content_image/430364.jpg content_image/430360.jpg content_image/430359.jpg content_image/430355.jpg content_image/430363.jpg content_image/430357.jpg content_image/430356.jpg content_image/430354.jpg content_image/430352.jpg content_image/430362.jpg content_image/430361.jpg content_image/430358.jpg content_image/430353.jpg
on the contrary, it produces a general equation that al- lows us to compute the correction terms in θ enclosed into G ( A$_{i}$, π$_{i}$, A$_{0}$, θ ), given in Eq. (2.21). To compute 

the correction term linear in θ , namely, G $^{(1)}$, we pick up the following terms from the general relation (2.21) given by, 

<img src='content_image/430364.jpg'>

After a straightforward calculation, the correction term linear in θ is, 

Substituting this result into the symplectic potential (3.15), we have that 

<img src='content_image/430360.jpg'>

However, the invariant formulation of the Proca model was not obtained yet because the contraction of the zero- mode (3.19) with the symplectic potential above does not generate a null value. Due to this, higher order correction terms in θ must be computed. For the quadratic term, we have, 

<img src='content_image/430359.jpg'>

and after a direct calculation, we can write that, 

Then, the first-order Lagrangian becomes, 

where the symplectic potential is 

<img src='content_image/430355.jpg'>

Since the contraction of the zero-mode (¯ ν $^{(0)}$) with the symplectic potential above does not produce a new con- straint, a hidden symmetry is revealed. 

To complete the gauge invariant reformulation of the Abelian Proca model, the infinitesimal gauge transfor- mation will be computed also. In agreement with the symplectic formalism, the zero-mode ¯ ν (0) is the generator of the infinitesimal gauge transformation ( δ O = ε ¯ ν $^{(0)}$), given by, 

<img src='content_image/430363.jpg'>

<img src='content_image/430357.jpg'>

<img src='content_image/430356.jpg'>

where ε is an infinitesimal time-dependent parameter. In- deed, for the above transformations the invariant Hamil- tonian, identified as being the symplectic potential ˜ V $^{(0)}$, changes as 

Now we will investigate the result from the Dirac point of view. The chains of primary constraints computed from the Lagrangian (3.29) are 

Next, these constraints will be introduced into the invari- ant Hamiltonian (3.30) through the Lagrange multipliers and so it can be rewritten as 

The time stability condition for the primary constraint φ$_{1}$ requires a secondary constraint, such as 

and no more constraint appears from the time evolution of φ$_{2}$ . Now, the total Hamiltonian is written as 

Since the time evolution of φ$_{1}$ just allows us to obtain the Lagrange multiplier λ$_{2}$ , and the constraint χ$_{1}$ has no time evolution ˙ χ$_{1}$ = 0, no more constraints arise. Hence, the gauge invariant model has three constraints ( φ$_{1}$, φ$_{2}$, χ$_{1}$ ). The nonvanishing Poisson brackets among these constraints are, 

<img src='content_image/430354.jpg'>

<img src='content_image/430352.jpg'>

<img src='content_image/430362.jpg'>

<img src='content_image/430361.jpg'>

<img src='content_image/430358.jpg'>

<img src='content_image/430353.jpg'>

