Collective effects in $\nu \bar{\nu}$ synchrotron radiation from neutron stars

We have considered collective effects in $\nu \bar{\nu}$ synchrotron radiation from an ultrarelativistic degenerate electron gas in neutron stars with strong magnetic fields. For this problem we apply a calculation method which explicitly makes use of the fact that the radiating electron moves semi-classically, but takes into account the interaction among particles in a quantum way. First we apply this method to calculate $\nu \bar{\nu}$ synchrotron radiation by an ultrarelativistic electron in vacuum and we compare this result with that obtained previously by other techniques. When a degenerate plasma is considered, we show that collective effects lead to an essential enhancement (about three times) of the vector weak-current contribution to neutrino pair emissivity.