  The low-energy structure of hadrons can be described systematically using
effective field theory, and the parameters of the effective theory can be
determined from lattice QCD computations. Recent work, however, points to
inconsistencies between the background field method in lattice QCD and
effective field theory matching conditions. We show that the background field
problem necessitates inclusion of operators related by equations of motion. In
the presence of time-dependent electromagnetic fields, for example, such
operators modify Green's functions, thereby complicating the isolation of
hadronic parameters which enter on-shell scattering amplitudes. The
particularly simple case of a scalar hadron coupled to uniform electromagnetic
fields is investigated in detail. At the level of the relativistic effective
theory, operators related by equations of motion are demonstrated to be
innocuous. The same result does not hold in the non-relativistic effective
theory, and inconsistencies in matching are resolved by carefully treating
operators related by equations of motion. As uniform external fields
potentially allow for surface terms, the problem is additionally analyzed on a
torus where such terms are absent. Finite-size corrections are derived for
charged scalar correlation functions in uniform electric fields as a useful
byproduct.
