Damping of Phonons at Low Temperature

Explore the decay mechanisms of quasiparticles in interacting Bose gas systems through this 36-minute mathematical physics lecture. Delve into the theoretical framework governing phonon behavior at low temperatures and momenta, examining how these quasiparticles predicted by Bogoliubov approximation become unstable due to third and fourth-order Hamiltonian terms. Learn about the two primary damping mechanisms: Beliaev damping, which occurs at zero temperature through phonon decomposition into two phonons, and Landau damping, which emerges only at positive temperatures involving the decay of left phonons into left and right phonon pairs. Discover how thermal state descriptions naturally incorporate the concept of "left phonons" and "right phonons" in understanding these decay processes. Follow the mathematical derivation based on the Bose gas Hamiltonian with c-number condensate, comparing two distinct analytical approaches: the W*-algebraic description inspired by Jaksic-Pillet methodology and the 2-body correlation function approach, both yielding identical formulas. Gain insights into how modern mathematical physics builds upon and elaborates the foundational work from the 1950s and 1960s by Beliaev and contemporaries, with results developed in collaboration with Lorenzo Pettinari that advance our understanding of quantum many-body systems and Bose-Einstein condensation phenomena.