  By considering AdS_5 x S^5 string states with large angular momenta in S^5
one is able to provide non-trivial quantitative checks of the AdS/CFT duality.
A string rotating in S^5 with two angular momenta J_1,J_2 is dual to an
operator in N=4 SYM theory whose conformal dimension can be computed by
diagonalizing a (generalization of) spin 1/2 Heisenberg chain Hamiltonian. It
was recently argued and verified to lowest order in a large J=J_1+J_2
expansion, that the Heisenberg chain can be described using a non-relativistic
low energy effective 2-d action for a unit vector field n_i which exactly
matches the corresponding large J limit of the classical AdS_5 x S^5 string
action. In this paper we show that this agreement extends to the next order and
develop a systematic procedure to compute higher orders in such large angular
momentum expansion. This involves several non-trivial steps. On the string
side, we need to choose a special gauge with a non-diagonal world-sheet metric
which insures that the angular momentum is uniformly distributed along the
string, as indeed is the case on the spin chain side. We need also to implement
an order by order redefinition of the field n_i to get an action linear in the
time derivative. On the spin chain side, it turns out to be crucial to include
the effects of integrating out short wave-length modes. In this way we gain a
better understanding of how (a subsector of) the string sigma model emerges
from the dual gauge theory, allowing us to demonstrate the duality beyond
comparing particular examples of states with large J.
