content_image/433345.jpg content_image/433349.jpg content_image/433346.jpg content_image/433344.jpg content_image/433348.jpg content_image/433347.jpg content_image/433343.jpg content_image/433340.jpg content_image/433339.jpg content_image/433338.jpg
The Dirac matrix given by 

is singular, indicating that the model has indeed, a gauge symmetry. However, it also has some nonvanishing Pois- son brackets among the constraints, suggesting that the model has both first and second-class constraints. It is easy to check that χ$_{1}$ is a first class constraint and φ$_{1}$ and φ$_{2}$ are second-class constraints. In accordance with the Dirac method, the set of second-class constraint must be taken equal to zero in a strong way, generating then the primary Dirac brackets among the phase space fields, given by 

The gauge invariant version of the Abelian Proca model is then governed by the following invariant Hamiltonian, 

<img src='content_image/433345.jpg'>

whose phase space metric is 

that has a first class constraint, χ$_{1}$ , which generates the infinitesimal transformations given in (3.31). 

## IV. THE O ( N ) INVARIANT NONLINEAR SIGMA MODEL 

In this subsection, the hidden symmetry present in the O ( N ) nonlinear sigma model will be disclosed enlarging the phase space with the introduction of WZ field via dual gauge-invariant formalism. We first apply the symplectic method to the original second-class model in order to show the second-class nature of the model, and also to obtain the usual Dirac’s brackets among the phase space fields. After that, we unveil the hidden gauge symmetry of the model which dwells on the original phase space. 

The O ( N ) nonlinear sigma model in two dimensions is a free field theory for the multiplet σ$_{a}$ ≡ ( σ$_{1}$, σ$_{2}$, . . . , σ$_{n}$ ) satisfying a nonlinear constraint σ 2 a = 1. This model has its dynamics governed by the Lagrangian density 

<img src='content_image/433349.jpg'>

<img src='content_image/433346.jpg'>

<img src='content_image/433344.jpg'>

where µ = 0 , 1 and “ a ” is an index related to the O ( N ) symmetry group. 

In order to implement the symplectic method, the orig- inal second order Lagrangian in the velocity, given in (4.1), is reduced into a first-order form, given by, 

with 

where prime represent spatial derivatives, respectively. The symplectic coordinates are ξ (0) α = ( σ$_{a}$, π$_{a}$, λ ) and the index (0) indicates the zeroth-iteration. The symplectic tensor given by Eq. (2.3) is computed in this case as 

This matrix is singular, thus, it has a zero-mode, 

Contracting this zero-mode with the gradient of the sym- plectic potential V $^{(0)}$, given in Eq. (4.3), the following constraint is obtained, 

In agreement with the symplectic formalism, this con- straint must be introduced into the canonical sector of the first-order Lagrangian (4.2) through a Lagrange multiplier ρ and then, we obtain the first-iteration La- grangian as 

with 

The symplectic coordinates are ξ (1) α = ( σ$_{a}$, π$_{a}$, ρ ) with the following one-form canonical momenta, 

<img src='content_image/433348.jpg'>

<img src='content_image/433347.jpg'>

<img src='content_image/433343.jpg'>

<img src='content_image/433340.jpg'>

<img src='content_image/433339.jpg'>

<img src='content_image/433338.jpg'>

