  The circular polarization of the cosmic microwave background (CMB) is usually
taken to be zero since it is not generated by Thomson scattering. Here we
explore the actual level of circular polarization in the CMB generated by
conventional cosmological sources of birefringence. We consider two classes of
mechanisms for birefringence. One is alignment of the matter to produce an
anisotropic susceptibility tensor: the hydrogen spins can be aligned either by
density perturbations or CMB anisotropies themselves. The other is anisotropy
of the radiation field coupled to the non-linear response of the medium to
electromagnetic fields: this can occur either via photon-photon scattering
(non-linear response of the vacuum); atomic hyperpolarizability (non-linear
response of neutral atoms); or plasma delay (non-linear response of free
electrons). The strongest effect comes from photon-photon scattering from
recombination at a level of $\sim 10^{-14}$ K. Our results are consistent with
a negligible circular polarization of the CMB in comparison with the linear
polarization or the sensitivity of current and near-term experiments.
