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arXiv:2509.00021 · International Journal of Modern Physics A

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

В. П. Незнамов

Published Apr 23, 2025• 1 citation
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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.

DOI
10.1142/s0217751x25500745

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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.
We show that the origin of the nonequivalence of Hamiltonians in different representations is a change of the form of the time-derivative operator at a time-dependent unitary transformation. This nonequivalence does not lead to an ambiguity of the energy expectation values of a particle in nonstationary fields but assigns the basic representation. It has been explicitly or implicitly supposed in previous investigations that this representation is the Dirac one. We prove the alternative assertion about the basic role of the Foldy-Wouthuysen representation. We also derive the general equation for the energy expectation values in the Dirac representation. As an example, we consider a spin-1/2 particle with anomalous magnetic and electric dipole moments in strong time-dependent electromagnetic fields. We apply the obtained results to a spin-1/2 particle in a plane monochromatic electromagnetic wave and give an example of the exact Foldy-Wouthuysen transformation in the nonstationary case.
The Dirac equation for a charged particle in a time-dependent electromagnetic field is reduced to the Pauli equation in the non-relativistic limit by making a time-dependent Foldy-Wouthuysen transformation in a completely gauge-invariant way. In a time-dependent field, the expectation value of the Hamiltonian is dependent on the gauge, and so it cannot be the energy. The energy is obtained by taking the expectation value of the energy operator, the time rate of change of which is the average of the quantum mechanical power operator. The Hamiltonian formulation of a charged particle in a time-dependent external electromagnetic field is satisfactory when this distinction between the Hamiltonian, which describes the time evolution of the system, and the energy operator, which is gauge invariant, is made.