Lagrangian Formulation of a Magnetostatic Field in the Presence of a Minimal Length Scale Based on the Kempf Algebra

Abstract

In the 1990s, Kempf and his collaborators Mangano and Mann introduced a $D$-dimensional $(\beta,\beta')$-two-parameter deformed Heisenberg algebra which leads to an isotropic minimal length $(\triangle X^{i})_{min}=\hbar\sqrt{D\beta+\beta'}\;,\forall i\in \{1,2, \cdots,D\}$. In this work, the Lagrangian formulation of a magnetostatic field in three spatial dimensions $(D=3)$ described by Kempf algebra is presented in the special case of $\beta'=2\beta$ up to the first order over $\beta$. We show that at the classical level there is a similarity between magnetostatics in the presence of a minimal length scale (modified magnetostatics) and the magnetostatic sector of the Abelian Lee-Wick model in three spatial dimensions. The integral form of Ampere's law and the energy density of a magnetostatic field in the modified magnetostatics are obtained. Also, the Biot-Savart law in the modified magnetostatics is found. By studying the effect of minimal length corrections to the gyromagnetic moment of the muon, we conclude that the upper bound on the isotropic minimal length scale in three spatial dimensions is $4.42\times10^{-19}m$. The relationship between magnetostatics with a minimal length and the Gaete-Spallucci non-local magnetostatics (J. Phys. A: Math. Theor. \textbf{45}, 065401 (2012)) is investigated.