  We derive dispersion relations for $K\to\pi\pi$ decay, using the
Lehmann-Symanzik-Zimmermann formalism, which allows the analytic continuation
of the amplitudes with respect to the momenta of the external particles. No
off-shell extrapolation of the field operators is assumed. We obtain
generalized Omn\`es representations, which incorporate the $\pi\pi$ and $\pi K$
$S$-wave phase shifts in the elastic region of the direct and crossed channels,
according to Watson theorem. The contribution of the inelastic final-state and
initial-state interactions is parametrized by the technique of conformal
mappings. We compare our results with previous dispersive treatments and
indicate how the formalism can be combined with lattice calculations to yield
physical predictions.
