Eikonal equation of the Lorentz-violating Maxwell theory

TL;DR

This paper derives the eikonal equation for light in Lorentz-violating Maxwell theory, showing how LIV affects wavefront propagation, dispersion relations, and causality, with implications for classical causality analysis in QED.

Contribution

It provides a self-consistent derivation of the eikonal equation in LIV Maxwell theory and links it to dispersion relations and causality considerations.

Findings

Wavefront velocity equals group velocity and energy flow velocity.

Existence of modes with superluminal signal velocity, indicating classical causality violation.

Derived dispersion relations consistent with other approaches.

Abstract

We derive the eikonal equation of light wavefront in the presence of Lorentz invariance violation (LIV) from the photon sector of the standard model extension (SME). The results obtained from the equations of $\mathbf{E}$ and $\mathbf{B}$ fields respectively are the same. This guarantees the self-consistency of our derivation. We adopt a simple case with only one non-zero LIV parameter as an illustration, from which we find two points. One is that, in analogy with Hamilton-Jacobi equation, from the eikonal equation, we can derive dispersion relations which are compatible with results obtained from other approaches. The other is that, the wavefront velocity is the same as the group velocity, as well as the energy flow velocity. If further we define the signal velocity $v_s$ as the front velocity, there always exists a mode with $v_s>1$, hence causality is violated classically. Thus our method might be useful in the analysis of Lorentz violation in QED in terms of classical causality .