  The quantum theory of the spin light of neutrino ($SL\nu$) exactly accounting
for the effect of the background matter is developed. The $SL\nu$ rate and
power, and also the emitted photon's energy are obtained for the different
values of the neutrino momentum and density of matter. The spatial distribution
of the radiation power and the dependence of the emitted photon's energy on the
direction of radiation are studied. It is also shown that, in a wide range of
the neutrino momentum and densities of matter, the $SL\nu$ radiation is
characterized by nearly total circular polarization. Conditions for the
effective neutrino spin-light radiation in the electron plasma are discussed.
