Formulation of an electrostatic field with a charge density in the presence of a minimal length based on the Kempf algebra
Department of Physics, Faculty of Sciences, Arak University - Arak 38156-8-8349, Iran
2 Department of Science, Campus of Bijar, University of Kurdistan - Bijar, Iran
Accepted: 7 May 2012
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).
PACS: 04.60.Bc – Phenomenology of quantum gravity / 03.50.-z – Classical field theories / 12.20.-m – Quantum electrodynamics
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