  We perform a first principle semiclassical quantisation of the general
finite-gap solution to the equations of a string moving on R x S^3. The
derivation is only formal as we do not regularise divergent sums over stability
angles. Moreover, with regards to the AdS/CFT correspondence the result is
incomplete as the fluctuations orthogonal to this subspace in AdS_5 x S^5 are
not taken into account. Nevertheless, the calculation serves the purpose of
understanding how the moduli of the algebraic curve gets quantised
semiclassically, purely from the point of view of finite-gap integration and
with no input from the gauge theory side. Our result is expressed in a very
compact and simple formula which encodes the infinite sum over stability angles
in a succinct way and reproduces exactly what one expects from knowledge of the
dual gauge theory. Namely, at tree level the filling fractions of the algebraic
curve get quantised in large integer multiples of hbar = 1/lambda^{1/2}. At
1-loop order the filling fractions receive Maslov index corrections of hbar/2
and all the singular points of the spectral curve become filled with small
half-integer multiples of hbar. For the subsector in question this is in
agreement with the previously obtained results for the semiclassical energy
spectrum of the string using the method proposed in hep-th/0703191.
  Along the way we derive the complete hierarchy of commuting flows for the
string in the R x S^3 subsector. Moreover, we also derive a very general and
simple formula for the stability angles around a generic finite-gap solution.
We also stress the issue of quantum operator orderings since this problem
already crops up at 1-loop in the form of the subprincipal symbol.
