continuation of the group eigenvalues iθ$_{i}$ → − βµ$_{i}$ . The group invariance is transformed into system’s fugacities Λ$_{i}$ = e iθ$_{i}$ → Λ$_{i}$ = e − $^{βµ$_{i}$}$. The internal structure of the composite is incor- porated by writing the canonical ensemble as a function of the group energy eigen-variables, namely, Z ( θ$_{1}$, · · · , θ$_{N}$ ). The easiest way to define the canonical ensemble of the composite in the Hilbert space is to write it as a tensor product of the Fock space canonical ensembles of the constituent quarks and gluons. The canonical ensemble for the quark-gluon bag is written as follows The canonical ensemble for quark gas in the Fock space reads while the canonical ensemble for gluon gas becomes The internal structure is incorporated by the group eigenvalues of the conserved charges. The fundamental and adjoint color charges are specified, respectively, by and where U and U$_{adj}$ are appropriate transformation matrices and they are reduced to the identity matrices (i.e. U = 1 and U$_{adj}$ = 1) for the unitary group representation. Finally, the canonical ensemble with a specific internal structure is projected using the standard orthogonal expansion procedure. The specific symmetry state is projected by the integration of a specific projection state over the symmetry group invariance Haar measure in the following way, 20 Z$_{χ}$ = ∫ dµ$_{G}$ ( θ$_{1}$, · · · θ$_{N}$ ) [ χ × Z ( θ$_{1}$, · · · θ$_{N}$ )] , (60) R$_{adj}$ ( g ) = U$_{adj}$ − 1 ( exp [ i diag( φ$_{1}$, · · · φ$_{N}$$_{adj}$ ) ]) U$_{adj}$, (59) R$_{fun}$ ( g ) = U − 1 (exp [ i diag( θ$_{1}$, · · · θ$_{N}$ )]) U, (58) tr [ R$_{g}$ ( g ) exp ( − βH$_{g}$ )] = exp [ − ∑ α tr ln ( 1 − R$_{adj}$ ( g ) e − βH$_{g}$$_{α}$ ) ] . (57) tr [ R$_{q}$ ( g ) exp ( − βH$_{q}$ )] = exp [ ∑ α tr ln ( 1 + R$_{fun}$ ( g ) e − βH$_{q}$$_{α}$ ) ] , (56) Z$_{qqg}$ ( R ( g )) = tr [ R$_{qqg}$ ( g ) exp ( − βH$_{qqg}$ )] , = tr [ R$_{q}$ ( g ) exp ( − βH$_{q}$ )] tr [ R$_{q}$ ( g ) exp ( − βH$_{q}$ )] tr [ R$_{g}$ ( g ) exp ( − βH$_{g}$ )] . (55)