  We extend a quantum kinetic approach to the description of hadronic showers
in space, time and momentum space to deep-inelastic $ep$ collisions, with
particular reference to experiments at HERA. We follow the history of hard
scattering events back to the initial hadronic state and forward to the
formation of colour-singlet pre-hadronic clusters and their decays into
hadrons. The time evolution of the space-like initial-state shower and the
time-like secondary partons are treated similarly, and cluster formation is
treated using a spatial criterion motivated by confinement and a
non-perturbative model for hadronization. We calculate the time evolution of
particle distributions in rapidity, transverse and longitudinal space. We also
compare the transverse hadronic energy flow and the distribution of observed
hadronic masses with experimental data from HERA, and find encouraging results.
The techniques developed in this paper may be applied in the future to more
complicated processes such as eA, pp, pA and AA collisions.
