where χ = χ ( θ$_{1}$, · · · θ$_{N}$ ) is an orthogonal basis. The integral element dµ$_{G}$ is the symmetry group invariance Haar measure for a specific symmetry group. The invariance Haar measure is essential in order to accommodate the composite internal symmetry. The color confine- ment is manifested by maintaining the color singlet state. Fortunately, the singlet state is the simplest case and its basis is given by χ$_{singlet}$ = 1. The prime objective is to confine the quarks in a color singlet state regardless of the composite internal group symmetry. In the context of the unitary symmetry group transformation, the physical Hamiltonian is subject to the Hermitian condition The unitary symmetry group subjects to the following constraint where the matrix U is a complex one and the Hermitian matrix is defined by U † = U ∗ $^{T}$. The unitary matrix U has 2 N 2 real parameters where N 2 comes from the real part of the unitary matrix and N 2 comes from the imaginary part. The unitary condition imposes [ N + 2 N ( N − 1) / 2] constraints on 2 N 2 real parameters to give N$_{adj}$ = N 2 essential real parameters. Hence, the unitary symmetry has N$_{fun}$ = N and N$_{adj}$ = N 2 for the fundamental and adjoint charges, respectively. The Gaussian unitary ensemble multiplied by the group volume element is defined in terms of Hermitian matrices as follows which is invariant under automorphism unitary transformation. The invariance Haar mea- sure is given by the volume element where H (0) kj and H (1) kj are real and imaginary parts of H$_{kj}$ . The partition function Z ( H ) is a function of only the eigenvalues θ$_{i}$ where i runs over 1 , · · · , N . This means that the 21 dH = ∏ k ≤ j dH (0) jk ∏ k<j dH (1) kj , (64) Z ( H ) dH (invariant ) , (63) U $^{†}$U = 1 , hence U − 1 = U $^{†}$, (62) H = H $^{†}$. (61) A. Unitary symmetry group U ( N )