  In this article, we shall develop and formulate two novel viewpoints and
properties concerning the three-point functions at weak coupling in the SU(2)
sector of the N = 4 super Yang-Mills theory. One is a double spin-chain
formulation of the spin-chain and the associated new interpretation of the
operation of Wick contraction. It will be regarded as a skew symmetric pairing
which acts as a projection onto a singlet in the entire SO(4) sector, instead
of an inner product in the spin-chain Hilbert space. This formalism allows us
to study a class of three-point functions of operators built upon more general
spin-chain vacua than the special configuration discussed so far in the
literature. Furthermore, this new viewpoint has the signicant advantage over
the conventional method: In the usual "tailoring" operation, the Wick
contraction produces inner products between off-shell Bethe states, which
cannot be in general converted into simple expressions. In contrast, our
procedure directly produces the so-called partial domain wall partition
functions, which can be expressed as determinants. Using this property, we
derive simple determinantal representation for a broader class of three-point
functions. The second new property uncovered in this work is the non-trivial
identity satisfied by the three-point functions with monodromy operators
inserted. Generically this relation connects three-point functions of different
operators and can be regarded as a kind of Schwinger-Dyson equation. In
particular, this identity reduces in the semiclassical limit to the triviality
of the product of local monodromies around the vertex operators, which played a
crucial role in providing all important global information on the three-point
function in the strong coupling regime. This structure may provide a key to the
understanding of the notion of "integrability" beyond the spectral level.
