content_image/409802.jpg content_image/409801.jpg content_image/409800.jpg content_image/409798.jpg content_image/409797.jpg content_image/409793.jpg content_image/409792.jpg content_image/409803.jpg content_image/409804.jpg content_image/409799.jpg content_image/409796.jpg content_image/409794.jpg
have 

<img src='content_image/409802.jpg'>

For a quadratic correction term, we have 

<img src='content_image/409801.jpg'>

From these equations, a recursive equation for n ≥ 2 can be chosen as, 

<img src='content_image/409800.jpg'>

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 

<img src='content_image/409798.jpg'>

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. 

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. 

## III. THE ABELIAN PROCA MODEL 

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 

<img src='content_image/409797.jpg'>

<img src='content_image/409793.jpg'>

<img src='content_image/409792.jpg'>

where the symplectic potential is 

<img src='content_image/409803.jpg'>

with 

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 

The zeroth-iteration symplectic matrix is 

which is a singular matrix. It has a zero-mode that gen- erates the following constraint, 

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 

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, 

<img src='content_image/409804.jpg'>

<img src='content_image/409799.jpg'>

<img src='content_image/409796.jpg'>

<img src='content_image/409794.jpg'>

