  We show that inconsistency between background field methods, which are
relevant for lattice QCD spectroscopy, and effective field theory matching
conditions, which are obtained from scattering amplitudes, can be resolved by
augmenting nonrelativistic QED with operators related by the equations of
motion. To determine the coefficients of such operators, we perform the
nonrelativistic expansion of QED for a spin-half hadron including non-minimal
electromagnetic couplings. As an effective field theory framework could provide
a valuable tool to analyze lattice QCD correlation functions in external
fields, we investigate whether nonrelativistic QED can be used to this end. We
argue, however, that the most desirable approach is a hybrid one, which
combines a relativistic hadron theory with operator selection based on
nonrelativistic QED power counting. In this hybrid framework, new results are
obtained for charged spin-half hadrons in uniform magnetic fields, including a
proper treatment of Landau levels both in infinite volume and on a torus.
