  We present a new technique for extracting decay and transition rates into
final states with any number of hadrons. The approach is only sensitive to
total rates, in which all out-states with a given set of QCD quantum numbers
are included. For processes involving photons or leptons, differential rates
with respect to the non-hadronic kinematics may also be extracted. Our method
involves constructing a finite-volume Euclidean four-point function, whose
corresponding spectral function measures the decay and transition rates in the
infinite-volume limit. This requires solving the inverse problem of extracting
the spectral function from the correlator and also necessitates a smoothing
procedure so that a well-defined infinite-volume limit exists. Both of these
steps are accomplished by the Backus-Gilbert method and, as we show with a
numerical example, reasonable precision can be expected in cases with multiple
open decay channels. Potential applications include nucleon structure functions
and the onset of the deep inelastic scattering regime, as well as semi-leptonic
$D$ and $B$ decay rates.
