  Studies of ${\cal N}=4$ super Yang Mills operators with large R-charge have
shown that, in the planar limit, the problem of computing their dimensions can
be viewed as a certain spin chain. These spin chains have fundamental
``magnon'' excitations which obey a dispersion relation that is periodic in the
momentum of the magnons. This result for the dispersion relation was also shown
to hold at arbitrary 't Hooft coupling. Here we identify these magnons on the
string theory side and we show how to reconcile a periodic dispersion relation
with the continuum worldsheet description. The crucial idea is that the
momentum is interpreted in the string theory side as a certain geometrical
angle. We use these results to compute the energy of a spinning string. We also
show that the symmetries that determine the dispersion relation and that
constrain the S-matrix are the same in the gauge theory and the string theory.
We compute the overall S-matrix at large 't Hooft coupling using the string
description and we find that it agrees with an earlier conjecture. We also find
an infinite number of two magnon bound states at strong coupling, while at weak
coupling this number is finite.
