The Most General Form of Deformation of the Heisenberg Algebra from the Generalized Uncertainty Principle

TL;DR

This paper proposes the most general deformation of the Heisenberg algebra inspired by the generalized uncertainty principle, affecting all quantum systems and leading to space discretization and nonlocal effects.

Contribution

It introduces a comprehensive deformation framework of the Heisenberg algebra based on the generalized uncertainty principle, connecting nonlocality and space fractional quantum mechanics.

Findings

Deformation affects all quantum mechanical systems.

Low energy effects include modifications to harmonic oscillator and Landau levels.

Deformation results in space discretization.

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 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.