  An axion background field induces tiny oscillating electric and magnetic
fields in an external static magnetic field. This signature is used to search
for axion dark matter. We use standard quantum field theory techniques to
obtain an expression for a transition amplitude, from which we identify the
classical electromagnetic fields induced by the background axion field. We
confirm previous results, that if the spatial size $R$ of the applied static
magnetic field is small compared to the axion Compton wavelength $\lambda$, the
induced electric and magnetic fields are parametrically suppressed by the small
numbers $(R/\lambda)^2$ and $R/\lambda$, respectively, relative to the case
when $R$ is larger than $\lambda$. Our approach allows an intuitive
interpretation in terms of 4-momentum conservation and momentum exchange via
the photon propagator.
