We study a cooling process of magnetars through the neutrino-antineutrino pair emission from electron and proton,
which occurs only under the strong magnetic field, in the exact quantum calculation. Our results show that the luminosity is proportional to T**(0:95-1:7) when the magnetic fields are B = 10**15 -101**6G. The powers of the temperature are much smaller than those in the direct Urca and moderate Urca processes. Thus, the nu nubar -pair emission processes by the strong magnetic field is expected to
contribute very largely to the cooling process of the magnetar.