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

## TL;DR

This paper generalizes the Foldy-Wouthuysen transformation to fermions in gauge-invariant, time-dependent electromagnetic fields, providing closed-form energy operators and linking to the Feynman-Gell-Mann representation.

## Contribution

It introduces a method to obtain closed-form energy operators for fermions in dynamic electromagnetic fields, extending previous static field results.

## Key findings

- Derived closed-form energy operators in time-dependent fields
- Established connection with Feynman-Gell-Mann representation
- Extended previous static field results to dynamic scenarios

## Abstract

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.