  We derive an alternative representation of the leading-order contribution to
the polarization operator in strong-field quantum electrodynamics with a
plane-wave electromagnetic background field, which is manifestly symmetric with
respect to the external photon momenta. Our derivation is based on a direct
evaluation of the corresponding Feynman diagram, using the Volkov
representation of the dressed fermion propagator. Furthermore, the validity of
the Ward-Takahashi identity is shown for general loop diagrams in an external
plane-wave background field.
