Abstract
In this article we use an electromagnetic Lagrangian constructed so as to include dispersive effects in the description of an electromagnetic wave propagating in the quantum electrodynamic vacuum. This Lagrangian is Lorentz invariant, includes contributions up to six powers in the electromagnetic fields, and involves both fields and their first derivatives. Conceptual limitations inherent to the use of this higher derivative Lagrangian approach are discussed. We consider the one-dimensional spatial limit and obtain an exact solution of the nonlinear wave equation recovering the Korteveg-de Vries type periodic waves and solitons given in S. V. Bulanov et al. [Phys. Rev. D 101, 016016 (2020)].
Highlights
WAVE EQUATIONS IN NONLINEAR QUANTUM ELECTRODYNAMICSField induced polarization and birefringence of the vacuum, see e.g., Refs. [1,2], are fundamental effects predicted by quantum electrodynamics (QED)
In this article we use an electromagnetic Lagrangian constructed so as to include dispersive effects in the description of an electromagnetic wave propagating in the quantum electrodynamic vacuum
These effects arise from the process of scattering of light by light: while in classical electrodynamics electromagnetic waves do not interact in vacuum, in QED photon-photon scattering can take place in vacuum via the generation of virtual electron-positron pairs that gives rise to polarization and magnetization currents that make the vacuum respond as a material medium
Summary
Field induced polarization and birefringence of the vacuum, see e.g., Refs. [1,2], are fundamental effects predicted by quantum electrodynamics (QED). Approximation using the well-known Heisenberg-Euler Lagrangian in the electromagnetic action functional [3,5] This approximation leads to nonlinear wave equations for the fields amplitude in vacuum that are not dispersive, i.e., that are homogeneous in the second order derivatives of the field four-vector potentials. In the present article we use a Lagrangian in the electromagnetic vacuum action that involves higher order derivatives of the wave vector potential and that is constructed so as to include the quantum nonlinearity parameter dependency of the invariant photon mass. In this formulation higher order derivatives enter in combination with nonlinear terms. L 1⁄4 LHE þ ∂λFαβ∂γFσδLλ1αβγσδðFμνÞ þ higher field derivative terms; ð6Þ where LHE is the Heisenberg-Euler Lagrangian and Lλ1αβγσδðFμνÞ is a local function of the electromagnetic field tensor
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