content_image/399231.jpg content_image/399250.jpg content_image/399249.jpg content_image/399247.jpg content_image/399236.jpg content_image/399237.jpg content_image/399246.jpg content_image/399248.jpg content_image/399230.jpg
Note that the Dirac brackets among the original phase space fields were obtained before. After this process, the model now have only one first-class constraint, 

identified as the Gauss law, that satisfies the Poisson al- gebra, 

In this way, the anomaly was eliminated, the symmetry is preserved, and the fundamental brackets among the original phase space fields were reobtained. Note that the Gauss law is the generator of the gauge symmetry given in (5.30). 

Once more, the number of the independent degrees of freedom matches with the result obtained in the second- class case. The invariant model has a phase space 

<img src='content_image/399231.jpg'>

<img src='content_image/399250.jpg'>

<img src='content_image/399249.jpg'>

( φ, π$_{φ}$, A$_{1}$, π$_{1}$, θ, π$_{θ}$ ), totaling six dependent fields, and has a first-class constraint which eliminates two fields. Consequently, the model has two independent degrees of freedom. 

At this point, we are interested in commenting about the consistency of the gauge invariant version of the bosonized CSM. To do this, the remaining symmetry will be eliminated with the introduction of the unitary gauge- fixing term, given by, 

Hence, both the noninvariant Hamiltonian and the cor- responding Dirac brackets computed in the beginning of this section are reobtained, then recovering the anomaly. In this way, we conclude that the new symplectic gauge- invariant formalism does not change the physics contents present inside the model. 

## VI. DUALITY FOR THE NON-ABELIAN EXTENSION OF THE PROCA MODEL 

The non-Abelian extension of the Proca model has its dynamics governed by the following Lagrangian density, 

with 

where the antisymmetric tensor C a bc ( C a bc = − C a $_{cb}$), are in fact a set of real constants, known as the structure constants of the gauge group, and satisfy the following property, 

Since we are interested in analyzing the non-Abelian Proca model from the symplectic point of view, the La- grangian will be reduced to its first-order form as follows, 

<img src='content_image/399247.jpg'>

where 

The symplectic variables are given by 

<img src='content_image/399236.jpg'>

<img src='content_image/399237.jpg'>

<img src='content_image/399246.jpg'>

<img src='content_image/399248.jpg'>

<img src='content_image/399230.jpg'>

