Diffusion Constant of a φ4-Kink: Quantum Corrections

A kink in a system of a φ 4 -chain undergoes a Brownian motion due to the collisions with thermally excited phonons. Deviations from the classical kink motion are associated with the Brownian motion. They are described by the quantum mechanics. Heisenberg equations of motion are solved, using an iteration method, to see how the kink behaves. With the help of the fluctuation-dissipation theorem, the diffusion constant is evaluated at low temperatures where the lowest order processes are important. The quantum effect becomes appreciable, when the temperature is lowered than the typical phonon energy. It reduces the diffusion constant drastically, since the number of thermally excited phonons is reduced. A diagram method is proposed to study the higher order processes which would become effective at higher temperatures.