  One way to relate semiclassical string states and dual gauge theory states is
to show the equivalence between their low-energy effective 2d actions. The
gauge theory effective action, which is represented by an effective
Landau-Lifshitz (LL) model, was previously found to match with the string
theory world-sheet action up to the first two orders in the effective parameter
$\tilde{\lambda} ={\lambda / J^2}$, where $\lambda$ is the `t Hooft coupling
and $J$ is the total $R$-charge. Here we address the question if quantizing the
effective LL action reproduces the subleading 1/J corrections to the spin chain
energies as well as the quantum corrections to the string energies. We
demonstrate that this is indeed the case provided one chooses an appropriate
regularization of the effective LL theory. Expanding near the BPS vacuum, we
show that the quantum LL action gives the same 1/J corrections to energies of
BMN states as found previously on the gauge theory and string theory sides. We
also compute the subleading $1/J^2$ corrections and show that these too match
with corrections computed from the Bethe ansatz. We also compare the results
from the LL action with a more direct computation from the spin chain. We
repeat the same computation for the $\beta$-deformed LL action and find that
the quantum LL result is again equal to the 1/J correction computed from the
$\beta$-deformed Bethe ansatz equations. We also quantize the LL action near
the rotating circular and folded string solutions, generalizing the known
gauge/string results for 1/J corrections to the classical energies. We
emphasize the simplicity of this effective field theory approach as compared to
the full quantum string computations.
