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## III. INTERNAL SYMMETRY 

The canonical ensemble for the quark-gluon bag with specific internal symmetry is con- structed in the standard way by defining the group’s eigenvalues of the conserved color charges which are commutated with the Hamiltonian. The group’s eigenvalues are defined by an appropriate transformation of the group’s generators to take a diagonal format as follows 

<img src='content_image/328412.jpg'>

where t$_{i}$ are the group’s generators in the fundamental representation. The continuous group’s parameters θ$_{i}$ (where i = 1 , · · · N ) are treated as variables of the conserved color charges which control the composite internal structure. In the same way, it is possible to write the diagonal form of the adjoint representation as follows 

<img src='content_image/328411.jpg'>

The conserved charges in the adjoint representation depend essentially on the fundamental charges and they are written as functions of the fundamental charges: 

<img src='content_image/328410.jpg'>

where the subscript index, namely, a runs over the adjoint index a = 1 , · · · , N$_{adj}$ . For instance with a suitable choice of the unitary representation U ( N ), the conserved adjoint charges φ$_{a}$ are related to the fundamental charges θ$_{i}$ by φ ( ij ) ︸︷︷︸ = ( θ$_{i}$ − θ$_{j}$ ). The adjoint index a is written in term of the double fundamental indexes i j . The group’s (energy-) eigenvalues which control the composite internal structure degrees of freedom correspond the conserved charges. The dual role of the group’s energy-eigenvalues and the conserved charges is the mechanism behind the symmetry breaking/restoration (i.e. symmetry de- composition/reduction from symmetry group to another). It is responsible of the phase transition from a composite with a specific internal structure to another state with dif- ferent symmetry configuration. This mechanism becomes obvious in the example of the phase transition scenario SU ( N ) → U (1) N − $^{1}$. The chemical reaction fugacity and the set of characteristic chemical potentials of the conserved charges are realized by the analytic 

