Altered Maxwell equations in the length gauge

TL;DR

This paper examines how the length gauge modifies Maxwell's equations by introducing source terms, highlighting differences from the transverse field description and implications for gauge transformations.

Contribution

It reveals that the length gauge leads to altered Maxwell equations with source terms, challenging assumptions of gauge equivalence in laser field descriptions.

Findings

Length gauge Maxwell equations include source terms.

Transverse fields propagate without sources.

Göppert-Mayer gauge transformation is not fully equivalent.

Abstract

The length gauge uses a scalar potential to describe a laser field, thus treating it as a longitudinal field rather than as a transverse field. This distinction is revealed in the fact that the Maxwell equations that relate to the length gauge are not the same as those for transverse fields. In particular, a source term is necessary in the length-gauge Maxwell equations, whereas the Coulomb-gauge description of plane waves possesses the basic property of transverse fields that they propagate with no source terms at all. This difference is shown to be importantly consequential in some previously unremarked circumstances; and it explains why the G\"oppert-Mayer gauge transformation does not provide the security that might be expected of full gauge equivalence.