Quantum corrections to the Larmor radiation formula in scalar electrodynamics

We use the semiclassical approximation in perturbative scalar quantum electrodynamics to calculate the quantum correction to the Larmor radiation formula to first order in Planck's constant in the nonrelativistic approximation, choosing the initial state of the charged particle to be a momentum eigenstate. We calculate this correction in two cases: in the first case the charged particle is accelerated by a time-dependent but space-independent vector potential whereas in the second case it is accelerated by a time-independent vector potential which is a function of one spatial coordinate. We find that the corrections in these two cases are different even for a charged particle with the same classical motion. The correction in each case turns out to be nonlocal in time in contrast to the classical approximation.