Formulation of an Electrostatic Field with a Charge Density in the Presence of a Minimal Length Based on the Kempf Algebra

TL;DR

This paper formulates an electrostatic field theory incorporating a minimal length based on Kempf algebra, revealing similarities to higher derivative electrostatics and providing bounds on the minimal length scale.

Contribution

It introduces a modified electrostatics model derived from Kempf algebra, connecting minimal length effects with higher derivative theories and estimating upper bounds on the minimal length.

Findings

Classical self-energy of a point charge becomes finite.

Two upper bounds on the minimal length are estimated.

Quantum bound is near the electroweak scale.

Abstract

In a series of papers, Kempf and co-workers (J. Phys. A: Math. Gen. {\bf 30}, 2093, (1997); Phys. Rev. D {\bf52}, 1108, (1995); Phys. Rev. D {\bf55}, 7909, (1997)) introduced a D-dimensional $(\beta,\beta')$-two-parameter deformed Heisenberg algebra which leads to a nonzero minimal observable length. In this work, the Lagrangian formulation of an electrostatic field in three spatial dimensions described by Kempf algebra is studied in the case where $\beta'=2\beta$ up to first order over deformation parameter $\beta$. It is shown that there is a similarity between electrostatics in the presence of a minimal length (modified electrostatics) and higher derivative Podolsky's electrostatics. The important property of this modified electrostatics is that the classical self-energy of a point charge becomes a finite value. Two different upper bounds on the isotropic minimal length of this modified electrostatics are estimated. The first upper bound will be found by treating the modified electrostatics as a classical electromagnetic system, while the second one will be estimated by considering the modified electrostatics as a quantum field theoretic model. It should be noted that the quantum upper bound on the isotropic minimal length in this paper is near to the electroweak length scale $(\ell_{electroweak}\sim 10^{-18}\, m)$.