  We present a study of the finite-volume two-pion matrix elements and
correlation functions of the I=0 scalar operator, in full and partially
quenched QCD, at one-loop order in chiral perturbation theory. In partially
quenched QCD, when the sea and valence light quark masses are not equal, the
lack of unitarity leads to the same inconsistencies as in quenched QCD and the
matrix elements cannot be determined. It is possible, however, to overcome this
problem by requiring the masses of the valence and sea quarks to be equal for
the u and d quarks while keeping the strange quark (s) quenched (or partially
quenched), but only in the kinematic region where the two-pion energy is below
the two-kaon threshold. Although our results are obtained at NLO in chiral
perturbation theory, they are more general and are also valid for non-leptonic
kaon decays (we also study the matrix elements of (8,1) operators, such as the
QCD penguin operator Q6). We point out that even in full QCD, where any
problems caused by the lack of unitarity are clearly absent, there are
practical difficulties in general, caused by the fact that finite-volume energy
eigenstates are linear combination of two-pion, two-kaon and two-eta states.
Our work implies that extracting Delta I=1/2, K-->pipi decay amplitudes from
simulations with m_s=m_{d,u} is not possible in partially quenched QCD (and is
very difficult in full QCD).
