# On constrained analysis and diffeomorphism invariance of generalised Proca theories

## TL;DR

This paper analyzes generalized Proca theories with specific constraints, deriving conditions for their degrees of freedom using Faddeev-Jackiw analysis, and highlights how diffeomorphism invariance simplifies the theoretical framework.

## Contribution

It provides a detailed analysis of the constraints and degrees of freedom in generalized Proca theories, emphasizing the role of diffeomorphism invariance in simplifying the conditions.

## Key findings

- Most conditions for three propagating degrees of freedom are trivial due to diffeomorphism invariance.
- A key non-trivial condition involves a complex combination of terms not being zero.
- Diffeomorphism invariance simplifies the structure of Faddeev-Jackiw brackets.

## Abstract

In this paper we consider generalised Proca theories coupled to any background field and with time-time and time-space components of Hessian of the vector sector are zero, whereas the space-space part is non-degenerate. By using Faddeev-Jackiw analysis, we derive the conditions that these theories have to satisfy in order for the vector sector to have three propagating degrees of freedom. Most of these conditions are trivialised due to diffeomorphism invariance requirements. This leaves only a condition that a complicated combination of terms should not be trivially zero. This condition is therefore easy to be fulfilled. For completeness, we have also investigated on how diffeomorphism invariance helps in simplifying Faddeev-Jackiw brackets.