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

In a series of papers, Quesne and Tkachuk (J. Phys. A: Math. Gen. \textbf{39}, 10909 (2006); Czech. J. Phys. \textbf{56}, 1269 (2006)) presented a $D+1$-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 space-time described by Quesne-Tkachuk algebra is studied in the special case $β'=2β$ up to 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 $(\ell_{electroweak}\sim 10^{-18}\, m)$, while the second one is near to the length scale for the strong interactions $(\ell_{strong}\sim 10^{-15}\, m)$. The relationship between the Gaete-Spallucci nonlocal electrodynamics (J. Phys. A: Math. Theor. \textbf{45}, 065401 (2012)) and electrodynamics with a minimal length is investigated.

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