  We compute the one-loop helicity amplitudes for low-energy
$\nu\gamma\to\nu\gamma$ scattering and its crossed channels in the standard
model with massless neutrinos. In the center of mass, with $\sqrt{s} =
2\omega\ll 2m_e$, the cross sections for these $2\to 2$ channels grow roughly
as $\omega^6$. The scattered photons in the elastic channel are circularly
polarized and the net value of the polarization is non-zero. We also present a
discussion of the optical activity of a sea of neutrinos and estimate the
values of its index of refraction and rotary power.
