  We study the nature of the phase transition of lattice gauge theories at high
temperature and high density by focusing on the probability distribution
function, which represents the probability that a certain density will be
realized in a heat bath. The probability distribution function is obtained by
creating a canonical partition function fixing the number of particles from the
grand partition function. However, if the Z_3 center symmetry, which is
important for understanding the finite temperature phase transition of SU(3)
lattice gauge theory, is maintained on a finite lattice, the probability
distribution function is always zero, except when the number of particles is a
multiple of 3. For U(1) lattice gauge theory, this problem is more serious. The
probability distribution becomes zero when the particle number is nonzero. This
problem is essentially the same as the problem that the expectation value of
the Polyakov loop is always zero when calculating with finite volume. In this
study, we propose a solution to this problem. We also propose a method to avoid
the sign problem, which is an important problem at finite density, using the
center symmetry. In the case of U(1) lattice gauge theory with heavy fermions,
numerical simulations are actually performed, and we demonstrate that the
probability distribution function at a finite density can be calculated by the
method proposed in this study. Furthermore, the application of this method to
QCD is discussed.
