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On the Coupling of Generalized Proca Fields to Degenerate Scalar-Tensor Theories^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

- (I)
- The constraint Ψ is lost, and no analogue of it exists.This will be the case when the rank of the Hessian matrix$${\mathcal{H}}_{IJ}:=\frac{{\partial}^{2}\mathcal{L}}{\partial {\dot{\psi}}^{I}\partial {\dot{\psi}}^{J}}\phantom{\rule{0.166667em}{0ex}},$$
- (II)
- The constraint Ψ (or some analogue of it) does exist, but it fails to Poisson-commute with one or more constraints present in the matter sector.In the absence of matter, the DHOST constraint Ψ is a primary, second-class constraint, and it Poisson-commutes with all the other primary constraints in the gravity sector. It therefore leads to a secondary constraint, which together with Ψ is responsible for removing the would-be ghost degree of freedom. If now the matter sector itself has some constraints, there is the risk that they may not commute with Ψ, implying the loss of the associated secondary constraint and the reappearance of the unwanted degree of freedom.

## 2. ADM Decomposition of DHOST and GP Theories

#### 2.1. DHOST Lagrangian

#### 2.2. GP Lagrangian

#### 2.2.1. ${\mathcal{L}}_{2}$ Term

#### 2.2.2. ${\mathcal{L}}_{3}$ Term

#### 2.2.3. ${\mathcal{L}}_{4}$ Term

#### 2.2.4. ${\mathcal{L}}_{5}$ Term

## 3. Constraint Analysis

#### 3.1. ${\mathcal{L}}_{3}$ Term

#### 3.2. ${\mathcal{L}}_{4}$ Term

#### 3.3. ${\mathcal{L}}_{5}$ Term

## 4. Discussion

## Funding

## Conflicts of Interest

## Appendix A. Full Results of the ADM Decomposition

## Appendix B. Inverse of DHOST Kinetic Tensor

## Notes

1 | |

2 | It is important to remark that the vanishing of the Hessian determinant implies the existence of a primary constraint, which is not by itself enough to remove a full Lagrangian degree of freedom. Nevertheless, the existence of an associated secondary constraint is guaranteed by the general covariance of the action, as shown explicitly in [15]. The same consideration holds for the situation wherein the DHOST constraint is modified in the presence of matter (as exemplified in Equation (43)), provided of course the latter admits consistent coupling according to the criteria explained in the introduction. |

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Garcia-Saenz, S.
On the Coupling of Generalized Proca Fields to Degenerate Scalar-Tensor Theories. *Universe* **2021**, *7*, 190.
https://doi.org/10.3390/universe7060190

**AMA Style**

Garcia-Saenz S.
On the Coupling of Generalized Proca Fields to Degenerate Scalar-Tensor Theories. *Universe*. 2021; 7(6):190.
https://doi.org/10.3390/universe7060190

**Chicago/Turabian Style**

Garcia-Saenz, Sebastian.
2021. "On the Coupling of Generalized Proca Fields to Degenerate Scalar-Tensor Theories" *Universe* 7, no. 6: 190.
https://doi.org/10.3390/universe7060190