I will discuss the decay rate of quasiparticles in interacting Bose gas at low temperatures and low momenta.
These quasiparticles, usually called phonons, are predicted by the Bogoliubov approximation. Due to 3rd and 4rth order terms in the Hamiltonian, they are unstable. I will consider the lowest nontrivial contribution, given by the Fermi Golde Rule, which is the sum of two terms. The so-called Beliaev damping is due to a decomposition of a phonon into two phonons, and is present at zero temperature. The so-called Landau damping is present only in positive temperatures. In the description of thermal states it is natural to introduce "left phonons" and "right phonons". The Landau damping involves a decay of a left phonon into one left and one right phonon.
The derivation will be based on the Hamiltonian of the Bose gas with a c-number condensate. I will describe two approaches to phonons: one inspired by the W^*-algebraic description in the spirit of Jaksic-Pillet, the other based on 2-body correlation functions. Both methods yield the same formula.
My talk will be based on the esults obtained with Lorenzo Pettinari. These results can be viewed as an elaboration of classic results from the 50's and 60's by Beliaev and others.