Formulation of an electrostatic field with a charge density in the presence of a minimal length based on the Kempf algebra

In a series of papers, Kempf and co-workers (J. Phys. A: Math. Gen., 30 (1997) 2093; Phys. Rev. D, 52 (1995) 1108; Phys. Rev. D, 55 (1997) 7909) introduced a D-dimensional (β, β′)-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 in which β′ =2β up to first order over the deformation parameter β. 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 (ℓelectroweak∼10− 18 m).