  The top-quark, $W$ and $Z^0$ bosons have widths that are a sizable fraction
of their masses and will be produced copiously at upcoming accelerators. Yet
S-matrix theory cannot treat unstable particles as external states. Dealing
with complete matrix elements involving their decay products complicates
calculations considerably and is unnecessary in many practical situations. It
is shown how to construct physically meaningful production cross-sections for
unstable particles by extracting that part of the matrix element that
corresponds to finite-range space-time propagation. This procedure avoids the
need to define unstable particles in external states and we argue its favour as
providing a solution to a long-standing problem in physics. As an example the
results are applied to the calculation of the cross-section $\sigma(e^+e^-\to
Z^0Z^0)$.

High Energy Physics - Phenomenology