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

Abstract

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