# The complete theory of Maxwell and Proca fields

## TL;DR

This paper develops the most general ghost-free Lagrangian for interacting Abelian spin-one fields, extending previous work to multiple fields with derivative self-interactions, and explores applications in holography and black hole physics.

## Contribution

It introduces a comprehensive framework for ghost-free multi-Proca theories with derivative interactions, generalizing existing models and establishing conditions for ghost-freedom.

## Key findings

- Reproduces known healthy interactions for single Proca fields
- Identifies conditions for ghost-freedom in multi-field setups
- Highlights that many existing multi-Proca interactions are ghostly

## Abstract

We present the most general ghost-free classical Lagrangian containing first-order derivatives and describing interacting real Abelian spin-one fields on Minkowski spacetime. We study both massive Proca and massless Maxwell fields and allow for a non-linear realization of mass, in the form of derivative self-interactions. Within this context, our construction notoriously extends the existing literature, which is limited to the case of a single Proca field and to multiple interacting Proca fields in the presence of a global rotational symmetry. In the limit of a single Proca field, we reproduce the known healthy interaction terms. We provide the necessary and sufficient conditions to ensure ghost-freedom in any multi-field setup. We observe that, in general, the said conditions are not satisfied by the rotationally symmetric multi-Proca interactions suggested so far, which implies that they propagate ghosts. Our theory admits a plethora of applications in a wide range of subjects. For illustrative purposes, we provide concrete proposals in holographic condensed matter and black hole physics.