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

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