A quantum field theory of the structure of phonons in amorphous solids and liquids

Atom and phonon Green functions are determined by the Ward-Takahashi relations associated with the spatially translational and rotational invariance. In the random phase approximation, the energy spectrum of atoms is constructed by introducing the hopping to atoms in their own harmonic potential. Phonon dispersion curves and their width are completely described by two parameters: the eigenfrequency of the harmonic potential of an atom and the magnitude of the hopping energy of atoms, and the pair distribution function. The phonon life time appears when the phonon frequency merges into the continuum of the particle-hole excitation; the particle-hole excitation appears above the band gap of atoms