  We argue the sign problem of the fermion determinant at finite density. It is
unavoidable not only in Monte-Carlo simulations on the lattice but in the
mean-field approximation as well. A simple model deriving from Quantum
Chromodynamics (QCD) in the double limit of large quark mass and large quark
chemical potential exemplifies how the sign problem arises in the Polyakov loop
dynamics at finite temperature and density. In the color SU(2) case our
mean-field estimate is in excellent agreement with the lattice simulation. We
combine the mean-field approximation with a simple phase reweighting technique
to circumvent the complex action encountered in the color SU(3) case. We also
investigate the mean-field free energy, from the saddle-point of which we can
estimate the expectation value of the Polyakov loop.
