Low-frequency failure of the G\"oppert-Mayer gauge transformation and consequences for the Strong-Field Approximation

The G\"oppert-Mayer (GM) gauge transformation, of central importance in atomic, molecular, and optical physics since it connects the length gauge and the velocity gauge, becomes unphysical as the field frequency declines towards zero. This is not consequential for theories of transverse fields, but it is the underlying reason for the failure of gauge invariance in the dipole-approximation version of the Strong-Field Approximation (SFA). This failure of the GM gauge transformation explains why the length gauge is preferred in analytical approximation methods for fields that possess a constant electric field as a zero-frequency limit.