  By generalizing the Green's function approach proposed by Beliaev [1, 2], we
investigate the effect of quantum depletion on the energy spectra of elementary
excitations in an F = 1 spinor Bose-Einstein condensate, in particular, of 87Rb
atoms in an external magnetic field. We find that quantum depletion increases
the effective mass of magnons in the spin-wave excitations with quadratic
dispersion relations. The enhancement factor turns out to be the same for both
ferromagnetic and polar phases, and also independent of the magnitude of the
external magnetic field. The lifetime of these magnons in a 87Rb spinor BEC is
shown to be much longer than that of phonons. We propose an experimental setup
to measure the effective mass of these magnons in a spinor Bose gas by
exploiting the effect of a nonlinear dispersion relation on the spatial
expansion of a wave packet of transverse magnetization. This type of
measurement has practical applications, for example, in precision magnetometry.
