The quantum theory for anomalous low-temperature thermal conductivity of noncrystalline dielectric solids

We discuss the interaction between two-level systems and the phonon field in noncrystalline dielectric solids, and we calculate the quantum transition matrix elements ofsuch an interaction. After studying the Boltzmann equation for phonon transport. we obtain the phonon distribution function. From a direct calculation of the heat current, and as a result of obtaining the thermal conductivity of non-crystalline dielectric solids, the P relation is obtained.