The low energy effective Lagrangian for photon interactions in any dimension

TL;DR

This paper derives a general, invariant effective Lagrangian for low energy photon interactions in arbitrary dimensions, extending the Euler-Heisenberg framework to include multiple photons and revealing connections to Bernoulli polynomials.

Contribution

It generalizes the low energy photon interaction Lagrangian to any dimension and multiple photons, linking coefficients to Bernoulli polynomials and space-time dimension.

Findings

Derived invariant form of the effective Lagrangian for arbitrary dimensions.

Extended the Euler-Heisenberg Lagrangian to include multiple photon scattering.

Connected the coefficients of invariant functions to Bernoulli polynomials.

Abstract

The subject of low energy photon-photon scattering is considered in arbitrary dimensional space-time and the interaction is widened to include scattering events involving an arbitrary number of photons. The effective interaction Lagrangian for these processes in QED has been determined in a manifestly invariant form. This generalisation resolves the structure of the weak-field Euler-Heisenberg Lagrangian and indicates that the component invariant functions have coefficients related, not only to the space-time dimension, but also to the coefficients of the Bernoulli polynomial.