content_image/384383.jpg content_image/384389.jpg content_image/384391.jpg content_image/384386.jpg content_image/384388.jpg content_image/384384.jpg content_image/384385.jpg content_image/384387.jpg content_image/384390.jpg
<img src='content_image/384383.jpg'>

This completes our proposal. 

At this stage, we would like to disclose the hidden sym- metry present inside the model using the Dirac point of view. To this end, we start with the set of primary con- straints, 

For the first set of constraints, the time stability con- dition is satisfied ( ˙ Ω a 1 = 0). For the second one, the following secondary constraints are required, 

Due to this, the total Hamiltonian is 

where λ 1 $_{a}$, ζ 1 a and ζ 2 a are Lagrange multipliers. Since the Poisson brackets among those constraints are 

no more constraints arise. Notice that some brack- ets above are null, indicating that there are both first and second-class constraints. Indeed, the first-class con- straint is Ω a 1 and the second-class are χ a 1 and χ a $_{2}$. In agreement with Dirac’s procedure, the second-class con- straints can be taken equal to zero in a strong way, this allows us to compute the primary Dirac brackets. Due to the Maskawa-Nakajima theorem [35], the primary Dirac brackets among the phase space fields are canonical. To demonstrate this, the brackets are computed explicitly. The Dirac matrix is 

<img src='content_image/384389.jpg'>

<img src='content_image/384391.jpg'>

<img src='content_image/384386.jpg'>

<img src='content_image/384388.jpg'>

with 

The inverse of the Dirac matrix is 

In accordance with the Dirac process, the Dirac brackets among the phase space fields are obtained as 

Finally, the infinitesimal gauge transformations are ob- tained, namely, 

<img src='content_image/384384.jpg'>

<img src='content_image/384385.jpg'>

<img src='content_image/384387.jpg'>

<img src='content_image/384390.jpg'>

