  An effective field theory of composite Dirac fermions was proposed by D. T.
Son as a theory of the half-filled Landau level with explicit particle-hole
symmetry. We compute the electromagnetic response of this Son-Dirac theory on
the level of the random phase approximation (RPA), where we pay particular
attention to the effect of an additional composite-fermion dipole term that is
needed to restore Galilean invariance. We find that once this dipole correction
is taken into account, unphysical interband transitions and collective modes
that are present in the response of the unmodified theory either cancel or are
strongly suppressed. We demonstrate that this gives rise to a consistent theory
of the half-filled Landau level valid at all frequencies, at least to leading
order in the momentum. In addition, the dipole contribution modifies the
Fermi-liquid response at small frequency and momentum, which is a prediction of
the Son-Dirac theory within RPA that distinguishes it from a separate
description of the half-filled Landau level by Halperin, Lee, and Read within
RPA.
