  We study many-particle phenomena of propagating multi-mode photons and
phonons interacting through Brillouin scattering-type Hamiltonian in nanoscale
waveguides. We derive photon and phonon retarded Green's functions and extract
their spectral functions in applying the factorization approximation of the
mean-field theory. The real part of the self-energy provides renormalization
energy shifts for the photons and the phonons. Besides the conventional leaks,
the imaginary part gives effective photon and phonon damping rates induced due
to many-particle phenomena. The results extend the simple spectral functions of
quantum optomechanics into continuum quantum optomechanics. We present the
influence of thermal phonons on the photon effective damping rates, and
consider cases of specific photon fields to be excited within the waveguide and
which are of importance for phonon cooling scenarios.
