  Large-$S$ and large-$N$ theories (spin value $S$ and spinor component number
$N$) are complementary, and sometimes conflicting, approaches to quantum
magnetism. While large-$S$ spin-wave theory captures the correct semiclassical
behavior, large-$N$ theories, on the other hand, emphasize the quantumness of
spin fluctuations. In order to evaluate the possibility of the non-trivial
recovery of the semiclassical magnetic excitations within a large-$N$ approach,
we compute the large-$S$ limit of the dynamic spin structure of the triangular
lattice Heisenberg antiferromagnet within a Schwinger boson spin
representation. We demonstrate that, only after the incorporation of Gaussian
($1/N$) corrections to the saddle-point ($N=\infty$) approximation, we are able
to exactly reproduce the linear spin wave theory results in the large-$S$
limit. The key observation is that the effect of $1/N$ corrections is to cancel
out exactly the main contribution of the saddle-point solution; while the
collective modes (magnons) consist of two spinon bound states arising from the
poles of the RPA propagator. This result implies that it is essential to
consider the interaction of the spinons with the emergent gauge fields and that
the magnon dispersion relation should not be identified with that of the
saddle-point spinons.
