limit the trace of the deformation matrix simplifies the bound to [Formula: see text] which can be considered as a rescaling of the uncertainty constant. Our work thus provides a geometric origin for modified uncertainty relations from symmetry breaking in the rotation group, and offers a framework in which quantum-gravity effects manifest themselves through state- and direction-dependent measurement limits, without necessarily introducing a universal minimal length.
A deformation of Heisenberg algebra induces among other consequences a loss of Hermiticity of some operators that generate this algebra. Therefore, these operators are not Hermitian, nor is the Hamiltonian operator built from them. In the present paper, we propose a position deformation of Heisenberg algebra with both maximal length and minimal momentum uncertainties. By using a pseudo-similarity transformation to the non-Hermitian operators, we prove their Hermiticity with a suitable positive-definite pseudo-metric operator. We then construct Hilbert space representations associated with these pseudo-Hermitian operators. Finally, we study the eigenvalue problem of a free particle in this deformed space and we show that this deformation curved the quantum levels allowing particles to jump from one state to another with low energy transitions.
A proper deformation of the underlying coordinate and momentum commutation relations in quantum mechanics provides a phenomenological approach to account for the influence of gravity on small scales. Introducing the squared momentum term results in a generalized uncertainty principle, which limits the minimum uncertainty in particle position to the Planck length. However, such a deformation of the commutator significantly changes the formalism, making it separate from the canonical formalism of quantum mechanics. In this study, it is shown that the deformed algebra of position and momentum operators can be incorporated into the framework of ordinary quantum mechanics.
Several theories of quantum gravity propose the existence of a minimal measurable length and maximum measurable momentum near the Planck scale. When integrated into the framework of quantum mechanics, such restrictions lead to the generalization of the Heisenberg uncertainty principle, which is commonly referred to as the generalized uncertainty principle (GUP). The GUP has been applied to a plethora of physical models in order to provide further insights on our understanding of the Universe, both in the macroscopic and the microscopic regime. Nevertheless, the corrections related to GUP are expected to be extremely small, which thus renders experimental verification a difficult task. Therefore, phenomenological approaches on a qualitative level are typically addressed with the aim of enhancing our possibility of detecting these tiny effects. Motivated by a recent work on a universal decoherence mechanism stemming from the standard formulation of GUP, we extend the ideas contained therein by considering a more general form of GUP which includes linear and quadratic momentum terms to examine its implications. To achieve such a picture, we equip the deformation parameter appearing in front of all momentum terms with a stochastic nature, so that the effective decoherence mechanism emerges by taking the average over fluctuations. We find that, despite the apparently small differences among the two generalizations of the Heisenberg uncertainty principle, the consequences at the level of the universal decoherence mechanism they entail is significant, as they predict decoherence times that are completely uncorrelated and distinct.