Abstract
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 non-linear 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 non-trivial 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 arXiv:1905.06968 [hep-th]. It puts on a solid footing the theory there introduced.
Highlights
A BRIEF HISTORY OF BEYOND ELECTRODYNAMICSIn the second half of the 19th century, the foundations of classical electromagnetism were laid
We begin by formalizing the notion of a Proca field in Definition 2, putting it on the same footing as that of the renowned Maxwell field
Based on these two concepts, we construct the Maxwell-Proca Lagrangian: the most general first-order theory in Minkowski spacetime for an arbitrary number of interacting Maxwell and Proca fields propagating the correct number of d.o.f
Summary
In the second half of the 19th century, the foundations of classical electromagnetism were laid. After Proca’s work, Bopp [6] and Podolsky [7] presented a model of a Uð1Þ-invariant Abelian vector field with a modified kinetic structure Such novel electrodynamics (and generalizations [8]) was introduced to amend some of the theoretical concerns at the time. Attempts have been made in pursuit of the most general theory of a massive vector field whose self-interactions extend the standard Proca mass term. Throughout the paper, we refer to any further d.o.f. carried by the action as a ghost The motivation behind these models goes beyond extending previous electromagneticlike theories: when considered over a cosmological background, generalized Proca fields can give rise to self-acceleration and fit data from late-time observations [15]. We focus on ghost-free interactions of multiple real Abelian fields on Minkowski spacetime Speaking, these can be either massless (Maxwell) or massive (Proca).
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