content_image/439384.jpg content_image/439377.jpg content_image/439385.jpg content_image/439379.jpg content_image/439386.jpg content_image/439381.jpg content_image/439383.jpg content_image/439382.jpg content_image/439380.jpg content_image/439378.jpg content_image/439387.jpg content_image/439388.jpg content_image/439376.jpg
Consequently, the first-order Lagrangian is rewritten as 

where the symplectic potential is 

<img src='content_image/439384.jpg'>

It completes the first step of the symplectic gauge- invariant formulation. 

To unveil the hidden symmetry inside the model, the zero-mode ¯ ν (0) does not generate new constraints, conse- quently, we have the following relation, 

From this relation we can compute the whole set of cor- rection terms as functions of θ $^{a}$. The linear correction term in θ is computed as being, 

After an integration we have that 

<img src='content_image/439377.jpg'>

Now, we will compute the quadratic term, namely, 

<img src='content_image/439385.jpg'>

<img src='content_image/439379.jpg'>

<img src='content_image/439386.jpg'>

<img src='content_image/439381.jpg'>

Integrating this relation in θ $^{f}$( w ), the quadratic correc- tion term is obtained as 

<img src='content_image/439383.jpg'>

where 

<img src='content_image/439382.jpg'>

In this way, two correction terms as functions of θ$_{a}$ ( G $^{(3)}$( x ) and G $^{(4)}$( x )) remain to be computed. Let us compute the first one. It can be done from the following relation, 

<img src='content_image/439380.jpg'>

and we can write that, 

<img src='content_image/439378.jpg'>

Finally, the last correction term is, 

<img src='content_image/439387.jpg'>

<img src='content_image/439388.jpg'>

Therefore, the gauge invariant first-order Lagrangian is 

where the gauge invariant Hamiltonian, identified as be- ing the symplectic potential, is given by 

<img src='content_image/439376.jpg'>

