Closed Foldy–Wouthuysen transformations for fermions moving in gauge-invariant time-dependent electromagnetic fields

Previously, we obtained closed expressions for energy operators in the Foldy–Wouthuysen representation in the presence of static electric fields. In this case, we also established a connection between the Foldy–Wouthuysen representation and the Feynman–Gell-Mann representation. In this work, we generalize these results for the case of fermions moving in time-dependent gauge-invariant electromagnetic fields.