Hamiltonian Expectation Values for Time-Dependent Foldy-Wouthuysen Transformations: Implications for Electrodynamics and Resolution of the External-FieldπNAmbiguity

Abstract

Although the time-dependent Foldy-Wouthuysen (FW) transformation of the "Schrödinger" equation $\frac{i\partial \psi }{\partial t}=H\psi$ yields the transformed equation $\frac{i\partial {\psi }^{\prime }}{\partial t}=H^{\prime }{\psi }^{\prime }$, where ${\psi }^{\prime }=U\psi$ and $H^{\prime }=UH{U}^{-1\mathrm{}}-\frac{\mathrm{iU}\partial \left(U^{-1\mathrm{}}\right)}{\partial t}$, the expectation values (${\psi }_1, H{\psi }_j$) of $H$ are not equal to (${{\psi }_i}^{\prime }, H^{\prime }{{\psi }_j}^{\prime }$), but rather, to (${{\psi }_i}^{\prime }, \mathrm{UH}{U}^{-1\mathrm{}}{{\psi }_j}^{\prime }$). But one still has $({{\psi }_i}^{\prime }, {{\psi }_j}^{\prime })=({\psi }_i, {\psi }_j)$. I discuss implications for perturbation calculations, scattering amplitudes, electrodynamics, and the external-field $\pi N$ ambiguity, and look at exact special cases.