  We study the neutrino anti-neutrino pair synchrotron emission from electrons
and protons in a relativistic quantum approach. This process occurs only under
a strong magnetic field, and it is considered to be one of effective processes
for neutron star cooling. In this work we calculate the luminosity of the
neutrino anti-neutrino pairs emitted from neutron-star-matter with a magnetic
field of about 10^{15} G. We find that the energy loss is much larger than that
of the modified Urca process. The neutrino anti-neutrino pair emission
processes in strong magnetic fields is expected to contribute significantly to
the cooling of the magnetars.
