  We consider, in Minkowski spacetime, higher-order Maxwell Lagrangians with
terms quadratic in the derivatives of the field strength tensor, and study
their degrees of freedom. Using a 3+1 decomposition of these Lagrangians, we
extract the kinetic matrix for the components of the electric field,
corresponding to second time derivatives of the gauge field. If the kinetic
matrix is invertible, the theory admits five degrees of freedom, namely the
usual two polarisations of a photon plus three extra degrees of freedom which
are shown to be Ostrogradski ghosts. We also classify the cases where the
kinetic matrix is non-invertible and, using analogous simple models, we argue
that, even though the degeneracy conditions reduce the number of degrees of
freedom, it does not seem possible to fully eliminate all potential
Ostrogradski ghosts.
