  The processes of neutrino (antineutrino) absorption and electron (positron)
capture on nucleons provide the dominant mechanisms for heating and cooling the
material between the protoneutron star and the stalled shock in a core-collapse
supernova. Observations suggest that some neutron stars are born with magnetic
fields of at least 10^15 G while theoretical considerations give an upper limit
of 10^18 G for the protoneutron star magnetic fields. We calculate the rates
for the above neutrino processes in strong magnetic fields of 10^16 G. We find
that the main effect of such magnetic fields is to change the equations of
state through the phase space of electron and positron, which differs from the
classical case due to quantization of the motion of electron and positron
perpendicular to the magnetic field. As a result, the cooling rate can be
greatly reduced by magnetic fields of 10^16 G for typical conditions below the
stalled shock and a nonuniform protoneutron star magnetic field (e.g., a dipole
field) can introduce a large angular dependence of the cooling rate. In
addition, strong magnetic fields always lead to an angle-dependent heating rate
by polarizing the spin of nucleons. The implications of our results for the
neutrino-driven supernova mechanism are discussed.
