  We study thermal transport in a chain of coupled atoms, which can vibrate in
longitudinal as well as transverse directions. The particles interact through
anharmonic potentials upto cubic order. The problem is treated quantum
mechanically. We first calculate the phonon frequencies self-consistently
taking into account the anharmonic interactions. We show that for all the
modes, frequencies must have linear dispersion with wave-vector $q$ for small
$q$ irrespective of their bare dispersions. We then calculate the phonon
relaxation rates $\Gamma_i(q)$, where $i$ is the polarization index of the
mode, in a self-consistent approximation based on second order perturbation
diagrams. We find that the relaxation rate for the longitudinal phonon,
$\Gamma_x(q) \propto q^{3/2}$, while that for the transverse phonon
$\Gamma_y(q) \propto q^2$. The consequence of these results on the thermal
conductivity $\kappa(N)$ of a chain of $N$ particles is that $\kappa(N) \propto
N^{1/2}$.
