$K \to ππ$ Decays in a Finite Volume

We discuss finite-volume computations of two-body hadronic decays below the inelastic threshold (e.g. $K\to\pi\pi$ decays). The relation between finite-volume matrix elements and physical amplitudes, recently derived by Lellouch and L\"uscher, is shown to have a natural interpretation in terms of the density of states. We present a detailed comparison of our approach with that of Lellouch and L\"uscher and discuss the limitations of the method, which are largely due to the presence of inelastic thresholds. We also demonstrate that for correlation functions such as $< 0 |T[\pi\pi{\cal H}_WK ]| 0 >$, the Lellouch-L\"uscher correcting factor is not appropriate for extrapolating the finite-volume result to infinite volume.