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

TL;DR

This paper develops a Lagrangian formulation for magnetostatics incorporating a minimal length scale based on Kempf algebra, revealing classical similarities to Lee-Wick models and deriving bounds on the minimal length from muon magnetic moment corrections.

Contribution

It introduces a novel Lagrangian framework for magnetostatics with minimal length effects and explores its classical and quantum implications, including bounds on the minimal length scale.

Findings

Derived modified Ampere's law and energy density for magnetostatics with minimal length.

Found the upper bound on the minimal length scale to be 4.42×10^{-19} meters.

Established a relationship between minimal length magnetostatics and non-local magnetostatics.

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.