Energy expectation values of a particle in nonstationary fields

TL;DR

This paper clarifies how the choice of representation affects energy expectation values for particles in nonstationary fields, emphasizing the importance of the Foldy-Wouthuysen representation over the Dirac one.

Contribution

It demonstrates that the Foldy-Wouthuysen representation is fundamental for energy calculations in nonstationary fields, correcting previous assumptions about the Dirac representation's primacy.

Findings

Derived the general energy expectation value equation in Dirac representation.

Showed the non-equivalence of Hamiltonians due to time-dependent transformations.

Provided an exact Foldy-Wouthuysen transformation example in nonstationary conditions.

Abstract

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.