A quantum field theory of localized and resonant modes in 3D nonlinear lattices at finite temperatures I

The effective interaction Hamiltonian in 3D nonlinear lattices is established taking into account the repetitions of the up and down transition of an atom between two levels at the same site. The effective interaction Hamiltonian leads to the Heisenberg equation for phonon operators, which yields the conventional dynamical equation for displacements of atoms in 3D nonlinear lattices in the tree approximation by the boson transformation method. Making the one-loop approximation to the nonlinear potential in the Heisenberg equation, we obtain a dynamical equation with a self-consistent potential created by a localized or a resonant mode. In paper II, we show that the dynamical equation yields solutions for localized and resonant modes at finite temperatures