5 Gauge invariance and dual equivalence of Abelian and non-Abelian actions via dual embedding formalism where the symplectic potential is have For a quadratic correction term, we have where ∂$_{i}$ = ∂ ∂x i and the dot denote space and time derivatives, respectively. The symplectic coordinates are ξ (0) α = ( A$_{i}$, π$_{i}$, A$_{0}$ ) with the corresponding one-form canonical momenta given by From these equations, a recursive equation for n ≥ 2 can be chosen as, which permits us to compute the remaining correction terms as functions of η . This iterative process is suc- cessively repeated until (2.21) becomes identically zero. Consequently, the extra term G ( a$_{i}$, p$_{i}$, η ) is obtained ex- plicitly. Then, the gauge invariant Hamiltonian, identi- fied as being the symplectic potential, is obtained from The zeroth-iteration symplectic matrix is and the zero-mode ˜ ν (0) ˜ θ is identified as the generator of an infinitesimal gauge transformation, given by where ε is an infinitesimal parameter. which is a singular matrix. It has a zero-mode that gen- erates the following constraint, In the following sections, we will apply the symplec- tic gauge-invariant formalism in some second-class con- strained Hamiltonian systems, Abelian and non-Abelian. We will see that the new results of the following sections demonstrate exactly that the dual (symplectic) embed- ding method can be seen as a mapping between the orig- inal action the final one. identified as being the Gauss law. We will include this constraint into the canonical part of the first-order La- grangian L (0) in (3.2) introducing a Lagrangian multi- plier ( β ). The first-iterated Lagrangian can be written in terms of ξ (1) α = ( A$_{i}$, π$_{i}$, A$_{0}$, β ) as with the following symplectic potential, The first-iterated symplectic matrix, computed as The analysis of the following model clarify the physics and gives a deeper insight into the general formalism de- scribed in the last section. To this end, let us start with a simple Abelian case which is the Proca model whose dynamics is ruled by the Lagrangian density, where m is the mass of the A$_{µ}$ field, g$_{µν}$ = diag (+ − −− ) and F$_{µν}$ = ∂$_{µ}$A$_{ν}$ − ∂$_{ν}$A$_{µ}$ . Observe that, as well known, the mass term breaks the gauge invariance of the usual Maxwell’s theory. Hence, the Lagrangian density above represents a second-class system. To perform the dual embedding formalism the La- grangian density is reduced to its first-order form as is a nonsingular matrix. Consequently, the Proca model is not a gauge invariant field theory. As settle by the method, the Dirac brackets among the phase space fields are obtained from the inverse of the symplectic matrix, namely, L (0) = π i ˙ A$_{i}$ − V $^{(0)}$, (3.2) L = − 1 4 F$_{µν}$ F µν + 1 2 m 2 A $^{µ}$A$_{µ}$ , (3.1) L (1) = π i ˙ A$_{i}$ + Ω ˙ β − V $^{(1)}$, (3.7) III. THE ABELIAN PROCA MODEL Ω = ∂$_{i}$π i + m $^{2}$A$_{0}$ , (3.6) δ ˜ ξ θ = ε ˜ ν (0) ˜ $^{θ}$, (2.27) ˜ H ( a$_{i}$, p$_{i}$, η ) = V $^{(0)}$( a$_{i}$, p$_{i}$ ) + G ( a$_{i}$, p$_{i}$, η ) , (2.26) a (0) A$_{i}$ = π$_{i}$, a (0) π i = a (0) A$_{0}$ = 0 . (3.4) µ θ ∂ G ( n − $^{1)}$( a$_{i}$, p$_{i}$, η ) ∂ξ (0) θ + ∂ G ( n $^{)}$( a$_{i}$, p$_{i}$, η ) ∂η = 0 , (2.25) µ θ ∂ G $^{(1)}$( a$_{i}$, p$_{i}$, η ) ∂ξ (0) θ + ∂ G $^{(2)}$( a$_{i}$, p$_{i}$, η ) ∂η = 0 . (2.24) µ θ ∂V $^{(0)}$( a$_{i}$, p$_{i}$ ) ∂ξ (0) θ + ∂ G $^{(1)}$( a$_{i}$, p$_{i}$, η ) ∂η = 0 . (2.23) 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$_{0}$ ) , (3.3) π$_{i}$ = ˙ A$_{i}$ − ∂$_{i}$A$_{0}$ , with