The free energy for the crystalline state and the liquid state is investigated at low temperatures and near the melting temperature from a quantum field theoretical point of view, starting with atoms interacting with each other. The order parameter is defined by the energy gaps on the Brillouin zone boundary. The gap equation is derived from the existence of acoustic phonons in the Bethe-Salpeter equation of the density-density correlation function. The free energy is represented as a function of the gaps. The melting temperature is related to the temperatures where the gaps of the crystalline state and the liquid state become zero in their gap equations, respectively. At low temperatures, the free energy reduces to that of the conventional lattice theory.
