6 Gauge invariance and dual equivalence of Abelian and non-Abelian actions via dual embedding formalism with with the following Hamiltonian, 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, where the phase space metric is 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, 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, Hence, the symplectic matrix becomes which is singular by construction. Due to this, the first- order Lagrangian is with 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 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, with ˜ V (0) given in Eq. (3.15). ν (0) α ∂ ˜ V (0) ∂ ˜ ξ$_{α}$ = 0 , (3.23) ˜ L (0) = π i ˙ A$_{i}$ − ∂$_{i}$π i ˙ θ − ˜ V $^{(0)}$, (3.22) f (0) =    0 − δ$_{ij}$ 0 0 δ$_{ji}$ 0 0 − ∂ y i 0 0 0 0 0 ∂ x j 0 0    δ $^{(3)}$( ⃗ x − ⃗ y ) , (3.21) ˜ L (0) = π i ˙ A$_{i}$ + ˙ θ Ψ − ˜ V $^{(0)}$, (3.14) ˜ V (0) = 1 2 π$_{i}$ $^{2}$+ 1 4 F 2 $_{ij}$+ 1 2 m 2 A$_{i}$ $^{2}$− A$_{0}$ ( ∂$_{i}$π $^{i}$+ 1 2 m 2 A$_{0}$ )+ G, (3.15) f (0) =    0 − δ$_{ij}$ 0 ∂ Ψ$_{y}$ ∂A x i δ$_{ji}$ 0 0 ∂ Ψ$_{y}$ ∂π x i 0 0 0 ∂ Ψ$_{y}$ ∂A x 0 − ∂ Ψ$_{x}$ ∂A y j − ∂ Ψ$_{x}$ ∂π y j − ∂ Ψ$_{x}$ ∂A y 0 f$_{θ}$$_{x}$$_{θ}$$_{y}$   $^{δ}$$^{(3)}$$^{(}$$^{⃗}$ x − ⃗ y ) , (3.16) Ψ = − ∂$_{i}$π $^{i}$. (3.20) ∫ d $^{3}$y ν(0) α ( ⃗ x ) f$_{αβ}$ ( ⃗ x − ⃗ y ) = 0 , (3.18) f$_{θ}$$_{x}$$_{θ}$$_{y}$ = ∂ Ψ$_{y}$ ∂θ$_{x}$ − ∂ Ψ$_{x}$ ∂θ$_{y}$ , (3.17) M i j = g i j − ∂ $^{i}$∂$_{j}$ m 2 , (3.13) H = V $^{(1)}$|$_{Ω=0}$ = 1 2 π 2 i − 1 2 m $_{2}$π$_{i}$ ∂ $^{i}$∂$_{j}$ π j + 1 4 F 2 ij + 1 2 m $^{2}$A2 i , = 1 2 π$_{i}$M i $_{j}$π j + 1 4 F 2 ij + 1 2 m $^{2}$A2 i , (3.12) { π$_{i}$ ( ⃗ x ) , π$_{j}$ ( ⃗ y ) } ∗ = 0 , (3.11) { A$_{i}$ ( ⃗ x ) , A$_{j}$ ( ⃗ y ) } ∗ = 0 , { A$_{i}$ ( ⃗ x ) , π$_{j}$ ( ⃗ y ) } ∗ = δ$_{ij}$δ $^{(3)}$( ⃗ x − ⃗ y ) , (3.10)