Wilson Fermion Determinant in Lattice QCD

We present a formula for reducing the rank of Wilson fermions from $4 N_c N_x N_y N_z N_t$ to $4 N_c N_x N_y N_z$ keeping the value of its determinant. We analyse eigenvalues of a reduced matrix and coefficients $C_n$ in the fugacity expansion of the fermion determinant $\sum_n C_n (\exp(\mu/T))^n$, which play an important role in the canonical formulation, using lattice QCD configurations on a $4^4$ lattice. Numerically, $\log |C_n|$ varies as $N_x N_y N_z$, and goes easily over the standard numerical range; We give a simple cure for that. The phase of $C_n$ correlates with the distribution of the Polyakov loop in the complex plain. These results lay the groundwork for future finite density calculations in lattice QCD.