Maxwell-Proca theory: Definition and construction

We present a systematic construction of the most general first order Lagrangian describing an arbitrary number of interacting Maxwell and Proca fields on Minkowski spacetime. To this aim, we first formalize the notion of a Proca field, in analogy to the well-known Maxwell field. Our definition allows for a nonlinear realization of the Proca mass, in the form of derivative self-interactions. Consequently, we consider so-called generalized Proca/vector Galileons. We explicitly demonstrate the ghost-freedom of this complete Maxwell-Proca theory by obtaining its constraint algebra. We find that, when multiple Proca fields are present, their interactions must fulfill nontrivial differential relations in order to ensure the propagation of the correct number of degrees of freedom. These relations had so far been overlooked, which means previous multi-Proca proposals generically contain ghosts. This is a companion paper to the paper by Diez et al. [arXiv:1905.06967]. It puts on a solid footing the theory there introduced.