  We study the intrinsic scattering of phonons by a general quantum degree of
freedom, i.e. a fluctuating "field" $Q$, which may have completely general
correlations, restricted only by unitarity and translational invariance. From
the induced scattering rates, we obtain the consequences on the thermal
conductivity tensor of the phonons. We find that the lowest-order diagonal
scattering rate, which determines the longitudinal conductivity, is controlled
by two-point correlation functions of the $Q$ field, while the off-diagonal
scattering rates involve a minimum of three to four point correlation
functions. We obtain general and explicit forms for these correlations which
isolate the contributions to the Hall conductivity, and provide a general
discussion of the implications of symmetry and equilibrium. We evaluate these
two- and four-point correlation functions and hence the thermal transport for
the illustrative example of an ordered two dimensional antiferromagnet. In this
case the $Q$ field is a composite of magnon operators arising from spin-lattice
coupling. A numerical evaluation of the required integrals demonstrates that
the results satisfy all the necessary symmetry restrictions but otherwise lead
to non-vanishing scattering and Hall effects, and in particular that this
mechanism leads to comparable thermal Hall conductivity for thermal currents
within and normal to the plane of the antiferromagnetism.
