content_image/392798.jpg content_image/392792.jpg content_image/392793.jpg content_image/392800.jpg content_image/392799.jpg content_image/392797.jpg content_image/392791.jpg content_image/392790.jpg content_image/392795.jpg content_image/392796.jpg content_image/392794.jpg content_image/392789.jpg content_image/392788.jpg
with the following Hamiltonian, 

<img src='content_image/392798.jpg'>

where the phase space metric is 

which completes the noninvariant analysis. 

At this point we are ready to carry out the symplectic gauge-invariant formulation of the Abelian Proca model in order to disclose the gauge symmetry present in the model. To this end, we will extend the symplectic gauge- invariant formalism [28], proposed by three of us in order to unveil, at that time, the gauge symmetry present on the Skyrme model. The basic concept behind the ex- tended symplectic gauge-invariant formalism lives on the extension of the original phase space with the introduc- tion of two arbitrary functions, Ψ and G , where both rely on both the original phase space variables and the WZ variable ( θ ). The former (Ψ) is introduced into the kinetic sector and, the later (G), within the potential sec- tor of the first-order Lagrangian. The process starts with the computation of Ψ and finishes with the calculation of G . 

In order to reformulate the Proca model as a gauge invariant field theory, we will start with the first-order Lagrangian L $^{(0)}$, given in Eq. (3.2), with the arbitrary terms, given by, 

with 

<img src='content_image/392792.jpg'>

where Ψ ≡ Ψ( A$_{i}$, π$_{i}$, A$_{0}$, θ ) and G ≡ G ( A$_{i}$, π$_{i}$, A$_{0}$, θ ) are the arbitrary functions to be determined. Now, the sym- plectic fields are ˜ ξ (0) α = ( A$_{i}$, π$_{i}$, A$_{0}$, θ ) while the symplec- tic matrix is 

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with 

where θ$_{x}$ ≡ θ ( x ), θ$_{y}$ ≡ θ ( y ), Ψ$_{x}$ ≡ Ψ( x ) and Ψ$_{y}$ ≡ Ψ( y ). 

In order to unveil the hidden U (1) gauge symmetry in- side the Proca model, the symplectic matrix above must be singular, then, Ψ ≡ ( A$_{i}$, π$_{i}$, θ ). As established by the symplectic gauge-invariant formalism, the corresponding zero-mode ν $^{(0)}$( ⃗ x ), identified as being the generator of the symmetry, satisfies the following relation, 

producing a set of equations that allows to determine Ψ explicitly. At this point, it is very important to notice that the extended symplectic gauge-invariant formalism opens up the possibility to disclose the gauge symmetry of the physical model. The zero-mode does not generate a new constraint, however, it determines the arbitrary function Ψ and consequently, obtain the gauge invariant reformulation of the model. We will scrutinize the gauge symmetry related to the following zero-mode, 

Hence, the symplectic matrix becomes 

<img src='content_image/392790.jpg'>

which is singular by construction. Due to this, the first- order Lagrangian is 

with ˜ V (0) given in Eq. (3.15). 

Now, we start with the second step of the formalism to transform the model into a gauge theory. The zero-mode ¯ ν (0) does not produce a constraint when contracted with the gradient of the symplectic potential, namely, 

<img src='content_image/392795.jpg'>

<img src='content_image/392796.jpg'>

<img src='content_image/392794.jpg'>

<img src='content_image/392789.jpg'>

<img src='content_image/392788.jpg'>

