Formulation of Electrodynamics with an External Source in the Presence of a Minimal Measurable Length

Abstract

In a series of papers,Quesne andTkachuk (2006) presented aD+ 1-dimensional (β,β)-two-parameter Lorentz-covariant deformed algebra which leads to a nonzero minimal measurable length. In this paper, the Lagrangian formulation of electrodynamics in a 3 + 1-dimensional spacetime described by Quesne-Tkachuk algebra is studied in the special case of β󸀠 = 2β up to the first order over the deformation parameter β. It is demonstrated that at the classical level there is a similarity between electrodynamics in the presence of a minimal measurable length (generalized electrodynamics) and Lee-Wick electrodynamics. We obtain the free space solutions of the inhomogeneous Maxwell’s equations in the presence of a minimal length. These solutions describe two vector particles (a massless vector particle and a massive vector particle). We estimate two different upper bounds on the isotropic minimal length. The first upper bound is near to the electroweak length scale (lelectroweak ∼ 10 −18 m), while the second one is near to the length scale for the strong interactions (lstrong ∼ 10 −15 m). The relationship between the Gaete-Spallucci nonlocal electrodynamics (2012) and electrodynamics with a minimal length is investigated.