17 Gauge invariance and dual equivalence of Abelian and non-Abelian actions via dual embedding formalism agreement with the symplectic method, this constraint is introduced into the kinetic sector of the first-order La- grangian through a Lagrange multiplier, namely, and the symplectic matrix is Since this matrix is singular, it has a zero-mode that generates the constraint Ω$_{a}$, given by equation (6.5). In The new group of symplectic variables is ξ a α = ( A a i , π a i , A a $_{0}$, η $^{a}$), and the new symplectic matrix is L (1) = π i a ˙ A a i + Ω$_{a}$ ˙ η a − 1 2 ( π i $_{a}$) 2 + A a $_{0}$Ω$_{a}$ − 1 2 m $^{2}$A a i A i a − 1 2 m $^{2}$A a $_{0}$A 0 a − 1 4 F a $_{kj}$F a $_{kj}$. (6.7) ξ a α = ( A a i , π a i , A a $_{0}$) , f (1) =    0 − δ$_{ji}$δ ba 0 − gC ab d π d $_{i}$( y ) δ$_{ij}$δ ab 0 0 δ $^{ab}$∂ y i − gC ab d A d $_{i}$( y ) 0 0 0 m $^{2}$δ ab gC ba d π d $_{j}$( x ) − δ $^{ba}$∂x j + gC ba d A d $_{j}$( x ) − m $^{2}$δ ba 0   $^{δ}$$^{(3)}$$^{(}$$^{⃗}$ x − ⃗ y ) . (6.8) Ψ$_{a}$ ≡ Ψ$_{a}$( A a i , π a i , A a $_{0}$, θ $^{a}$) , G ≡ G ( A a i , π a i , A a $_{0}$, θ $^{a}$) = ∞ ∑ n =0 G $^{n}$( A a i , π a i , A a $_{0}$, θ $^{a}$) , (6.11) { A a $_{i}$( x ) , A b $_{j}$( y ) } = 0 , { A a $_{i}$( x ) , π b $_{j}$( y ) } = δ $^{ab}$δ$_{ij}$ δ $^{(3)}$( ⃗ x − ⃗ y ) , { π a i ( x ) , π b $_{j}$( y ) } = 0 , (6.9) { A a $_{i}$( x ) , A b $_{0}$( y ) } = − 1 m $_{2}$δ $^{ab}$∂x i δ $^{(3)}$( ⃗ x − ⃗ y ) − g m $_{2}$C ab e A e $_{i}$( x ) δ ( ⃗ x − ⃗ y ) , { A a $_{0}$( x ) , A b $_{0}$( y ) } = − 2 gm $^{2}$C ab e A e $_{0}$( x ) δ $^{(3)}$( ⃗ x − ⃗ y ) , { π a i ( x ) , A b $_{0}$( y ) } = − g m $_{2}$C ab e π e $_{i}$( x ) δ $^{(3)}$( ⃗ x − ⃗ y ) . G ≡ ( A a i , π a i , A a $_{0}$, θ a = 0) = G $^{0}$( A a i , π a i , A a $_{0}$, θ a = 0) = 0 . (6.12) f (0) =    0 − δ$_{ji}$δ ba 0 ∂ Ψ$_{b}$( y ) ∂A a $_{i}$( x ) δ$_{ij}$δ ab 0 0 Ψ$_{b}$( y ) ∂π a i ( x ) 0 0 0 ∂ Ψ$_{b}$( y ) ∂A a $_{0}$( x ) − ∂ Ψ$_{a}$( x ) ∂A b $_{j}$( y ) − ∂ Ψ$_{a}$( x ) ∂π b $_{j}$( y ) − ∂ Ψ$_{a}$( x ) ∂A b $_{0}$( y ) 0   $^{δ}$$^{(3)}$$^{(}$$^{⃗}$ x − ⃗ y ) . (6.13) ∫ w ¯ ν $^{(0)}$( ⃗ x ) f$_{αβ}$ ( ⃗ x − ⃗ w ) = 0 . (6.15) ˜ L (0) = π i a ˙ A a i + Ψ$_{a}$ ˙ θ a − 1 2 ( π i $_{a}$) 2 + A a $_{0}$Ω$_{a}$ − 1 2 m $^{2}$A a i A i a − 1 2 m $^{2}$A a $_{0}$A 0 a − 1 4 F a $_{kj}$F a kj − G, (6.10) In order to determine the Ψ$_{a}$ functions, we analyze the symmetry related to the following zero-mode, This condition produces a set of differential equations which allows us to compute the Ψ$_{a}$ function as where the arbitrary functions are The model now can be reformulated as a gauge in- variant field theory. This will be carried out in the context of the symplectic gauge-invariant formulation. In agreement with this formalism, the first-order La- grangian (6.4) can be rewritten as This completes the analysis of the noninvariant descrip- tion of the model. In this context, the corresponding symplectic matrix is and where the G function obeys a boundary condition given by, This matrix is nonsingular and its inverse leads to the commutation relations among the dynamical variables, given by