  We investigate the quantum effect on the Larmor radiation from a moving
charge in an expanding universe based on the framework of the scalar quantum
electrodynamics (SQED). A theoretical formula for the radiation energy is
derived at the lowest order of the perturbation theory with respect to the
coupling constant of the SQED. We evaluate the radiation energy on the
background universe so that the Minkowski spacetime transits to the Milne
universe, in which the equation of motion for the mode function of the free
complex scalar field can be exactly solved in an analytic way. Then, the result
is compared with the WKB approach, in which the equation of motion of the mode
function is constructed with the WKB approximation which is valid as long as
the Compton wavelength is shorter than the Hubble horizon length. This
demonstrates that the quantum effect on the Larmor radiation of the order
e^2\hbar is determined by a non-local integration in time depending on the
background expansion. We also compare our result with a recent work by Higuchi
and Walker [Phys. Rev. D80 105019 (2009)], which investigated the quantum
correction to the Larmor radiation from a charged particle in a
non-relativistic motion in a homogeneous electric field.
