Generalization of the Proca Action
  We consider the Lagrangian of a vector field with derivative
self-interactions with a priori arbitrary coefficients. Starting with a flat
space-time we show that for a special choice of the coefficients of the
self-interactions the ghost-like pathologies disappear. For this we use the
degeneracy condition of the Hessian. This constitutes the Galileon-type
generalization of the Proca action with only three propagating physical degrees
of freedom. The longitudinal mode of the vector field is associated to the
usual Galileon interactions for a specific choice of the overall functions. In
difference to a scalar Galileon theory, the generalized Proca field has more
free parameters and purely intrinsic vector interactions. We then extend this
analysis to a curved background. The resulting theory is the Horndeski Proca
action with second order equations of motion on curved space-times.
