Fractional formulation of Podolsky Lagrangian density

Amer D Al-Oqali

doi:10.21833/ijaas.2022.02.015

Abstract

Lagrangians which depend on higher-order derivatives appear frequently in many areas of physics. In this paper, we reformulate Podolsky's Lagrangian in fractional form using left-right Riemann-Liouville fractional derivatives. The equations of motion are obtained using the fractional Euler Lagrange equation. In addition, the energy stress tensor and the Hamiltonian are obtained in fractional form from the Lagrangian density. The resulting equations are very similar to those found in classical field theory.

Highlights

Higher derivative field theories have been gaining popularity in recent years
In order to avoid singularities in electromagnetic fields and to have a finite and positive self-energy of point charges, Bopp and Podolsky proposed a gradient theory representing a classical generalization of Maxwell electrodynamics towards generalized electrodynamics with fourth-order linear field equations (Lazar, 2019)
Let us consider the Lagrangian proposed by Podolsky (Bertin et al, 2017; Podolsky and Schwed, 1948): L

Summary

Introduction

Higher derivative field theories have been gaining popularity in recent years. Many models, including renormalizable quantum gravity, Podolsky's generalized electrodynamics, the Lee–Wick model, and others, include higher derivative field equations. Podolsky’s theory was introduced in the early 1940s by Bopp and Podolsky (Lazar and Leck, 2020). In order to avoid singularities in electromagnetic fields and to have a finite and positive self-energy of point charges, Bopp and Podolsky proposed a gradient theory representing a classical generalization of Maxwell electrodynamics towards generalized electrodynamics with fourth-order linear field equations (Lazar, 2019). The purpose of this research is to reformulate the Lagrangian, proposed by Podolsky, in fractional form and to obtain conjugate momenta and energy stress tensor. The energymomentum tensor is constructed, and the Hamiltonian is obtained.

Riemann-Liouville fractional derivative

Formulation

Fractional Lagrangian density

Energy-momentum tensor

Conclusion