content_image/329303.jpg content_image/329302.jpg content_image/329301.jpg content_image/329300.jpg content_image/329299.jpg
The canonical ensemble with specific internal symmetry group and that is projected to a certain quantum wavefuntion is given by 

where Z$_{0}$ ( β, V ) = exp ( a (0) q ¯ qg ) . The symmetry group structure is defined by G = 1, 2, 4 for the orthogonal, unitary and symplectic ensemble, respectively. The color-singlet-state quantum wave-function is given by χ$_{singlet}$ = 1 regardless of the symmetry group. In general the physical state in QCD is assumed to satisfy the condition det( G ) = 1 or equivalently it is restricted to the unimodular-like constraint, namely, ∑ N$_{c}$ i =1 θ$_{i}$ = 0. The unimodular-like constraint modifies the extrapolated color-singlet-state canonical Gaussian ensemble with a specific group such as unitary, orthogonal and symplectic symmetry groups to the following quantity 

<img src='content_image/329303.jpg'>

<img src='content_image/329302.jpg'>

where N = N$_{U}$ , N$_{O}$ and N$_{Sp}$ for the unitary, orthogonal and symplectic symmetry groups, respectively. Eq.(84) is derived by imposing the unimodular-like constraint using the fol- lowing trick 

The integral part in Eq.(84) is evaluated as follows 

<img src='content_image/329301.jpg'>

<img src='content_image/329300.jpg'>

The exponent α is determined by 

<img src='content_image/329299.jpg'>

