  The Wigner time delay is a measure of the time spent by a particle inside the
scattering region of an open system. For chaotic systems, the statistics of the
individual delay times (whose average is the Wigner time delay) are thought to
be well described by random matrix theory. Here we present a semiclassical
derivation showing the validity of random matrix results. In order to simplify
the semiclassical treatment, we express the moments of the delay times in terms
of correlation functions of scattering matrices at different energies. In the
semiclassical approximation, the elements of the scattering matrix are given in
terms of the classical scattering trajectories, requiring one to study
correlations between sets of such trajectories. We describe the structure of
correlated sets of trajectories and formulate the rules for their evaluation to
the leading order in inverse channel number. This allows us to derive a
polynomial equation satisfied by the generating function of the moments. Along
with showing the agreement of our semiclassical results with the moments
predicted by random matrix theory, we infer that the scattering matrix is
unitary to all orders in the semiclassical approximation.
