content_image/377301.jpg content_image/377303.jpg content_image/377305.jpg content_image/377302.jpg content_image/377306.jpg content_image/377308.jpg content_image/377307.jpg content_image/377309.jpg content_image/377310.jpg
The corresponding symplectic matrix (5.16) is rewritten as 

<img src='content_image/377301.jpg'>

which is obviously singular. Consequently, it has a zero- mode that, by construction, is given by equation (5.19). Now we start the second step of the method reformu- lating the model as a gauge invariant theory. At this stage, the correction terms as functions of θ , embraced by the arbitrary function G , given in Eq. (2.11), will be computed. It is achieved just imposing that no more constraints arise from the contraction of the zero-mode, given in Eq. (5.19), with the gradient of the symplectic 

potential, 

The first-order correction term in θ , G $^{(1)}$, is determined by, 

<img src='content_image/377303.jpg'>

after an integration. Substituting this expression into the equation (5.13), the new Lagrangian is obtained as 

with 

<img src='content_image/377305.jpg'>

<img src='content_image/377302.jpg'>

<img src='content_image/377306.jpg'>

The Lagrangian in (5.24) is not yet gauge invariant be- cause the zero-mode ¯ ν (0) still generates new constraints, given by 

<img src='content_image/377308.jpg'>

indicating that it is necessary to obtain the rest of the cor- rection terms G ( n ) as functions of θ . This is accomplished just imposing that no more constraints are generated by the contraction of the zero-mode with the gradient of ex- tended symplectic potential. It allows us to determine the second order correction term G (2) as, 

<img src='content_image/377307.jpg'>

Hence, the first-order Lagrangian (5.24) becomes 

with the new symplectic potential 

<img src='content_image/377309.jpg'>

<img src='content_image/377310.jpg'>

The contraction of the zero-mode ¯ ν (0) with the new sym- plectic potential above does not produce a new con- 

straint. Consequently, the model has a symmetry and this zero-mode is the generator of the infinitesimal gauge 

