content_image/425801.jpg content_image/425800.jpg content_image/425802.jpg content_image/425804.jpg content_image/425808.jpg content_image/425803.jpg content_image/425805.jpg content_image/425806.jpg content_image/425807.jpg
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 

<img src='content_image/425801.jpg'>

agreement with the symplectic method, this constraint is introduced into the kinetic sector of the first-order La- grangian through a Lagrange multiplier, namely, 

<img src='content_image/425800.jpg'>

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 

<img src='content_image/425802.jpg'>

This matrix is nonsingular and its inverse leads to the commutation relations among the dynamical variables, given by 

<img src='content_image/425804.jpg'>

This completes the analysis of the noninvariant descrip- tion of the model. 

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 

<img src='content_image/425808.jpg'>

where the arbitrary functions are 

<img src='content_image/425803.jpg'>

and where the G function obeys a boundary condition given by, 

<img src='content_image/425805.jpg'>

In this context, the corresponding symplectic matrix is 

<img src='content_image/425806.jpg'>

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 

<img src='content_image/425807.jpg'>

